J. Berndt RIEMANNIAN GEOMETRY OF COMPLEX TWO-PLANE ... › ... › 55-1 › 19.pdf · Riemannìan geometry of complex two-plane Grassmannians 23 Let p be any point in an almost Hermitian

Rend. Sem. Mat. Univ. Poi. Torino Voi. 55, 1 (1997)

J. Berndt

RIEMANNIAN GEOMETRY OF COMPLEX TWO-PLANE GRASSMANNIANS

Abstract. We develop a theory for the Riemannian geometry of the complex Grassmann manifolds CG2,m = <?2(Cm+2) whi|ch is similar to the familiar theories for the complex and quaternionic projective spaces CPm and HPm respectively. The cruciai point is the construction of a suitable model for CG2,m- Each point in this model represents a closed geodesie in the focal set Qm+1 of C P m + 1 in MPm+1 . The Riemannian submersion Q m + 1 —• CG2)m might be viewed as an analogue of the Hopf map s2m+1 —*• CPm . Applications include new and simple formulas for the curvature of CG2,m, the explicit classification of ali maximal totally geodesie submanifolds of €G2)m» the classification of ali isometric reflections of CG2)m in submanifolds, and the classification of ali cohomogeneity one isoparametric systems in CG2,m with a totally geodesie leaf. It is worthwhile to mention that the Riemannian geometry of CGk,™ is of particular interest since CCx2,m is equipped with both a Kahler and a quaternionic Kahler structure.

Contents

1. Introduction 20

2. Kahler manifolds 22

3. Quaternionic Kahler manifolds 23

4. Sasakian manifolds and Sasakian ^>-symmetric spaces 25

5. The quaternionic projective space 26

6. The focal set of C P m + 1 in H P m + 1 : 27

7. Extrinsic geometry of the focal set — 29

8. Intrinsic geometry of the focal set 35

9. The model for CG2 ,m 37

10. Curvature 41

11. Singular tangent vectors and flats 47

12. Maximal totally geodesie submanifolds 48

20 J. Bemdt

13. The totally geodesie C P m , part 1 . 51

14. Adapted quaternionic orthonormal bases 53

15. The totally geodesie C P m , part 2 . 54

16. Holomorphic isometries 55

17. The totally geodesie £Pa x CPb 58

18. The totally geodesie CG2)jb • 61

19. The totally geodesie MG2,m • 64

20. Complex-quaternionic orthonormal bases 71

21. The totally geodesie MPn 72

22. Isometric reflections in submanifolds 76

23. Isoparametric systems of codimension one 77

24. A remark on the dual symmetric space CGl m • ••••.. 82

References 83

1. Introduction

Since the celebrated wòrk of E. Cartan on the classification of Riemannian symmetric spaces, these manifolds are renowned for their richness of beautiful properties, which turned them into a major subject for studies in geometry, anàlysis, and other branches of mathematics. As regards Riemannian geometry, there is a striking difference between the methods used for tackling symmetric spaces of rank one and of rank greater than one, respectively. The symmetric spaces of rank one consist of the spheres, and the projective and hyperbolic spaces over the real, complex, quaternionic, and Cayley numbers, ali of them equipped with their naturai metrics. In ali these spaces an explicit expression for the curvature tensor in terms of suitable geometrical structures, like Kàhler or quaternionic Kàhler structures, is available. By looking at the enormous amount of literature dealing with particular rank one symmetric spaces, one gets the feeling that this fact explains, at least partially, the difference. As concerns studies on symmetric spaces of higher rank, the major working method is a skilful handling of the associated algebraic objects. For example, root systems are important for questions related to curvature and totally geodesie embeddings. In principle, the roots and their eigenspaces are known, but geometrical interpretations as for rank one symmetric spaces are stili missing. In this article we consider the complex Grassmann manifolds CG .m of ali two-dimensional complex linear subspaces in C m + 2 . These particular Grassmann manifolds have rank two and are characterized among ali compact, non-Ricci fiat, symmetric spaces by the feature of being equipped with both a Kàhler structure and a quaternionic Kàhler structure. We shall illustrate that these two

Riemannian geometry of complex two-plane Grassmannians 21

structures are sufficient in order to treat the Riemannian geometry of CG .m in a way as one is familiar with for rank one symmetric spaces. It is worthwhile to mention here that CG^/i is isometric to the two-dimensional complex projective space CP2, and CC?2,2 is isometric to the real Grassmann manifold KGt4 of ali oriented two-dimensional linear subspaces of IR6.

The first part of these notes is devoted to the construction of a particular model for GG2,m- We derive this model in the following manner. Consider the (m + 1)-dimensional complex projective space C P m + 1 embedded in the canonical way as a totally geodesie submanifold in the (m -f l)-dimensional quaternionic projective space H P m + 1 . The focal set Qm+1 of C P m + 1 in MP m + 1 is a submanifold of H P m + 1 with codimension three. At each point of Qm+1 the nuli space of the shape operator is independent of the direction of the normal vectors and determines a one-dimensional Riemannian foliation T on Qm+1 by closed geodesics. The orbit space Bm+1 := Qm+1 /f, equipped with the Riemannian structure for which the canonical projection Qm+1 —»• Bm+1 becomes a Riemannian submersion, is isometric to the Riemannian symmetric space CG2,m. Thus, a two-dimensional complex linear subspace of C m + 2 , which represents a point in CG^m in the usuai model, is here replaced by a closed geodesie in the focal set of C P m + 1 in H P m + 1 . The main tools involved in this construction are the theories of focal sets and Riemannian submersions. Essentially ali basic information about CG^m is encoded in the intrinsic and extrinsic structure of the focal set Qm+1 of C P m + 1 in MP m + 1 . Therefore, we start with a thorough study of Qm+1 by means of focal set theory. The Kàhler and quaternionic Kàhler structure on CG .m will then be deduced from corresponding objects on Qm+1 by means of theory of Riemannian submersions.

In the second part, the above model is used to study certain topics of Riemannian geometry on CG^^z- The main results may be briefly summarized as follows:

1. An explicit expression of the Riemannian curvature tensor of CG2,m in terms of its Riemannian metric, Kàhler structure and quaternionic Kàhler structure is derived. This leads to a complete spectral description of the Riemannian Jacobi operators on CG2,m« As applications we provide simple formulas for the (holomorphic) sectional curvature of CGf2,m- Moreover, the location of the singular tangent vectors and flats of CGf2,m can be easily described in terms of the Kàhler and quaternionic Kàhler structure. (Sections 10 and 11.)

2. B.Y. Chen and T. Nagano classified in [11] ali Riemannian symmetric spaces which can be embedded in CG2)m as a maximal totally geodesie submanifold. For m > 3 the list consists of CP m , CPa x CPb (a, b > 0, a + b = m), CG2,m-i, ^G2,m and MPn (m = In). To each of these spaces there is associated a set of classical totally geodesie embeddings. We prove here that any totally geodesie embedding of one

22 J. Bemdt

of the above spaces into CG .m ls m âct a classica! one. Moreover, for each type any two totally geodesie embeddings are shown to be holomorphically congruent. We determine also the structure of the tangent spaces to any of these embeddings. (Sections 12,13,15,17-19 and 21.)

3. Two characterizations of the group of holomorphic isometries of CG .m in terms of its action on certain kinds of bases of the tangent spaces. (Sections 16 and 20.)

4. The classification of ali isometric reflections in submanifolds of CG^.m- (Section 22.)

5. The classification of ali isoparametric systems of codimension one on CG2,?n which have a totally geodesie submanifold as a singular orbit. (Section 23.)

I found no literature dealing with aspeets of this paper just for CG .m* Of course, curvature, totally geodesie submanifolds, and groups of isometries have been topics for research in the more general framework of Grassmann manifolds or symmetric spaces. In this regard, I just refer to the beautiful survey article by A.A. Borisenko and Y.A. Nikolaevskii [8].

2. Kàhler manifolds

A basic reference for Kàhler manifolds is [5], Chapter 2. We summarize nere some notions which are used frequently throughout this paper.

Let (M, g) be a Riemannian manifold with Riemannian metric g, Levi Civita connection V and Riemannian curvature tensor R with the convention R(X,Y) = [Vx,Vy] - V[X,Y]. By TM we denote the tangent bundle of M and by TPM the tangent space of M at p E M. If X is a non-zero vector tangent to M, we denote by RX the real line spanned by X. If not stated otherwise, the dimension of a real, complex, or quaternionic vector space or manifold will always mean its real, complex, or quaternionic dimension, respectively.

An almost Hermitian structure on (M, g) is a tensor field J of type (1,1) satisfying p - -idTM and g(JX,JY) = g(X,Y) for ali X,Y e TPM, p e M. An almost Hermitian manifold (M,g,J) is a Riemannian manifold (M,g) equipped with an almost Hermitian structure J. A Hermitian manifold (M,g, J) is a Riemannian manifold (M,g), such that M is a complex manifold, for which its naturai complex structure J is almost Hermitian. If the complex structure J of a Hermitian manifold (M, g, J) is parallel with respect to V, then J is called a Kàhler structure on (M, g) and (M, g,J)is called a Kàhler manifold. A Riemannian symmetric space, which is also a Hermitian manifold and for which the geodesie symmetry at any point is a holomorphic isometry, is called a Hermitian symmetric space.

Riemannìan geometry of complex two-plane Grassmannians 23

Let p be any point in an almost Hermitian manifold (M,g,J) and V a vector subspace of TPM. Then V is called a real subspace of TPM if JV 1 V, that is, if JV is orthogonal to V, and V is called a complex subspace of TpM if V is invariant under J. À rea/ submanifold (resp. a complex submanifold) of M is a submanifold for which at each of its points the tangent space is a real (resp. complex) subspace of the respective tangent space of M. Note that in literature the notion totally real is often used instead of real. We use the latter one in order to avoid confusion with the notion of totally real in the context of quaternionic Kàhler structures.

For any non-zero vector X tangent to an m-dimensional almost Hermitian manifold (M,g,J) we denote by CX the complex line spanned by X and JX. Some unit vectors Xi,..., Xm 6 TPM, p 6 M, are said to be a complex orthonormal basis of TPM, if the complex lines VXi,..'., CXm are mutually orthogonal.

A standard example of a Kàhler manifold is the m-dimensional complex projective space CPm equipped with the Fubini Study metric. For later use we recali here the classification of totally geodesie submanifolds in CPm. Let p G CP m , k 6 { 1 , . . . , m}, and V a fe-dimensional real (resp. complex) subspace of TpCPm. Then there is a unique connected, complete, real (resp. complex) totally geodesie submanifold M of CP m with p E M and TpM = V. Moreover, M is holomorphically congruent to the standard embedding of MPk (resp. CPk) in CPm. Any connected, complete, totally geodesie submanifold of CPm is one of the above.

3. Quaternionic Kàhler manifolds

Also here, a basic reference is [5], Chapter 14. We shall provide here the basic

definitions and fix some notations.

A quaternionic Kàhler structure on a Riemannian manifold (M,g) is a rank three

vector subbundle J of the endomorphism bundle End(TM) over M with the following

properties:

(i) for each p £ M there exists an open neighborhood U of p in M and sections

h, h, h of J over U such that for ali i/ e {1,2,3}:

(a) Jv is an almost Hermitian structure on V\

(b) JvJv+\ = J„+2 = —Ju+i Jv (index modulo three);

(ii) J is a parallel subbundle of End(TM), that is, if J is a section in J and X a vector

field on M, then VxJ is also a section in J.

Any triple J i , Jo, J3 of the above kind is called a canonical locai basis of J", or, if restricted to the tangent space TpM of M at p, a canonical basis of Jp. A quaternionic Kàhler manifold (M,g, J) is a Riemannian manifold (M, #) equipped with a quaternionic

24 J. Berndt

Kàhler structure J. A Riemannian symmetric space equipped with a quaternionic Kàhler structure is called a quaternionic symmetric space (or quaternionic Kàhler symmetric space).

Any quaternionic Kàhler manifold (M,g,J) is orientable and has real dimension Am with some m e N+. Furthermore, if m > 2, M is always an Einstein manifold. An equivalent definition for a 4m-dimensional Riemannian manifold to be quaternionic Kàhler is to require that the holonomy group of (M,g) is contained in the subgroup Sp(m)Sp(l)

ofSO(4m).

Throughout this paper we shall use on several occasions the following general fact in the special cases of WPm and CG ,™; see [1] for details. Suppose (M,g,J) is a quaternionic Kàhler manifold of real dimension > 8 and with noh-vanishing scalar curvature (note that the latter one is Constant since M is an Einstein manifold). Then any isometry / of (M,g) is an automorphism of (M,g,J), that is, for any p 6 M and any almost Hermitian structure J\ E Jp there exists an almost Hermitian structure J[ € Jf(p) with J*Ji — J\J*'

We consider the quaternionic Kàhler structure J of a quaternionic Kàhler manifold always as an oriented Riemannian vector bundle, equipped with the bundle metric and orientation such that any locai orthonormal frame field of J is a canonical locai basis of J. Note that the elements of unit length in J correspond to the almost Hermitian structures mj.

Let p be any point in a quaternionic Kàhler manifold (M,g,J) and V a vector subspace of TPM. Then

(i) V is called a quaternionic subspace of TPM, if JV C V for ali J £ Jp;

(ii) V is called a totally complex subspace of TPM, if there exists a one-dimensional

subspace Jp of Jp such that JV C V for ali J e Jp and JV _L V for ali J in the

orthogonal complement of Jp in Jp ;

(iii) V is called a tota//;y rea/ subspace of TPM, if JV L V for a\\ J e Jp.

In an obvious way we may now define quaternionic, totally complex and totally real submanifolds of a quaternionic Kàhler manifold.

Let p be any point in an m-dimensional quaternionic Kàhler manifold (M, g,J). For any non-zero vector X G TPM we define JX := {JX \ J € Jp} and denote by MX the quaternionic line spanned by X and JX. Some unit vectors X\,..., Xm £TpM are said to be a quaternionic orthonormal basis ofTpM, if the quaternionic lines MXi,... ,MXm

are mutually orthogonal, or equivalently, if XiìJiXiìJ2XiìJ^X\ì...ìJzXm is an orthonormal basis of TPM for some (and hence for any) canonical basis Ji,J2,J3 of Jp. Suppose (M,g,J) is also equipped with a Kàhler structure J. Then we denote by MCX the vector subspace of TPM spanned by MX and HJX. If JX e JX, we denote


by C X the orthogonal complement of the complex line CX in the quaternionic line MX.

A standard example of a quaternionic Kàhler manifold is the quaternionic projective space MPm equipped with the Fubini Study metric, which we shall discuss in more detail in section 5.

4. Sasakian manifolds and Sasakian ^-symmetric spaces

Roughly speaking, Sasakian manifolds are the odd-dimensional analogues of Kàhler manifolds, and Sasakian y-symmetric spaces that of Hermitian symmetric spaces. We refer to [6] and [18] for details, further references, and motivation.

A Sasakian structure (<p,(, rj) for a Riemannian manifold (M, g) consists of a tensor field <p of type (1,1), a vector field £, and a one-form rj on M satisfying

»?(0 = i , <p2x = -x + r,(x)(t

g(<pXì<pY) = g(XtY)-ri(X)fi(Y)t

(Vx<p)Y = g(X,Y)t-i1(Y)X

for ali vector fields X,Y on M. A Sasakian manifold (M, #, y?,£, 77) is a Riemannian manifold (M,g) equipped with a Sasakian structure (^, £,??). Note that on any Sasakian manifold the vector field £ is always a Killing vector field, and the curvature operator R(Xt Y)( is that of a Riemannian unit sphere, that is, R(X, Y)( = g(Y,£)X - g(X,£)Y for ali vector fields X,Y on M. Conversely, we have

PROPOSITION 1. [6] If £ is a unit Killing vector field on a Riemannian manifold (M,p), and ifR(X,Y)t = g{Y,£)X - g(Xt()Y holds for ali X,Y eTpM,p£M, then (—V£,£, g(.,£)) is a Sasakian structure on (M, g).

Now, let (M,5f,^,£,^) be a Sasakian manifold. As £ is a Killing vector field, there exists for each p £ M an open neighborhood U of p in M, such that £\U induces a regular fibration of U. The corresponding orbit space Bu can be equipped with a Riemannian structure such that the canonica! projection TT :U —*• Bu becomes a Riemannian submersion, that is, the length of horizontal vectors (vectors perpendicular to £) is preserved by the differential 7r* of ir. The restriction of <p to the horizontal subspaces of TU induces via 7r* an almost Hermitian structure J on Bu-

PROPOSITION 2. [16] If(M, g) is a Sasakian manifold, then for any locai fibering U —» Bu as above the induced almost Hermitian structure on Bu is a Kàhler structure.

If X is a non-zero vector tangent to a Sasakian manifold (M,g.f^.rf) and orthogonal to £, then the (p-sectional curvature with respect to X is the sectional curvature with respect to the span of X and <pX. A Sasakian space form is a Sasakian manifold for which the ^-sectional curvature is globally Constant. The connected, complete, simply

26 J. Berndt

connected Sasakian space forms are classified (see [6], and [2] for their geometrical realizations).

A Sasakian manifold (M,g,(p,^,rj) is called a Sasakian locally (p-symmetric space, if <p2(Vw R)(X, Y)Z = 0 holds for ali W, X,Y,Z £TPM orthogonal to (p and ali p e M.

PROPOSITION 3. [7], [18] For any Sasakian manifold (M, g,<p,€,r)) the following statements are equivalenti

(i) (M, g) is a Sasakian locally (p-symmetric space;

(ii) each Kàhlerian manifold, which is the base space of a locai fibering U —* B\j {see above), is a Hermitian locally symmetrìc space;

(Hi) the locai reflections of M in the flow lines of£ are ìsometries.

A Sasakian locally £>-symmetric space (M, g,(p,£,r)) is called a Sasakian globally (p-symmetric space, if the Killing vector field £ generates a global one-parameter group of isometries of M, and if any locai reflection of M in a flow line of £ can be extended to a global automorphism of (M, g,<p,£,ii).

PROPOSITION 4. [18] Any complete, simply connected, Sasakian locally ip-sym-metric space is a globally (p-symmetric space.

5. The quaternionic projective space

We refer to [4] for details. As usuai, we shall write any quaternion z G i a s z = a + bi + cj + dk with a, 6, e, e? E E and imaginary units t , j , k satisfying ij = k. We consider the complex numbers C as the subalgebra {a + bi \ a,b G ì } of M. For vectors z = (zo,..., zm+i) in C m + 2 or H m + 2 we adopt the analogous notation. By e„ we denote the (v + l)-st standard vector in C m + 2 (i/ = 0 , . . . , m + 1). Finally, by <•, •><£ and <•, > J J we denote the Hermitian and quaternionic inner produci on C m + 2 and IHT+2

given by

m + l m + 1

<Z>V>C= È V " ' ^ £ C m + 2 ; <z1v>m=Y^zltvlt1 z,veMm+\,

respectively. We always assume m > 1.

We denote by H P m + 1 the (m + 1)-dimensionai quaternionic projective space equipped with the Fubini Study metric 'g of Constant quaternionic sectional curvature four and by J its quaternionic Kahler structure. Let r : S4m+7 —> MPm+1 be the Hopf map from the (4m+7)-dimensional Riemannian unit sphere S4m+7 C H m + 2 onto H P m + 1 . It is well-known that r is a Riemannian submersion whose fibers are three-dimensional spheres. These spheres are the orbits of the action of S3 « Sp(l) on S4m+7 defined by multiplication


from the right by unit quaternions. At each point z G S4rn+7 the canonical isomorphism jjm+2 _> j^jjm-f-2 j n c j u c e s a i j n e a r isometry iz from the orthogonal complement Hz of zW := {z\ | A G H} in H m + 2 onto a subspace of the tangent space TzS

4m+7 of S4m+7 at z, namely the horizontal subspace at z with respect to the Riemannian submersion r. The map hz := r*z oiz, where T*Z denotes the differential of r at z, is a linear isometry from Hz

onto the tangent space TT(z)WPm+1 of H P m + 1 at r(z). If X G TT(z)MPm+1, we cali the vector h~l(X) the horizontal lift of X at z. Note that this definition corresponds up to the isomorphism tz with the usuai one for Riemannian submersions. For ali v £ Hz and A G S3

we have hz(v) = hz$v$y and for X := hz(v) we have h~l{JX) = {uA | A G I m i } . The geodesie 7 : M -»• H P m + 1 in MP m + 1 with 7(0) = r(z) and 7(0) = X is the curve

£ »-+ r(cos(i)^ + •sin(i)v) ,

where X e r r ( , ) H P m + 1 has length one and v = h'l(X).

The Riemannian curvature tensor R of M P m + 1 is locally given by

R(X, Y)Z = g(Y, Z)X - g(X, Z)Y 3

+ ^(?(J„y,z)jI/A'-?(jl,x,z)jl,y-2?(jl,x,y)jl,z), v-l

where J\,J2, J3 is any canonical locai basis of J.

Finally, we mention the following classification and rigidity result for totally geodesie

submanifoldsinMPm+1:

PROPOSITION 5. Let p be any point in H P m + 1 , k 6 {l,...,m-h 1}, and V a k-dimensionai totally real (resp. totally complex, resp. quaternionic) vector subspace of TpM. Pm+1 . Then there exists a unique connected, complete, totally geodesie submanifold M ofM pm+l with p G M and TpM = V. Moreover, M is congruent to the standard totally geodesie embedding ofRPk (resp. CPk, resp. MPk) in HLPm+1. Any connected totally geodesie submanifold o / H P m + 1 is congruent to a totally geodesie submanifold of the standard totally geodesie embedding ofCPm+1 or MPm in WPm+l.

6. The focal set of C P m + 1 in H P m + 1

The standard isometric action of SU(m + 2) on H P m + 1 is as follows. First, SU(m + 2) acts on S4m+7 by isometries via

SU(m + 2) x S4m+7 - • S4m+7 , (A,z + vj) » (Az) + (Av)j ,

where z,v G Cm+2 with \z\2 + \v\2 — 1. This action commutes with the 5p(l)-action on S4m+7 described above and hence descends to an action on MP m + 1 by isometries. In what

28 J. Bemdt

follows we shall need a more explicit description of the resulting action of SU(m -f 2) on H P m + 1 .

The canonical totally geodesie embedding of C P m + 1 into H P m + 1 is induced by the embedding

C m + 2 _^ H m + 2 ? fl + W , _ fl + W + Qj + Q&

Henceforth we shall denote by C P m + 1 the totally geodesie submanifold in H P m + 1 which is the image of this embedding C P m + 1 -+ MP m + 1 . Let S2m+3 be the unit sphere in C m + 2 . We regard S2m+3 as a submanifold in S4m+7 by means of the embedding C m + 2 -> 1T+2 . Clearly, 5'2m+3 is the orbit of the action of SU(m+2) on 54 m+7 through e0, which implies that CPm+1 is the orbit of the action of SU(m + 2) on H P m + 1 through r(e0). A simple computation yields

/ i ;1 ( i r ( , ) C P m + 1 ) = {vj | v e C m + 2 , <Zì v>£ = 0}

for each z G S2m+3, where _LT(2)CPm+1 denotes the normal space of C P m + 1 at r(z). From this one sees readily that SU(m + 2) acts transitively on the unit normal bundle of C P m + 1 i n H P m + 1 .

For r (E M+ we now define

Nr := {r(cos(r)z + sin(r)vj) \ z,v E S2m+3 , <z,v>£ = 0} .

Geometrically, Nr is the set of ali points in H P m + 1 which can be reached by running the distance r along any geodesie emanating orthogonally from C P m + 1 . We choose z,v e S2m+3 with <z,v>£ = 0 and consider the geodesie 7 : [0,co[-»- M P m + 1 , t •-»• r(cos(t)^ + sin(t)t;j) in H P m + 1 . This geodesie emanates orthogonally from C P m + 1 at the point r(z) in direction hz(vj). At the parameter t = 7r/2 the geodesie hits C P m + 1

again at the point r(vj) = r(v) and, because of <z,v>£ ='0, interseets C P m + 1 at that point orthogonally in direction hvj(—z) = hv(zj). This implies TV^ = C P m + 1 and Nr = Nn/2-r f°r ali 0 < r < 7r/4. Furthermore, for any such r, the isotropy subgroup of SU(m + 2) at r(cos(r)eo + sin(r)eij) is £7(1) x SU(m), where we identify U(l) with the Lie group of ali matrices of the form

Co ' e - « ) ' < G R -

It is easy to see that SU(m -f 2) acts transitively on the unit normal bundle of CPm+1. Thus TV,., the tube of radius 0 < r < 7r/4 around €Pm+1, is in fact a principal orbit of the action of SU (ni -f 2) on H P m + 1 with codimension one.

As a consequence of the considerations so far we get, that there is only one more singular orbit, namely

Q-+ 1 := NT/4 = {r((l/V2)(z + vj)) \z,v£ S2m+3 , <z,v>£ = 0} .


The isotropy subgroup of SU(m + 2) at f((l/y/2)(e0 + eij)) e Qm+1 is 5/7(2) x SU(m), whence Qm+1 has codimension three in H P m + 1 . Since Qm+1 is the image under the normal exponential map of C P m + 1 of ali vectors with length 7r/4 orthogonal to C P m + 1 , we also see that Qm+1 is the focal set of C P m + 1 in H P m + 1 . It is now obvious that C P m + 1

is the focal set of Qm+1 and that SU(m-\-2) acts transitively on the unit normal bundle of Qm+i since S t / ( m i 2 ) is simply connected and the isotropy subgroup SU(2) x SU(m) is connected, it follows that Qm+1 is simply connected. Everìtually, for m = 1 we have Q2 = SU(Z)/(SU(2) x SU(Ì)) = SU(Z)/SU(2) S S5 . Altogether we have now proved

PROPOSITION 6. The singular orbits of the action of SU(m -f 2) on H P m + 1

are C P m + 1 and Qm+1, the principal orbits of this action are the tubes around these singular ones. SU(m -f 2) acts transitively on the unìt normal bundles of CPm+1 and Qm+1 in H P m + 1 . Qm+1 is the focal set ofCPm+1 in H P m + 1 , and vice versa. Qm+1 is simply connected, has codimension three in MPm + 1 , and is isometric to the Riemannian homogeneous space SU(m + 2)/(SU(2) x SU(m)) equipped with a suìtable invariant Riemannian metric. The space Q2 is diffeomorphic to S5.

REMARK 1. We could also consider the combined action of SU(m + 2) x Sp(ì) on S4m+7, where Sp(l) acts by multiplication from the right with unit quaternions. The induced action on HLPm+1 is the same as before, but in this case the orbits in 54m+7 are precisely the inverse images under the Hopf map of the orbits in MP m + 1 . The collection of ali these orbits in S4m+7 is just the homogeneous isoparametric system on 54m+7 with four distinct principal curvatures and multiplicities (2,2ra -f 1) (see 13).

7. Extrinsic geometry of the focal set

The extrinsic geometry of Qm+1 in MPm+1 is basically described by the shape operator of the submanifold Qm+1 in H P m + 1 . Starting from the well-known extrinsic geometry of CPm+1 in MP m + 1 we shall now calculate the shape operator of its focal set Qm+i by m e a n s 0f Jacobi field theory.

Let l 1 C P m + 1 be the unit normal bundle of C P m + 1 . For each £ G J_1CPm+1 we denote by 7^ : M -^ H P m + 1 the geodesie in H P m + 1 with 7^(0) = £. The focal map of C P m + 1 is the differentiable map

$ : ^ C P ^ 1 -+ Qm+1 , e H» 7^(TT/4) .

We define a vector field rj along $ by

ri(() := 7^(^/4) •

Let a : l C P m + 1 -* C P m + 1 be the canonical projection and

K : T(L£Pm+l) -H. l C P m + 1 , u ^ Vû

30 J. Berndt

be the normal connection map of CP m + 1 , where V 1 is the induced covariant derivative on l C P m + 1 from the Levi Civita connection V on HP m + 1 , and u is the identity transformation of l C P m + 1 considered as a section in l C P m + 1 aìong <r.

We now fix a unit normal vector £ G ^CP™*1 and put 7 := 7 , p := 7(0) and q := $(£) = 7(?r/4)- As CP m + 1 is a totally complex submanifold of HP m + 1

(see Proposition 5), there exists a canonical basis Ji, J2) J3 of Jp such that TpCPm+1 is invariant under JL and mapped onto l p C P m + 1 by J2 and J3. Then Ji£ is normal to £pm+i a n d j 2 ^ j 3 ^ are t a n g e n t t 0 cpm+i at p W e decompose TpHPm+1 orthogonally

info subspaces by

TpHPm+1 = Ti e T4 e l i e i 4 e i o , where

Ti := {X € TpCPm+1 | ?(X, J2f) = 0 = g(X, M)ì ,

T4 := MJ2f e MJ3£ C TpCPm+1 ,

±1 := {x e ipCPm+11 ?(x,o = 0 = ?(x, JiO},

J_4 := MJif C l p CP m + 1 , lo := M C lpCPm + 1 .

Note that these spaces are the tangential and normal components of the eigenspaces of the Jacobi operator R% := R(-,£)£. The index refers to the corresponding eigenvalue, and we have dimTi = 2ra = d iml i , dimT4 = 2 and diml4 = 1 = dimlo. Moreover, Ti, T4 and l i are invariant under J\. For each vector u G T^(l}CPm+1) we denote by Zu the Jacobi field along 7 which is uniquely determined by the initial values Zu(0) = a*u and Z'U(Q) — Ku, where the prime denotes covariant differentiation with respect to 7. Taking into account that the eigenspaces of the Jacobi operator R^ := R(., 7)7 are invariant under parallel translation along 7, we can calculate some basic Jacobi fields explicitly, namely

Zu(t) - cos(t)Pu(t), if <r*u G Ti and Ku = 0 ,

Zu(t) ~ cos(2t)Pu(t), if <r*u e T4 and Ku = 0 ,

Zu(t) = sin(t)Pu(t), if <r*u = 0 and Ku G l i ,

Zu(t) = (l/2)sin(2t)Pu(t), if <r*u = 0 and Ku G 1 4 .

Here, Pu denotes the parallel vector field along 7 with initial value P«(0) == <r*u -f Ku. Any other Jacobi field Zu arises as a linear combination of these basic ones.

PROPOSITION 7. In the situation described above we have:

(i) The tangent space ofQm+1 at q is obtained by parallel translation ofT\ © l i © I4 along 7 from p to q.

Riemannian geometry ofcomplex two-plane Grassmannians 31

(ii) The normal space ofQm+1 at q is obtained by parallel translation ofT^ 0 1 0 = JpJ-4 — {Jv | J £ Jp, v E -L4} along 7 from p to q.

(iti) T)(£) is a unit normal vector of Qm+1 at q, and each unit normal vector ofQm+1

is obtained in this way.

(iv) The principal curvatures ofQm+1 at q with respect to r)(£) are -fi, —1 and 0. The corresponding eigenspaces are obtained by parallel translation of Ti, l i and I4 along 7 from p to q, respectively.

Proof The differential $* of $ at p can be computed in terms of Jacobi fields by

**u = Zu(v/4) , u€Ts(l}CPm+1) .

From the above explicit expressions of the basic Jacobi fields Zu we deduce that TqQm+1 —

$*T^ ( l 1 C P m + 1 ) is obtained by parallel translation of 7\ 0 l i 0 1 4 along 7 from p to q. Thus (i) is proved. Statement (ii) is a consequence of (i) and the fact that parallel translation preserves orthogonality. Since £ G l o , it follows from (ii) that 7/(£) is a unit normal vector of Qm+1 at q. That each unit normal vector of Qm+1 is obtained in this way is a consequence of the fact that SU(m + 2) acts transiti vely on the unit normal bundle of Qm+i Tne s n a p e operator A?(£) of Qm+1 at q with respect to rj(€) can be calculated by using the Jacobi fields Zu and the Weingarten equation by means of

AmZu(7r/4) = Amû = ( -V U T?) T = {-Z'u(w/4))T ,

where u £ T^(±1CPm+1) and ( )T refers to the component tangential to Qm+1. Using again the above explicit expressions of the basic Jacobi fields Zu and putting X := Pu(ir/4) we get

AV^X = X, if cr*u e Ti and Ku = 0 ,

A^yX = —X, if cr*w = 0 and Ku € l i ,

A^^X = 0, if a*u = 0 and Ku E I4 .

This implies statement (iv). •

An immediate consequence of part (iv) is

COROLLARY 1. Qm+l is a minimal submanifold ofE.Pm+1.

We identify r~1(Qm+1) with the complex Stiefel manifold CF2,m of ali complex orthonormal two-frames (z,v) in C m + 2 by means of (z,v) := (l/y/2)(z + vj). We shall frequently write T(Z,V) instead of r((z,v)). The Lie group (7(1) acts on S4m+7 C Mm+2 = cm+2 0 Cm+2j b y i s o m e t r i e s v i a l e f t muitiplication with eH, t E M. Since

CV2,m is invariant by this action, it induces an action of U(ì) on CVb,™ by isometries. The latter one commutes with the induced action of Sp(l) on CV2 m and hence descends

32 J. Bemdt

to an action of U(ì) on Q m + 1 by isometries. We denote the corresponding Killing vector field on Qm+1 by U. Explicitly, we have

UT(Z,V) = h(2ìV)(i(z,v))

for ali (z, v) G CV .m» which shows that U has length one at each point. Furthermore, the maximal integrai curve a of U with a(0) = r(z,v) is

a : t »-• r(e'*(z,v)) .

For the normal bundle ± Q m + 1 of Q m + 1 in HLPm+1 we have the following nice characterization:

PROPOSITION 8. ±Qm+1 = JU := {«/[/«, | J G J g , ? G Q m + 1 } .

Proof. Let (2,v) G CVb.m'be arbitrary and q := T(2 ,V) . Since diml.qQm+1 = -

3 = dimJ'L/g, it suffices to prove that JUq is contained in lqQm+1. According to the

results in Section 6 we have to show, that each geodesie 7 in MLPm+1, parametrized by are length, and with 7(0) = q and 7(0) G JUq, satisfìes 7(71-/4) G C P m + 1 . Any such geodesie is of the form

jx : t K-• r(cos(t)(z, v) + sm(t)i(z, v)X)

with some purely imaginary unit quaternion A G ImH. We easily get

7i(Tr/4) = T(vj)eCPm+1 ,

7 i (TT/4) = r((z- vi)(l + ib)/2) G C P m + 1 ,

Tfc(7r/4) = T((Z - v)(l - i)/2) G CPm+1 ,

which shows that

h(2,v)(i(z,v)X) , A G {i,j,fc} ,

is a unit normal vector of Qm+1 at g. This implies

JUq = span{h(z>v)(i(z,v)\) | A G {i, j,&}} C ±qQm+1 ,

by which the proposition is proved. •

We denote by V the rank 3 Riemannian vector subbundle of End(TQm+1) induced by restriction and projection from J. Locai sections Pi,P2,P3 of V are said to be a canonical locai basis of V if they stem from a canonical locai basis Ji,J2, J3 of J. The Levi Civita connection of Qm+1 will be denoted by V, and the induced Riemannian metric on Qm+1 by #. Since U is a Killing vector field,

<p := - W


is a skew-symmetric (l,l)-tensor field on Qm+1. We denote by % the orthogonal complement of MU in TQm+1.

LEMMA 1. Lei J he an almost Hermitian structure in J and P the induced element in V. Then

(i) AJUU = <PU = PU = QÌ

(ti) AjuTi = <pn = PH=:H , P\H = J\H ,

(iti) -A2jVX = <p2X = P2X = -X + ?(X, (7)C/ ,

(iv) Am = P<p=z<pP ,

(v) <p = -PAjc ; = -AJUP , P = - ^ A J C ; = -Aju<p ,

(vi) PAJU = —AjuPfor ali P in V orthogonal to P.

REMARK 2. Part (v) and Proposition 7 (iv) show, that any eigenspace of the shape operator of Qm+1 with respect to any unit normal vector is invariant under (p.

Proof. PU = 0 is a consequence of Proposition 8, and P\H = J\H follows from HL = ±Qm+1 ®RU = JU ®MU = MU. In particuiar, PH - H. As P\H = J[H, it follows that P2\H - -idn, and since P2U = 0, we obtain P2X = -X + </(X, */)£/.

The equations AJUU = 0 and AJJJH = Ti are consequences of Propositions 7 and 8.

Suppose g e Qm+1 such that J G Jg , and let X e T?<2m+1. We extend J to a locai section (also denoted by J) in J" with | |J | | = 1 everywhere. Since J is parallel in End(THPm+1), there exists an almost Hermitian structure J E Jq orthogonal to J, such that VxJ = J"• Using this and the equation of Weingarten we have

V*f7 = -Vx{J2U) = -(VxJ)JU - JVx(JU)

= -JJU + JAJUX-JVX(JU).

Since JJ € Jq (recali that J and J are orthogonal), Proposition 8 shows that JJU is perpendicular to Qm+1 at </. Further, from

-g(JVx(JU), U) = g(Vx{JU\ JU) = (1/2)X • g(JU,JU) = 0

and Proposition 8, we obtain that JVX{JU) is perpendicular to Qm+1. The tangential component to Qm+1 of the above equation therefore yields

-<pX = PAJUX ,

or equivalently,

P<pX = -P2AjVX = AJUX - g(AjuX, U)U = AJVX ,

34 J. Berndt

since AJUU = 0.

We now easily get

<pU = -PAjuU = 0

and

g(<p2X, Y) = -g(<pX, ipY) = -giPAjuX, PAJVY) = g(P2AJUXì A3UY )

= -g(AjuX, AJUY) + g(AjuX, U)g(AjuY, U) = -?(i4j„ X, y ) .

Thus <p2 — —A'JU, and from Proposition 7 and (pU = 0 we conclude

(p2X = -X + g(XtU)U

and also 7tf = W. Thus (i), (ii) and (iii) are proved and for (iv) it remains to show that P and (p commute. But this follows from

g(<pPX,Y) = -g(PX,<pY) = g(X,P<pY) = g^X.AjuY)

= 9(AJUX, Y) = g(P?X, Y) .

Next, we have

giPAjuX, Y) = -g(<pX, Y) = g(<pP2X, Y) = g(AjuPX, Y)

and

g{<pAjuX\ Y) = -g(PX, Y) = g(P<p2X, Y) = g{Aju<pX, Y) ,

by which (v) is proved. Eventually, (vi) follows from the fact that the two eigenspaces of AJU\H with respect to +1 and -1 are invariant under J (see Proposition 7). •

From parts (i), (iii) and (iv) of the lemma we get another characterization of the eigenspaces of A JU, namely

COROLLARY 2. For any almost Hermitian structure J in Jq, q € Qm+1, and any X e TqQ

m+1 we have

Amx = o< > x emù ,

AmX = X <=• <pX = -PX and X 1RU ,

AJUX = -X <=> <pX = PX and X 1 RU .


8. Intrinsic geometry of the focal set

We start with a detailed study of the curvature of Qm+1. Using Proposition 8 and Lemma 1, we get for the second fundamental form h of Qm+1 the locai expression

3 3 3

h(Y,z) = J tAfy .z) , ;^ ) /^ = ^2g(^jvuYtz)jvu = ^ ( P ^ Z ^ U t / = l J / = l I / = l

with any canonica! locai basis J\,J2,J3 of J. Using once again Lemma 1, we obtain 3 3

MY,Z)X = E KP»<PY> Z)AjvUX = £ ? ( P ^ , Z)Pv<pX . V = l l/ = l

From this and the explicit expression for the curvature tensor of H P m + 1 (see Section 5) we get, by using the Gauss equation of second order,

PROPOSITION 9. The Riemannian curvature tensor R of Qm+1 is locally given by

R{X, Y)Z = g(Y, Z)X - g(X, Z)Y

3

+ Yf3(PvY> Z)PUX - g(PuX, Z)PVY - 2g(PuX, Y)PVZ)

3

.+ £(y(P„y>y, Z)Pu<pX - g(Pu<pX, Z)Pv<pY) ,

where Pi, P2, P3 is any canonica! locai basis ofV.

The Riemannian metric g on Qm+1 induces a bundle metric on A2TQm+1, which we also denote by g. Let X,Y G T ?Qm + 1 , g G Q m + 1 , be two orthonormal vectors and a = span{A",y}. The sectional curvature K(<r) of Qm+1 with respect to <r is

3 3

K(a) = 1 + 3 ^ p A y , ? , I A P„y) + Y^g(<pX A^Y,PUX A P„y) .

In particular, we have

t/g <E e = » K(<r) = 1 ,

and

I 6 ^ J ^ I = > A > ) = l + 4cos2Z(<£>X,PX) ,

where L(tpX% VX) is the angle between >X and the three-dimensional subspace VX =

{PX\PeVq}ofTqQm+1.

Let T be the foliation on Qm+1 induced by the flow of the unit Killing vector field

U.

36 J. Berndt

PROPOSITION 10. The maximal integrai curves of U are closed geodesics in Qm+1 as well as in HPm + 1 , and T is a totally geodesie Riemannian foliation on Qm+1.

Proof. From VuU = —<pU — 0 we get that the integrai curves of U are geodesics in Q m + 1 . Furthermore, using Proposition 8 and Lemma 1, the Gauss equation implies

3

$uu = Y^y(Aj»uU>u)J>'u = ° >

which shows that each integrai curve of U is also a geodesie in H P m + 1 . Since U is complete and of unit length, and as ali maximal geodesics in MP m + 1 are closed, the first assertion follows. Next, as U is a Killing vector field, its maximal integrai curves are the orbits of a one-parameter group of isometries and hence locally equidistant to each other. This just means that T is a Riemannian foliation on Qm\x (see for example [19]). • -

REMARK 3. Let L be any leaf of T and q £ L. According to Propositions 8 and 5 there exists a totally geodesie submanifold M of H P m + 1 congruent to H P 1 such that q e M and TqM = MUq 0 LqQ

m+1 = UUq. As M is totally geodesie in H P m + 1 , the situation inside M « HP1 is exactly as described in Sections 6 and 7 by putting m = 0. In particular, N := M f) C P m + 1 is the totally geodesie submanifold of C P m + 1 congruent to CP1 which is uniquely determined by r(z) E N and TT^N = hz(Cv), where z,v are any orthonormal vectors in C m + 2 with r(z, v) E L. Furthermore, the leaf L is the focal set of N in M and is just the intersection of M and Qm+1. Given any orthonormal vectors z,v in Cm+2 with T(Z,V) € L, the geodesie 7 in M given by 7(0) = r(z) and 7(0) = hz(e

2ltvj) emanates orthogonally from ÀT in M and interseets L at

r(z, e2itv) = r(e"(z, v)e~it) =• r(ez'*(z, v)) .

This shows that, given any two points p E N and q € L, there exists a unique geodesie 7 in M with 7(0) = p and y(ir/4) = q.

PROPOSITION 11. The reflection of Qm+1 in any leaf of T is a well-defined global isometry of Qm+1.

Proof Let L be any leaf of JF, M as in Remark 3, and q = r(z,v) 6 L. Then p = r(z) G M D CP m + 1 , { = hz(vj) e TpM fi _L^CPm+1, and q = 7^(ir/4). It was proved in [3] that the reflection / of H P m + 1 in M w HP1 is a well-defined global isometry of MP m + 1 . We decompose TpMPm+1 orthogonally into TPM 0 V. The differential of / at p is the map

TpM 0 V -» TPM 0 V, w;i + iy2 •-+ ^1 - 12 , and clearly maps

T p CP m + 1 = />;(£>) e /»a((Cz 0 Ct;)1)


into itself, where the orthogonal complement has to be taken in C m + 2 . Since / is an isometry of H P m + 1 and C P m + 1 is totally geodesie in H P m + 1 , this implies / ( C P m + i ) = C P m + 1 . According to the results in Section 6, the manifold Qm+1 consists precisely of ali points in H P m + 1 at distance 7r/4 from CP m + 1 . Since isometries preserve distances, it follows f(Qm+1) = Qm+1, whence / induces an isometry / on Qm+1. By construction, / flxes L pointwise and f*X = —X for ali X perpendicular to L and tangent to Qm+l. Therefore, / is the reflection of Qm+1 in L, by which the assertion is proved. •

PROPOSITION 12. Qm+1 is a Sasakian globally cp-symmetric space with Sasakian strueture (—V?7, t/,(7(•,£/)).

Proof. We already know that U is a unit Killing vector field on Qm+1 and, by means of Proposition 9,

R(X, Y)U = J(Y, U)X - g(X, U)Y .

According to Proposition 1, Qm+1 is a Sasakian manifold with Sasakian strueture (—VU,U, <7(-,t/)). It now follows from Propositions 3 and 11 that Qm+1 is a Sasakian locally <p-symmetric space. Finally, since Qm+1 is complete and simply connected, we conclude with Proposition 4 that Qm+1 is in fact a Sasakian globally ^-symmetric space. •

For m = 1 we have even a better result.

PROPOSITION 13. Q2 is a Sasakian space forni with Constant ip-sectional curvature five and isometric to a distance sphere with radius 7r/4 in CP 3 equipped with the Fubini Study metric of Constant holomorphic sectional curvature four.

Proof If X £ 7ìq, q 6 Q2, then also <pX G 7iq and VX C 7{q. Since the real dimension of 7iq is four and X is orthogonal to both ipX and VX, we must have <f>X € VX. The discussion of the sectional curvature K above thus shows that K(cr) = 5 for any ^-invariant two-plane a C Ti. This proves the first assertion. As Q2 is complete and simply connected, it is isometric to a standard model for such a Sasakian space form. The second assertion then follows from the second table in [2]. •

9. The model for €G2,m

Since the leaves of T are compact orbits of an isometric action, we can equip the space of leaves Bm+1 := Qm+1/Jr with the strueture of a Riemannian manifold so that the canonical projection -K : Qm+1 —> Bm+1 becomes a Riemannian submersion. As Qm+1 is complete and simply connected and the fibers of 7r are connected, it follows that Bm+1 is also complete and simply connected. We denote by g the Riemannian metric and by V the Levi Civita connection on Bm+1. If X is a vector or a vector field tangent to Bm+1, we denote its horizontal lift by X.

38 J. Berndt

LEMMA 2. V%Y = V x Y + g(<pX, Y)U .

Proof. According to the fundamental equations of a Riemannian submersion (see for instance [17]) we ha ve

V^Y^VxY + \v[X,Y],

where V denotes the projection onto the component tangent to the fibers of w. As

V[X,Y] = g([X,Y],U)U

= g(V-Y,U)U-g(V-X,U)U

= -g(Y^zU)U + g(X,V?U)U

= g(Yt<pX)U-g(X,<pY)U

= 2g(<pXiY)U,

the assertion follows. •

Let (<È><) be the one-parameter group of isometries of Qm+1 generating the Killing vector fìeld U. As

$u<pX = -*uVxU = -Vx&uU = -VX(U o $t) = -V*tmXU = <P$t*X

for ali X e TQm+1, there exists a tensor fìeld J of type (1,1) on Bm+l characterized by

As <p2X = —X for ali X 6 Ti, it follows that J is an almost Hermitian structure on (Bm+\g).

THEOREM 1. (Bm+1, g, J) is isometric to the Hermitian globally symmetric space ^G2>m, the complex Grassmann manifold ofall two-dimensional complex linear subspaces ofCm+2. Moreover, any isometry f E SU(m + 2) on Qm+1 descends to a holomorphic isometry on Bm+1.

Proof. As the flow lines of U are compact, the Killing vector fìeld U induces a regular fibration of Qm+1. Thus Propositions 2 and 12 imply that J is a Kàhler structure on (Bm+1,g). We then apply Propositions 3 and 12 to get that (Bm+1,g, J) is a Hermitian locally symmetric space. As (Z?m+1,</, J) is complete and simply connected, it is in fact a Hermitian globally symmetric space. Since the action of the one-parameter group (<!><) commutes with the one of SU(m + 2) on Qm + 1 , we obtain a transitive action of SU(m + 2) on Bm+1 by isometries. The isotropy group of SU(m + 2) at 7r(r(e0,ei)) is S(U(2) x U(m)), which shows that, as a homogeneous space, Bm+1 can be identified


with SU(m + 2)/S(U(2) x U(m)). The latter space is a model for CG2,m, a n d a s tnere is only one (up to homothety) Riemannian metric on CG2,m making it into a Hermitian globally symmetric space, the first statement follows.

Next, we consider an isometry / E SU(m + 2) on Qm+1. As ®t and /commute, / maps any integrai curve of U into another integrai curve of U. This implies f*U = U o / , and therefore

for ali horizontal vectors X. Thus, if / denotes the induced isometry on Bm+1, the definition of J gives

which shows that / is holomorphic. •

REMARK 4. Of course, one would like to see an explicit isometry from CG2)m

to Bm+1. Fortunately, the obvious map

F : CG2|m -* Bm+1 , Cz®£v^ TT(T(Z, «))

0 , v E C m + 2 complex orthonormal) is an isometry. To start with, using Remark 3 one gets easily that F is a well-defined bijection. Now, consider the standard Riemannian submersions / i : CV2,m —• CG2|m and /2 : CV .m —• Q m + 1 . The latter one is just |CV2,m when considering CV^m m 5 4 m + 7 in the usuai way. Then F o fi = ir o /2 ,

and because / i and 7r O /2 are smooth submersions, the maps F and F " 1 are also smooth. Therefore, F is a smooth diffeomorphism. By construction, at any point (z, u) £ CV^>m the horizontal subspaces of T^^CV^.m with respect to / i and 7r O / 2 coincide. This implies that F is in fact an isometry. So, geometrically, in our model for CG2)m the role of a complex two-dimensional linear subspace in C m + 2 is taken over from a closed geodesie in Q m + 1 .

We shall now proceed and construct a quaternionic Kàhler structure on Bm+1. For any (z, v) E CVb.m» t £ M and P E VT(Z)V), there exist some A E I m i and P E V$tT(Z)V), so that

for ali w E h^v){HT^tV)). Thus $**?> = P$t* for ali * E R. This implies that there exists a rank three vector subbundle J of End(TjBm+1) such that

40 J. Berndt

PROPOSITION 14. J is a quaternionic Kàhler stmcture for (Bm+1,g).

Proof. We first prove that J is parallel. Let J be a locai section in J> q G Qm+1

such that J is defìned at ir(q), and X G T^q)Bm+1. We have to show that VxJ G J^qy

Let P be the locai section in V with n*P = Jn*. If X denotes the horizontal lift of X at q, then

ir*PX = Jir*X = JX = TT, JX

implies

PX = JX .

Now, using Lemma 2 we get

(V^)Y = V^{JY) - JV^Y

= V~7Y - g(<pX,lY)U - JV^Y

= V~PY-^g{PiPX)Y)U-PVstY

= (V-P)Y + g(P^X,Y)U .

We extend P\7i-to a locai section J in the quaternionic Kàhler structure J of MPm+l.

Since the latter one is parallel in Er\d(TM.Pm+l), there exists an element J G Jq so that V~ J = J. We denote the induced element from J in Vq by P. Using the equation of Gauss, Lemma 1, and keeping in mind that PY is tangent to Qm+1, we get

PY = JY = (V%J)Y = V~(JY)-JV%Y

= V~(PY) - PVfY + y(ATuX, Y)U

= (VzP)Y + g(P<pX,Y)U.

Combining the last two equations yields (VxJ)Y = PY, and hence

(VXJ)Y = TT*PY = J0TC*Y = J0Y

with some J0 G JTTO) and because of 7r*"P = «7 fi*. SO we have proved that J is a parallel subbundle of End(T5m + 1).

It remains to prove the existence of canonical locai bases for J. Let Pi,P2, P3 be a canonical locai basis of V in some open neighborhood of some point q G Qm+l and s a locai section of ir in some open neighborhood of 7r(q) with s(7r(q)) = q. Then we obtain a canonical locai basis J\,Ji,h of J" by defìning Jv := 7r* O P„ O S*, */ = 1,2,3. •


10. Curvature

From now on we identify Bm+1 with CG2,m by means of Remark 4.

PROPOSITION 15. Let J\ be an almost Hermitian structure in J. Then 3J\ — J\J, and JJ\ is a symmetric endomorphism with (J'J\)2 = I and trJJi = 0.

Proof. This is a consequence of the construction of J and J, Lemma 1 (i,iv) and Corollary 1. •

Thus, at each point p of CG2,m> the almost Hermitian structure Jp is orthogonal to each almost Hermitian structure in Jp. This fact is well-known, see for example Remark 14.53 in [5].

We continue with calculating the Riemannian curvature tensor R of CG2,m. Let X, Y, Z be vector fields tangent to CG2,m- Using Lemma 2 we obtain

(V xVyZ) o 7T = T T * ( V ~ V ? Z + K<PY> %)<pX) •

Since U is 7r-related to the zero vector field on CG2,m> the Lie bracket [U,Z] is tangent to the fibers of ir. Using that the integrai curves of U are geodesics in Qm+1 and that U has Constant length, we get [U, Z] = 0, and hence

VuZ = V~U = -ipZ .

Therefore, using again Lemma 2, we obtain

[X~Y] = VxY - VTX = [X,Y] - 2g(<pX,Y)U

and

Altogether we now get

(R(X, Y)Z) o 7T = ir*(R(X, Y)Z + g(<pY, Z)<pX - g{tpX, Z)pY - 2g(^X, Y)<pZ) .

Inserting here the explicit locai expression for R (see Proposition 9), and using the defining relations for J and J, we eventually arrive at

42 J. Berndt

THEOREM 2. The Riemannian curvature tensor R o/CG^,™ I5 locally given by

R(X,Y)Z=g(Y,Z)X-g(X,Z)Y

+ g(JY,Z)JX-g(JX,Z)JY -2g{JX,Y)JZ

3

+ Y^WvY, Z)JVX - g(JuX, Z)JVY - 2g(Jì/X, Y)JVZ)

3

+ J^WuJY, Z)JVJX - g(J„JX, Z)JVJY) ,

where Ji , J2, >h is any canonical locai basis of J.

Contracting this expression and using Proposition 15, we get

COROLLARY 3. The Ricci tensor S and the scalar curvature s ofCG2,m satisfy

SX - A(m + 2)X and s = 16ra(m + 2) .

In particular, CG2,m is an Einstein manifold.

We equip À2TCGf2,m with the induced bundle metric, which we also denote by g. Theorem 2 then implies for the sectional curvature K of CG2,m

PROPOSITION 16. Let X,Y e TpCG2,m> P G CG2,m, be orthonormal and a = span{X,Y}. Then the sectional curvature K(<r) ofCG2,m with respect to a is given by

3

K(<r) = 1 + 3g(X A Y, JX A JY) + 3 £ ] g(X A Y, JVX A JVY)

3

+ ^g{JX/\JY,JVX f\JuY) .

Note that

g(X AY,JXA JY) = cos2 Z(F, JX) ,

where l(Y, JX) is the angle between Y and JX, which is just the Kàhler angle of a, and

3

J2 9(x A ^ U A JVY) = cos2 L{Y, JX) ,

where l(Y,JX) is the angle between Y and J'X. From Proposition 16 we easily derive


COROLLARY 4. CGî is isometric to the two-dimensional complex projective space equipped with the Fubini Study metric of Constant holomorphic sectional curvature eight.

Owing to this fact, we assume from now on

ra> 2

In order to get more information about the sectional curvature we compute now the eigenvalues and eigenspaces of the Jacobi operators Rx := R(.,X)X. From Theorem 2 we immediately get

3

RXY" = Y - g(Y, X)X + 3g(Y, JX)JX + 3 ] £ g(Y, JVX)JVX

3 3

+ Y,3{XJUJX)JUJY -Y,9(YJuJX)JvJX v-\ u = l

for any unit vector X 6 TCG2,,

THEOREM 3. Let X he a unit vector tangent to CG2,m- In each of the following cases we list ali eigenvalues K, of Rx and the corresponding eigenspaces TK with its dimensions.

(i) JX J- JX. Then we have

K, TK

RX e JJX ( H O : ) 1

MJX e jx

dimTK

. 4

4ra-8 4

(ii) JX € JX. Let J\ be the almost Hermitian structure in J so that JX = J\X. Then we have

K TK

mx e {Y | y i mx, JY = -J{Y} C 1 X e [Y | Y 1 MX, JY = JxY}

RJX

dimTK

2 m - l

2m

1

(Hi) otherwise. There exist an almost Hermitian structure J\ £ J and a unit vector Z orthogonal to MX so that

JX = cos a J\ X -f sin a J\ Z ,

44 J. Berndt

where a = i{j X, JX) 6]0, TT/2[. Then we have

K, TK d i m i ;

0 mxemz 2

1 — cos a {Y\Y 1 (MX e WZ), JY = - JiY} 2 m - 4

1 -f cos a {y 1 y i ( M e M ) , j y = Jiy} 2 m - 4

2(1 — sin a) {cos ! (f + f ) J2X + sin (f + J) J2£ | J2 e (MJi)1} 2

2(1 + sin a) {sin (f + f ) J2X - cos ( f + f ) J2Z | J2 G (KJi)1} 2

4(1 — cosa) M(sin(a/2)JiX - cos(a/2)Ji^) 1

4(1 + cosa) M(cos(a/2)JiX + sin(a/2)JiZ.) 1

r/ie number ofdistinct eigenvalues K, is severi unless cos(a) = 4/5 or 3/5, in which case there are sìx orfive distinct eigenvalues, respectively.

Note that a partial study of Jacobi operators for CG2)2 from a quaternionic point of view can be found in [15].

REMARK 5. In standard symmetric space theory, the Jacobi operator Rx is just —ad(X)2. So the eigenvalues K and eigenspaces TK correspond in an obvious way to the roots and root spaces associated with CGf2,m» and a/2 is the angle between X and the line determined by a singular tangent vector of type JX € JX and bordering the Weyl chamber X belongs to. We do not work out this correspondence here because it is not needed for the further treatment. But we want to point out that the above geometrie realization of the eigenspaces TK provides new information which cannot be extracted from the familiar root space decomposition associated to CG2,m.

Proof. Since al way s RxX = 0, we assume from now on that Y is perpendicular to X. We consider the three cases:

(i) JX J- JX. Then we have the orthogonal decomposition

RX e WLJX e j x e JJX e ( H O : ) 1

of the respective tangent space. For the Jacobi operator Rx we have 3 3

RXY = Y + 3</(y, JX)JX + 3 Y, 9(Y, JVX)JUX - J^ ^(y» JVJX)JVJX .

The assertion follows now readily by inserting vectors from the various components in the above decomposition.


(ii) JX € JX. In this case we have

RxY = Y- KJY + 6</(Y, hX)hX + 2</(Y, J2X)J2X + 2<?(Y, J3X)J3X ,

and the assertion follows easily.

(iii) otherwise. In the situation described in the theorem we have

RxY = Y — cos a J\ JY — sin a cos a g(Y, Z)X — sin2 a g(Y, Z)Z

+ 3(1 + cos2 a)g(Y, JxX)JiX + 3sin a cos ag(Y,J1Z)J1X

4- (3 — cos2 a)#(Y, J2X)J2X — sin a cos a <7(Y, J2Z)J2X

+ (3 — cos2 a)<7(Y, J3X) J3X — sin a cos a #(Y, J3Z)J3X

•+ 3sin2afif(Y,JiZ)J1Z + 3 sin a cos a </(Y, J\X)J\ Z

— sin2 a g(Y, J2Z)J2Z — s inacosa#(Y, J2X)J2Z

— sin2 a <7(Y, J3Z)J3Z — sin a cos a g(Y, J3X)J3Z .

First, if Y G (WX&MZ)1, then i f rY = Y - c o s a J i JY and the assertion for RX\(MX®

MZ)1 follows easily. A simple computation also yields JZ = s i n a / i X — cosaJiZ, and

therefore RxZ = 0. Next, we have

RxJ\X = 4(1 -f- cos2 a)JiX + 4 sin a cos a JxZ

and

Rx J\Z — 4sin a cos aJ\X -f 4sin2 OLJ\Z.

A straightforward calculation yields that

cos(a/2) JiX + sin(a/2) JXZ

and

sin(a/2) JiX - cos(a/2) JXZ

are eigenvectors of Rx with corresponding eigenvalues 4(1 + cosa) and 4(1 - cosa),

respectively. Final ly, we have

RXJ2X = (4-2 cos2 a) J2X-2 sin a cosa J2Z

and

RXJÌZ — —2 sin a cos a J2X -f 2 cos a J2Z .

Once again, a straightforward calculation shows that

. (a 7 r \ T A r / a 7 r \ T r 7

s : n ( - + - ) j 2 X - c o s ( - + i)j^

46 J. Berndt

and

are eigenvectors of Rx with corresponding eigenvalues 2(1 4- sin a) and 2(1 - sino;), respectively. Thus part (iii) of the assertion is proved. •

We now continue the discussion about the sectional curvature of CG .m • For a unit vector X tangent to CG .m we denote by Kx the sectional curvature function with respect to ali two-planes containing X, and by Khoi(X) the (holomorphic) sectional curvature with respect to CX.

PROPOSITION 17. Let X e TCG2,m be a unit vector and a = l(JX,JX). Then we have

0 < Kx < 4(1 +cosa)

and

Kh0Ì(X) = 4(1 + cos2 a) .

Moreover, both inequalities are sharp. )

Proof. Theorem 3, and the fact that maximal and minimal values for the sectional curvature always arise as maximal and minimal eigenvalues of the respective Jacobi operator, imply the assertion for Kx- The formula for Khoi(X) follows from Theorem 3; if X is non-singular one might use the expressions for JX in (iii) and for RxJ\X and Rx J\ Z in its proof. •

By varying with X we obtain

PROPOSITION 18. For the sectional curvature K and the holomorphic sectional

curvature Khoi ofCG2>m we have

0 < K < 8

and

4 < Khol < 8 .

Moreover, ali inequalities are sharp, and we have

K(a) = 8 <=> a = Ja C J<r ,

Khol(X) = 4 <=> JX 1 JX ,

Khoi(X) = '8<=>JX£JX.

The case K(&) = 0 will be the subject of the next section.


11. Singular tangent vectors and flats

Recali that a fiat in CG .m is a two-dimensional connected, complete, totally geodesie, fiat submanifold. From general theory of symmetric spaces it is known, that if the sectional curvature with respect to some two-plane <x e TpCG2lm vanishes, then the exponential map of CG2,m at p maps a onto a fiat of CG2,m, and every fiat arises in this way. A non-zero vector X E TCG^.m is called singular, if it is contained in more than one fiat of CG .m- From Theorem 3 and Proposition 18 we thus get the following characterizations of the singular tangent vectors of CG2,m-

PROPOSITION 19. For each non-zero vector X e TCG2,m the following statements are equivalenti

(i) X is singular;

(ii) JX A. JX or JX e JX;

(Hi) the holomorphic sectional curvature determined by X is minimal or maximal.

The preceding result shows that there are two types of singular tangent vectors, those X for which JX 1 JX, and those for which JX e JX. The set of flats to which a singular tangent vector X is tangent to can be identified in an obvious manner with the real projective space induced from the sphere of ali unit vectors Y orthogonal to X with RxY = 0. From the tables in Theorem 3 we therefore get

PROPOSITION 20. Let X be a non-zero tangent vector of CG2,ìn and a be a two-plane spanned by X and some tangent vector Z orthogonal to X.

(i) If JX _L JX, then a determines a fìat if and only if Z £ J JX. In particular, the set of flats to which X is tangent is an M.P2.

(ii) If JX E JX, say JX — J\X, then a determines a fiat if and only if Z L WX and JZ = —J\Z. In particular, the set of flats to which X is tangent is an MP2m~3.

(Hi) If X is non-singular, then there exist an almost Hermitian structure J\ £ J and a tangent vector Z orthogonal to M.X so that JX — cosaJiX 4- sin a J\Z, where a = l(JX,JX). Then the span of X and Z determines the unique fìat to which X is tangent.

The next lemma describes a useful property of singular tangent vectors.

LEMMA 3. Let X be a singular tangent vector ofCG2,m-

(i) If JX £ JX, say JX = J\X, then each non-zero vector Y in CX is singular and satisfies JY = J\Y, and each non-zero vector Y in CLX is singular and satisfi.es JY = - JXY.

http://satisfi.es

48 J. Bemdt

(ii) If JX _L JX, then each non-zero vector Y in MX is singular and satisfies JY 1 JY.

Proof. Any vector Y in CX is a linear combination of X and JX — J\X. Thus the first statement in (i) follows from JJ\X = J\JX = J\J\X. Now let J2 be an almost Hermitian structure in J orthogonal to J\. Then we have

JJ2X = J2JX = J2J1X = —J1J2X .

As any unit vector in CLX is of the form J2X with some suitable almost Hermitian structure J2 orthogonal to J\, the second statement in (i) follows easily. For (ii), we write Y in the form

Y = a0X + aiJiX + a2J2X + q3J3X

with some canonical basis Ji,J2,J3- Using this and the assumption JX JL JX, a straightforward calculation yields JY _L JY. •

An immediate consequence of Proposition 7 and Corollary 2 is

PROPOSITION 21. Thefollowing statements are equivalentfor any non-zero vector

XeTir(q)CG2>m,qeQm+1:

(i) X is singular oftype JX 6 JX;

(ii) the horizontal lift X of X at q is a principal curvature vector ofQm+1 with respect to some unit normal vector ofQm+1 at q;

(Hi) there exists a point p £ CPm+1 and a unit normal vector £ £ ±.pCPm+l, so that q = " (71-/4) and the parallel translate along 7 from q to p of the horizontal lift X of X at q belongs to T\ or ±1 (see section 1).

Furthermore, Proposition 8, Lemma 1 (iv), and the construction of J and J yield

PROPOSITION 22. For each non-zero vector X £ Tw^CG2,m, q € Qm+1, the following statements are equivalent:

(i) X is singular of type JX J_ JX;

(ii) the horizontal lift X of X at q satisfies ^(A^X.X) = Ofor ali unit normal vectors V ofQm+1 atq.

12. Maximal totally geodesie submanifolds

Our main goal in the subsequent sections is to derive the classification and to develop a structure theory for the maximal totally geodesie submanifolds in CG2,m- Owing to the

Rìemannian geometry of complex two-plane Grassmannians 49

correspondence CG2,2 = MC?^, ** t u r n s o u t t n a t m e c a s e s m = 2 and m > 3 have to be treated seperately. In case m = 2, there are three types of maximal totally geodesie submanifolds, namely MG^3 (embedded as a complex submanifold), (Sa x Sb)/Z2 with a + 6 = 4 (embedded as a real submanifold), and Ci72 (embedded neither as a complex nor as a real submanifold). Any maximal totally geodesie submanifold of CCr2,2 is congruent to one of these. We omit the discussion of this exceptional case here and refer to [10], where a detailed account on totally geodesie submanifolds in M G ^ is given. For this reason, we assume from now on

ra> 3

B.Y. Chen and T. Nagano have classifìed in [11] ali symmetric spaces which can be realized as a maximal totally geodesie submanifold in a given symmetric space. For CG^m they obtained the following list:

(1) the m-dimensional complex projective space £Pm;

(2) the Riemannian product CPa x CPb of an a-dimensional complex projective space and a ò-dimensional complex projective space, where a, 6 are any positive integers with a + b = m;

(3) the complex Grassmann manifold CG2,m-i;

(4) the real Grassmann manifold MG2,m of ali two-dimensional linear subspaces in M m+2.

(5) the quaternionic projective space MPn, in case m is even and m = 2rc.

To each of these spaces there is associated a set of classical totally geodesie submanifolds of CG2,m which we shall describe now:

(1) Let t be a complex linear line in Cm+2. Then the orthogonal complement i1 of l in C m + 2 determines an m-dimensional complex projective space CPm(£L). The image of the embedding

Fi : cpm{£L) -+ cG2,m , a ~ i e a

is a totally geodesie submanifold of CG2,m- Two embeddings Fi and Fv have the same image if and only if t — £'. Hence the set of ali classical totally geodesie submanifolds of CG2,m attached to CPm corresponds in a naturai way to a C P m + 1 .

(2) Let V be an (a + 1)-dimensionai complex linear subspace of C m + 2 and V1 its orthogonal complement in C m + 2 , which is a (b + 1)-dimensionai complex linear subspace of C m + 2 . The image of the embedding

Fv :£Pa(V)(BCPb(VL)->CG2>m , (£,?)»£&£!

50 J. Berndt

is a totally geodesie submanifold of CG^m^ If a ^ &, ali these submanifolds are distinct and the set of ali such submanifolds corresponds in a naturai way to the complex Grassmann manifold CGa+i,6+i of ali (a -f 1)-dimensionai complex linear subspaces in C m + 2 . If a = b, the maps Fv and Fy< determine the same submanifold if and only if V' is either V or V 1 .

(3) Let V-be an (m -f l)-dimensional complex linear subspace of C m + 2 . Then the image of the embedding

Fv : €G2,m-i{V) -+ CG2,m , t®t'^l®t'

is a totally geodesie submanifold of CG^m- The set of ali such submanifolds corresponds in a naturai way to a CPm+1.

(4) Let V be an (m + 2)-dimensional real linear subspace of C m + 2 . Then the image of the embedding

Fv : m2,m(v) -+ CG2im , % e 4 ~ £c e t'€ ,

where £j^ denotes a real linear line in V and #£ denotes the induced complex linear line in C m + 2 , is a totally geodesie submanifold of CG^m- Two embeddings Fv

and Fv have the same image if and only if V = eltV for some t E M. The set of ali such submanifolds corresponds in a naturai way to SU(m + 2)/SO(m + 2).

(5) Let.? be a quaternionic structure on C 2 n + 2 compatible with its Hermitian structure

i. Thus *, ,; and ij make £2n+2 into a right quaternionic vector space Hn+1(,;).

Then the image of the embedding

F3 : WPn(j) -H. CG2)2n , zW^Cz® Czj

is a totally geodesie submanifold of CG2,2n- Two embeddings F ; and Fy have the same image if and only if / = eltj for some < G l . The set of ali such submanifolds corresponds in a naturai way to SU(2n + 2)/Sp(n -f-1).

In the following we will always identify the submanifold with the space on which

the embedding is defìned.

Our studies are motivated by the following questions: What is the structure of

these classical totally geodesie submanifolds relative to the Kàhler and quaternionic Kàhler

structure of CG2,m? If M is a manifold of any of the above rive types, are there any

other totally geodesie embeddings of M into CG2,m apart from the classical ones? And

are ali the (classical) totally geodesie embeddings of M (holomorphically) congruent to

each other? What subspaces of TpCG2,m c a n be realized as the tangent space of a totally

geodesie submanifold of CG2)m which is isometric to M?


As regards holomorphic congruence, a further remark is useful. According to [9], the isometry group of CG .m consists of two connected components (for m = 2 there are four). The complex conjugation on C m + 2 induces an anti-holomorphic isometry on C ^ m - Geometrically, this conjugation is the reflection of CG2,m in KG ,™, where the latter one is embedded according to the canonical inclusion Rm+2 —> C m + 2 (see also Section 22). The group of holomorphic isometries of CG .m is connected and makes up the entire identity component of the full isometry group of CG2,m (see, for instance, [14], p. 303). The second component then consists precisely of ali anti-holomorphic isometries of CG2,m.

We start with a general lemma which is useful to produce totally geodesie submanifolds of submanifolds in Riemannian manifolds.

LEMMA 4. Let M be a Riemannian manifold, M\ÌM2 submanifolds of M, and suppose that N = M\ D M2 is a submanifold of Mi and M2. If Mi is totally geodesie in M and (±.qN fi TqM2) C l.qM\ holdsfor ali q e N, then N is totally geodesie in M2.

Proof. Let X, Y be any vector fields tangent to N. Using repeatedly the Gauss equation, we obtain

V^Y=V^Y

= V™2Y + hM^M(X,Y)

= V%Y + hN^M*(X,Y) + hM^M(X,Y)

£TN e (liV n TM2) e 1M2 (pointwise) ,

where h denotes the second fundamental form of the respective inclusion map. The terms on the right-hand side are pairwise orthogonal to each other. Thus N is totally geodesie in M2 if and only if V^Y is orthogonal to IN n TM2 for ali vector fields X, Y tangent to N. As Mi is totally geodesie in M and'JJV D TM2 C LM\ by assumption, the latter property holds and the assertion is proved. •

13. The totally geodesie CPm , part 1

Let p be an arbitrary point in CPm+1 C H P m + 1 and denote by CPm+1(p) the totally geodesie complex projective space in MP m + 1 tangent to the normal space of CPm+1

at p. Since Qm+1 is the set of ali points at distance 7r/4 from C P m + 1 , and every geodesie in MP m + 1 is closed with length 7r, the intersection N of £Pm+1(p) and Qm+1 is precisely the distance sphere with radius 7r/4 and center p in CPm+1(p). Let q e N be arbitrary and 7 the geodesie in €Pm+1(p) with 7(0) = p and y(w/4) - q. Using the notations of Section 7, the tangent space T«N of N at q is obtained by parallel translation of ±1 ® ±4

52 J. Berndt

along 7 from p to q. As Uq is tangent to the parallel translate of _l_4, we get, by varying

with q, that U is tangent to N at ali points of N. Since N is complete, because it is

compact, it follows that any leaf of the Riemannian foliation T through a point of N is

entirely contained in N. Therefore, w(N) is a submanifold of CG2,m •

Next, to see that N is totally geodesie in Qm+1, we apply Lemma 4 with

M = MLPm+1, Mi = CPm+1(p) and M2 = Q m + 1 . We just nave to prove that

±qN fi TqQm+1 C ±qCPm+1(p) for ali ? G N. So let ? G N be arbitrary and 7

the geodesie in fflLPm+1 with 7(0) = p and 7(TT/4) = #. We use the notations of Section

7. Then, XgiV is obtained by parallel translation of Ti © T4 0 JLo along 7 from p to q,

and T g Q m + 1 by parallel translation of Ti © l i © X4. Therefore, LqN n TqQm+1 is the

parallel translate of Ti along 7. On the other hand, ±qCPm+1(p) is easily seen to be the

parallel translate of Ti ©T4, which clearly contains that of Tf. Thus we may apply Lemma

4 and get, that N is totally geodesie in Qm+1.

As 7r is a Riemannian submersion, ir(N) is totally geodesie in CG2,m- To give a

precise argument, let a be a geodesie in w(N). Any horizontal lift fi of a with respect to

the Riemannian submersion N —• TT(N) is a geodesie in N, and hence also in Qm+1. The

geodesie fi is also horizontal with respect to the Riemannian submersion Qm+1 —» CG ,™

and hence projeets to a geodesie in CG .m» which clearly coincides with a considered as

a curve in CG2,m- This shows that ir(N) is totally geodesie in CG2,m-

Next, using the above notations, TqN is obtained by parallel translation of ±1 © ±4

along 7 from p to q, and the horizontal subspace ?{? HTqN of TgAT is just the parallel

translate of Xi. Since Xi is totally complex, we deduce that 7iq fi TqNy and hence by

constructiòn of J, also TT(?)7r(AT) is totally complex. Therefore, ir(N) is a totally complex

submanifold of CG2,m-

Further, since %q fi TqN is the parallel translate of Xi, Proposition 7 (iv) and

Corollary 2 (see also Proposition 21) imply, that for any p G 7r(iV) there exists an almost

Hermitian structure J\ G Jp leaving Tpir(N) invariant, such that JX = J\X for ali

X G Tpir(N). Therefore, ir(N) is a complex submanifold of CG ,™, and any non-zero

vector X tangent to ir(N) is a singular vector of CG2,m of type JX G JX.

As N is diffeomorphic to S'2m+1 and the leaves of the Riemannian foliation F\N

are closed circles, the manifold TT(N) is diffeomorphic to S2m+1/S1 « CPm. Further,

7r(N) is a complex submanifold of the Kàhler manifold CG2,m and hence by itself a

Kàhler manifold via the induced structure. Proposition 18 implies that ir(N) has Constant

holomorphic sectional curvature eight. As ir(N) is also complete and simply connected,

we now conclude that w(N) is in fact isometric to the m-dimensional complex projective

space with Constant holomorphic sectional curvature eight. According to the identiflcation

of our model with CG2)m as described in Remark 4,7r(N) is just the image of the classical


embedding Ft : CPm(£L) -+ CG2,m with £ := p.

Furthermore, for any pi,P2 € CP m + 1 , there exists an isometry in S'Erra -f- 2) e Sp(m + 2) mapping p\ to p2 (and which leaves CPm+1 automatically invariant). This isometry maps CPm + 1(pi) onto CPm+1(p2) and induces an isometry of CG2,m mapping the totally geodesie complex projective space induced from p\ onto the one induced from

P2-

We summarize the preceding discussion in the following

THEOREM 4. Let £,£' be complex linear lines in Cm + 2 . Then CPm(£L) is a totally geodesie, complex, totally complex submanifold of CG2,m- Furthermore, for any p £ CPm{£L) there exists an almost Hermitian structure J\ £ Jp, such that JX = 3\X for ali X £ TpCPm(£L). In particular, any non-zero vector X tangent to CPm(£L) is a sìngular tangent vector of CG2,m of type JX £ JX. The submanifolds CPm(£-L) and CPm (i'1 ) are holomorphically congruent to each other.

14. Adapted quaternionic orthonormal bases

A quaternionic orthonormal basis X\ì... ,Xm of TpCG .m» P E CG^m» ls said to be adapted, if there exists an almost Hermitian structure J\ 6 Jp so that JXV = J\XV for ali v. The existence of quaternionic orthonormal bases is clear. For the adapted ones we have

THEOREM 5. Let p e CG2)m and X\,..'., Xm e TpCG2,m- Then X\,..., Xm

is an adapted quaternionic orthonormal basis of TpCG2,m if and only if there exists a complex linear line £ £ C m + 2 such that p £ CPm(£L) and Xi,... ,Xm is a complex orthonormal basis ofTp£Pm{£L).

Proof. Suppose that X\,..., Xm is an adapted quaternionic orthonormal basis of TpCG2,m' Denote by Xi,..., Xm the horizontal lifts of X\,..., Xm with respect to the Riemannian submersion ir : Qm+1 —• CG2,m at some point q G Qm+1 with ir(q) = p. Let Ji be the almost Hermitian structure in Jp so that JXV — J\XV for ali v. Then there is a unique endomorphism P\ £ Vq so that 7r* O P1 = ^ o ir*. Further, by construction of V, P\ is the restriction and projection to TqQ

m+1 of an almost Hermitian structure J\ £ Jq. Let M\ be the unique totally geodesie (m + 1)-dimensionai complex projective space in MPm+1 tangent to the span of XiìJiXi1...iXmìJiXm,UqìJiUq

(see Proposition 5), and 7 the geodesie in H P m + 1 with 7(71-/4) = q and 7(71-/4) = J\Uq. Then 7(0) £ CPm+1 and, using the notation of the previous section, Corollary 2 and Proposition 7 (iv), Mi = CPm+1(7(0)). The intersection of Mi and Qm+l then induces via 7r a totally geodesie m-dimensional complex projective space M in CG .m with p E M

54 J. Berndt

and TpM = CX\ ® •.. 0 CXm. In particular, X\,..., Xm is a complex orthonormal basis of TPM. By construction, we have M = CP™^1) with £ := 7(0).

Conversely, suppose there exists some M := CPm(£1) such that p € M and -Xi,... ,X m is a complex orthonormal basis of TPM. According to Theorem 4, there exists an almost Hermitian structure J\ G Jp such that JX = J\X for ali X e TpM. We extend «/1 to a canonical basis Ji, J2> J3 of Jp. Since TpM is complex, we then get J2X,JsX e 1PM for ali X e TPM. A simple argument now gives WX„ 1 MX^ for ali v ^ p, that is, A"i,... ,Xm is a quaternionic orthonormal basis of TpCG .m- As JX = J\X for ali X e TPM, this basis is also adapted. •

15. The totally geodesie CPm , part 2

We continue the discussion about the totally geodesie CPm in CG2,m- An immediate -consequence of Theorem 5 and Lemma 3 is

COROLLARY 5. For any p G CPm(£L) there exists some t' such that p G CPm(£fl) and TpCPm(£'L) = Lp£Pm(lL).

We now prove that the classical totally geodesie embeddings of an m-dimensional complex projective space into CG .m are in fact the only possible total geodesie embeddings.

THEOREM 6. Any totally geodesie submanifold M ofCG2,m, which is isometric to an m-dimensional complex projective space with some Fubini Study metric, is equal to CPm(£L) for some complex linear line £ in Cm + 2 .

REMARK 6. The corresponding statement for m = 2 does not hold. In fact, according to [10], there exists a non-complex totally geodesie embedding of CP2 into

KGJ|4 = CG2|2..

Proof. Let M be as in the assertion, J the complex structure of M, 4c the value of the holomorphic sectional curvature of M, and p G M. Then, for any unit vector X G TPM, the Jacobi operator Rx restricted to TPM has the eigenvalues 0,c,4c with multiplicities 1,2ra — 2,1, respectively. The eigenspaces with respect to 0 and 4c are MX and UJX, respectively. From Theorem 3 we deduce the following possibilities:

(1) e = 1 and JX X JX for ali X G TPM\

(2) e = 2 and JX e JX for, ali X G TPM;

(3) e = 1 - cosa, a = l(JX,JX) for ali non-zero X G TPM, and a G]0,TT/2[

satisfìes 1 — cosa = 2(1 — sin a).


The table in Theorem 3 (iii) shows that JY G JY for some non-zero vector Y £ TPM orthogonal to some unit vector X in case of (3), which is a contradiction. Now suppose that (1) holds. Let X±,... ,Xm be a complex orthonormal basis of TPM. Then the table in Theorem 2 (i) proves that X^ J_ HCX„ for ali v ^ p, which implies M£XV 1 HCXM for ali v / JI. Therefore,

m = dimM 'TpCG2,m > dimjj HCXi e . . . 0 HCXm = 2ra ,

which is a contradiction.

Thus we have to study the final possibility (2). Let X\,..., Xm be a complex orthonormal basis of TPM. The table in Theorem 3 (ii) implies JXV = ±JXV for ali v. Now let Jx G i7p be an almost Hermitian structure with JX\ = J\X\. According to the table in Theorem 3 (ii) and Lemma 3 we may decompose X2 = X2 + X2 so that JX\ = -JiX\ and JX2

2 = JXX%, where X£ G C 1 ^ and X | _L HXi. There exists some almost Hermitian structure J2 £ *7P so that JX 2 = /2X2. Then

—J\X2 4- «/iX2 = JX2 = J2X2 4 </2X2

implies 32X\ = — JiX2 and J2X2 = JiX2 . The first equation yields X\ = 0 or </2 =' — «/i, the second one X2 = 0 or J2 = «Ti. Thus either X2 or X2 vanishes. This means that either X2 G C^Xi and hence JX2 = - J\X2, or X2 1 HXi and hence JX 2 = J iX 2 . Suppose that X2 G CxXi, and hence JX2 = -J\X2. Since X3 is complex orthonormal (with respect to / and hence also with respect to J) to X\ and X2, and since X2 G C 1 ^ ! and X\ G C1X2 , it must be perpendicular to HXi = MX2. We now apply the table in Theorem 3 (ii) to Rxx and Rx2 to get JX 3 = J1X3 and JX 3 = —JiX3, which is a contradiction. Thus we must have X2 _L HXi. An analogous argument implies X^ _L HX„ for ali 1/ t, and hence HXy ± HX^ for ali * / / / J . Therefore, TpM = C X i 0 . . .0CX m , and X i , . . . , Xm is an adapted quaternionic orthonormal basis of TpCG2)m. Using Theorem 5, we finally get that M is equal to CPm(^1) for some complex linear line £ in Cm + 2 . •

16. Holomorphic isometries

As is well-known, the isometry group of a (standard model for a) real space form M is characterized by the property that for any p,q G M, any orthonormal basis X\,...., Xm

of T^M and any orthonormal basis Y\,... ,Ym of TqM, there exists an isometry f of M with /(p) = <j and f*Xu = Yv for ali v. By using only complex orthonormal bases, the group of holomorphic isometries of a complex space form can be characterized in a similar way. We now describe in a similar fashion the group of holomorphic isometries of CG2,m.

THEOREM 7.

(i) Let f be a holomorphic isometry ofCG2tm, p G <CG2>rn, X i , . . . , Xm G TpCG2im an adapted quaternionic orthonormal basis of TpCG2ìTn with, say, JXU = J\XV

56 J. Bemdt

for ali v, and J2 € Jp an almost Hermitìan structure orthogonal to J\. Then f*X\,... ,f*Xm is an adapted quaternionic orthonormal basis of TqCG2,m at q := f(p) with, say, Jf*Xv =.J[f*Xv, and there exists an almost Hermitìan structure J2 E Jq orthogonal to J[ such that /* J2 = J2f*.

(iij Conversely, let pi,£>2 E CGf2,m> Ai , . . . ,X m an adapted quaternionic orthonormal basisofTPlCG2,m with JXU = J\Xuforall v, Y\,..., Ym an adapted quaternionic orthonormal basis of TP2CG2>m with JYV = J[Y„ for ali v, J2 G JPl an almost Hermitìan structure orthogonal to Ji, and J2 E JP2 an almost Hermitìan structure orthogonal to J[. Then there exists a unique holomorphic isometry f of CG2,m with f(pi) = p2, f*Xv = Yv for ali v, and f+J2 = J2f*.

Proof. Ad (i): According to Theorem 5, Xi,..., Xm determines a totally geodesie M := CPm(£L) in CG2,m> Since / is a holomorphic isometry, M1 := f(M) is also a totally geodesie submanifold of CG2ym holomorphically congruent to M and therefore, by means of Theorem 6, equal to some CPm(£,J-). Further, as Ai, . ' . . ,Xm is adapted, it is a complex orthonormal basis of TPM. Since / is holomorphic, f*Xi,..., f*Xm is a complex orthonormal basis of Tj^M' and hence, applying once again Theorem 5, an adapted quaternionic orthonormal basis of T^CG^.m with Jf*Xv — J[f*Xv for ali v and some almost Hermitìan structure J[ E Jf(P)- For the existence of J'2 as in the assertion see Section 3.

Ad (ii): We first prove uniqueness. Let / be a holomorphic isometry of CG2>m

with the properties as in the assertion. Then we have

fM=P2 ,

j^yi-i/ — i v ,

fifj\Xv = f*JXu = Jjit.Xv = JYU = J\YV ,

J + J2X.V = J2j^Xl/ = J2YV ?

f*J\JlXv — —f*J2JlXu = —f*J2JX,s = —f*JJ2Xl/ = — J f*J2Xv

=-~JJ2)*XV = —J2JjifX.v = —J2f*J Xv = —J2j^JiJiv

= —J2JijifXu = JiJ2j*Xl/ = J^J2YU

for ali v. As X\,..., Xm is a quaternionic orthonormal basis of TPl£G2,m, / is uniquely determined by the above equations. This proves uniqueness.

Next, we prove existence. Let pi,p2,Xi,...,Xm, Yì,...,Ym, J\, J2, J[, J2 be as in the assertion. By means of (i) we may assume p := pi = P2 = n(q) with q := r(e0 ,ei) E Qm+1. Moreover, according to Theorems 5 and 6, we may also


assume that both X\,..., Xm and Y\,..., Ym are complex orthonormal bases of TPM with M := CPm(£L) and £ := Ce0, and hence J\- J[. Let J2 G Jq be the almost Hermitian structure determined by ir*J2\Hq = fon+l'Hq- We define two complex orthonormal vectors z,v G Ce o 0Ce i by

2z = (e"eb + ie-"ei)(l-+ t) , 2u = (ie"e0 + c-,'*ci)(l - i) ,

where 2 G M so that

hUq = /i(e0)ei)(i(e0)ei)e2itj) .

A straightforward calculation shows that

T(z,v) = T((eiie0,e-itel)(l + i)(l+j)/2) = q

and foUq = 7^(7r/4) with £ = hz(vj). Now, any unit vector X G TPM has a unique horizontal lift X E7iq which, by construction, is obtained by parallel translation along 7 from T(Z) to q of some vector of the forni hz((l/y/2)(X' + X'j)) with some unit vector X' G Ce2 © . . . © Cem + 1 . Applying the above procedure to X\,...,Xm, we obtain a complex orthonormal basis z, v, X[,..., X^ of C m + 2 .

We now choose an arbitrary point q' G Qm+1 with 7r(q) = ir{q'). The analogous procedure as before yields a complex orthonormal basis z'ìv'ìY{ì...,YJn of C m + 2 . Let A G U(m + 2) be defined by

Az = z' , Av = v' ,AXj = y ; for ali v .

By a suitable choice of <?' we may even assume that A G SU(m + 2). Then A induces an isometry g of MPm+1 with p(CPm + 1) = C P m + 1 (and hence g(Qm+1) = Qm+1), g(r(z)) = T(z'),

g*jt(0) = g*hz(vj) = / * A * ( ( ^ ) J ) = M v ' i ) = 7*'(°) >

and

</**„ = g*hz((l/V2)(Xl + Xjj))) = hAz((l/V2)((AXl) + (AXj)j)))

==M(Vv^)«+rI//i)))==yl,.

From #*7É(0) = 7£'(0) we get g oj^ = 7^, hence

.</(</) = 0(7* (*/4)) = 7*'(*/4) = «'

and

^ J 2^ ? = 0*7^ /4 ) = OH71"/4) = ^ t fy = «^2^(?) - ?29*Uq .

As (/ is an isometry, the last equation implies (see also Section 3)

g* J2 = </2#* •

58 J. Berndt

As it is an isometry, g maps parallel vector fìelds along 7 to parallel ones along 7^/, which implies g*Xv = Yv for ali v, Therefore, g induces an isometry / of Qm+1 with

/(<?) = q' , lh\Hq = J'jAHq , UXV = % .

We now apply Theorem 1 to get a holomorphic isometry / of CG2,m with

f(p) = P > Uh - J'2f* , f*Xv = Yv .

By this (ii) is now proved. •

17. The totally geodesie CPa x CPb

Let a,b be positive integers with a -f- b = m, V an (a + l)-dimensional complex linear subspace of Cm + 2 , and VL its orthogonal complèment in Cm + 2 . The subspace V determines an a-dimensional complex projective space Ma(V) embedded totally geodesically in CPm+1. We fìx a point pò G Ma(V), choose an almost Hermitian structure Jo in JPo with J0TPoCPm+1 = l P o C P m + 1 , and decompose TP oCPm + 1 orthogonally into

TP oCPm + 1 = ] / 0 e ^ o , V0:=TPoMa(V) .

Let Mi be the totally geodesie (m + 1)-dimensionai complex projective space in H P m + 1

determined by p0 e Mi and TPoMi = Vo 0 JoW0. Then N := Mi fi Qm+1 is the tube of radius 7r/4 around Ma(V) in Mi.

Next, let p be any point in Ma(V) and £ a unit normal vector of Ma(V) in Mi at p. The vector £ is also perpendicular to CPm+1 at p, whence we may adopt the notations as in Section 7. Moreover, we put Vp := TpM

a(V), and denote by Wp the orthogonal complèment of Vp in TpCPm+1. Let Jp G Jp be an almost Hermitian structure with JpTpCPm+1 = lpCPm+1. The normal space LqN of N at g is obtained by parallel translation of JpVp ® Wp © M along 7 from p to q. Hence (see Proposition 7) ±qN C\TqQ

m+1 is the parallel translate of JPVP @ (Wp n Ti) and is clearly contained in -LgMi, the parallel translate of JpVp © Wp. So Lemma 4 yields that N is a totally geodesie submanifold of Qm+1. As the parallel translate of ±4 is tangent to N at q, we conclude that {/ is tangent to N at ali points of N. Moreover, since N is compact and hence complete, we have 7r~1(ir(N)) = N, and we now get that 7r(N) is a totally geodesie submanifold of CG ,™-

The horizontal subspace 7iq C\TqN of TqN is obtained by parallel translation of Vp © (JPWP fi l i ) along 7 from p to g. As both Vp and JPWP fi ±1 are invariant with respect to any almost Hermitian structure in Jp orthogonal to Jp, the horizontal subspace of TqN is totally complex. This shows that 7r(N) is a totally complex submanifold of


Using Proposition 7 and Corollary 2 we get that the parallel translate of both Vp

and JpWp fi ± i are invariant under cp. Therefore, also the horizontal subspace of TqN is invariant under <p. This implies that ir(N) is a complex submanifold of £G2,m-

We now choose z,v G S2™*1 so that r(z) £ Ma(V) and /^(uj) is a unit normal vector of Ma(V) at r(z) in Mi. Then

a : < H-> r(e'*(2, v)) = r(^, e2itv)

is the maximal integrai curve of U with a(0) = r(z, v). This shows that the fìbers of the submersion

N ->£Pa(V)x£Pb(VL) , T(z,v)t-+(CzXv)

are precisely the leaves of J7 restricted to N. From this we conclude that 7r(N) is diffeomorphic to the product £Pa x CP6. According to our identification described in Remark 4, TT(N) coincides with the image of the classical embedding Fy described in Section 12. Therefore we identify from now on TT(N) with CPa(V) x CP^V1). The argument in the preceding paragraph shows that each factor of £Pa(V) x £Pb(V±) is itself a complex submanifold of £G2,m-

Next, let p = (pa,Pb) è CPa(V) x €Pb{VL) C CG2,m and Jx £ Jp an almost Hermitian structure leaving Tp(£Pa(V) x CPb(VL)) invariant. Let Xu ... ,Xa

be a complex orthonormal basis of Tp(CPa(V) x {pb}) and Xa+i,. •. ,Xm a complex orthonormal basis of Tp({pa} x £Pb(VL)). Then we have eithér

JX„ = J\XV {y = 1 , . . . , a) and JXV = - JiXv (i/ = a + 1 , . . . , m)

or

JXV = —J\XV {y = 1, . . . , a) and JX„ = J i ^ {y — a + 1 , . . . , ra) .

Without loss of generality we may assume that the first statement holds (just replace J\ by —3\ if necessary). Let Yv be any unit vector in C 1 ^ , v — a + 1 , . . . ,m. Then, using Lemma 3, XL, . . . , Xa,Ya+i,..., Ym is an adapted quaternionic orthonormal basis of Tp€G2ìm and

Tp(CPa(^) x CP^Ì/ 1 ) ) = cxi e . . . e cxa e cLYa+1 e. . . e c-'-Ym .

In particular, we also see that any non-zero vector X tangent to CPa(V) x {p&} or {pa} x £Ph(VL) is singular of type JX e JX. Thus Theorem 3 implies that each factor in the product manifold £Pa(V) x £Pb(VL) has Constant holomorphic sectional curvature eight. As any complete totally geodesie submanifold of a symmetric space is itself a symmetric space, it follows now that £Pa(V) x CPÒ(V1) is in fact isometric to the Riemannian product of £Pa(V) and £Pb(V1), where each factor is equipped with the Fubini Study metric of Constant holomorphic sectional curvature eight.

60 J. Berndt

Next, let p' € CG .m be arbitrary and Zi,...,Zm an adapted quaternionic orthonormal basis of Tp>£G2,m with, say, JZU = 3\ZV. Let q' E Qm+1 with Tr(q') = p' and Ji e Jq> be the almost Hermitian structure such that it*J\\Hq> = JiTr*\Hq>. Furthermore, letV,V e€m+2 be the orthonormal vectors in C m + 2 so that q' = T(Z',v') and 7^(7r/4) = J\Uq> with £ := hz>(v'j). Then, by using Proposition 7 (iv) and Corollary 2, it follows that there exist complex orthonormal vectors Z[,...tZ

fa in £m+2 perpendicular

to €zf 0 Cv' and characterized by the property that the parallel translate of hzi(Z'vj), v = 1, . . . ,a, along 7 from r(^) to #' is the horizontal lift of Zv at </'. Then, putting

V := e*' e cz[ e... e cz'a,

we get

Tp,{CPa(V') x£Pb(V'1)) = CZ1®...e£ZaeC1Za+1®...®£1Zm .

According to Theorem 7, there exists a holomorphic isometry / of CG2,m with /(p) = //, /* A"„ = Zv for ali ^ = 1 , . . . , a, and /.«Y^ = Z„ for ali */ = a + 1 , . . . , m. Then M := / (€P a (V) x C P 0 ^ 1 ) ) is a totally geodesie submanifold of CG2>m holomorphically congruent to CPa(V) x CPb(VL) so that p e M and

TPM = ezi e... e cza e€1za+i-e... e cLzm .

Rigidity of totally geodesie submanifolds then implies M = CPa(V) x £Pb(V'-L).

We now summarize the results obtained so far in this section.

THEOREM 8. Let a,b be positive integers with a + b = m.

(ì) Let V be an (a -j- 1)-dimensionai complex linear subspace of C m + 2 . Then CPa(V) x CPb(VL) is a totally geodesie, complex, totally complex submanifold ofCG2im. Any non-zero vector X tangent to any of the two factors is singular oftype JX e JX. More precisely, for any p = (pa,Pb) e CPa(V) x C P ^ F 1 ) there exists an almost Hermitian structure J\ 6 JP so that JX = J\X for ali X G Tp(CPa(V) x {Pb}) andJX = -JxXforallX e Tp({pa} x £Pb{VL)). In particular, there exists an adapted quaternionic orthonormal basis X\,..., Xm of TpCG2,m, so that

Tp(cpa(v) x C P 6 ^ 1 ) ) = exi e... e cxa e cH-Xa+i e... e cLxm .

(ii) Conversely, ifp is any point in CG2,m and Xi,..., Xm any adapted quaternionic orthonormal basis ofTpCG2tm> then there exists an (a + ì)-dimensional complex linear subspace V of€m+2, so that p e CPa(V) x CPb(VL) and

Tp(cpa(v) x cpb(vL)) = exi e... © cxa e cLxa+1 e... 0 cxArm .


(Hi) For any two (a + 1)-dimensionai complex linear subspaces V and V of C m + 2

the induced totally geodesie submanifolds CPa(V) x CPb(VL) and CPa(V) x

m are holomorphically congruent to each other.

From Theorem 8 and Lemma 3 we get

COROLLARY 6. Let V be an (a -f- 1) -dimensionai complex linear subspace of GG2,m- For any p E CPa(V) x CP6(VAl) there exists an (a + 1)-dimensionai complex linear subspace V of C m + 2 such that p E £Pa{V) x CPb(V,L) and TP(CP"(V) x CPt'iV'1)) = lp(CPa(V) x CP^V1)).

We finally prove that the classical submanifolds CPa(V) x CPb(VL) are the only totally geodesie submanifolds of CG2)m isometric to some Riemannian product CPa x £Pb.

THEOREM 9. Let a,b be positive integers with a + b = m and M a totally geodesie submanifold of CG2)m isometric to a Riemannian symmetric space of the form CPa x CPb. Then M is equal to CPa(V) x CPb(VL) for some (a + 1)-dimensionai complex linear subspace V ofCm+2.

Proof Without loss of generality we may assume b > 2. We write M = Mi x M2, fix a point p = (pi,j>2) ^ Mi x M2, and identify TpM with TPlM\ © TP2M2 in the usuai manner. Let X e TPlMi be a unit vector. As b > 2 and RxY = 0 for ali Y e TP2M2, Theorem 3 implies JX e JX, say JX = JXX, and JY = -JXY for ali Y ETP2M2. lì X' e TPlMi is another unit vector, the same argument as before yields JX' = J[X' and JY = -J[Y for ali Y e TP2M2. Hence Jx = J[, and we conclude JX = JiX for ali X E TPlMi. Note that for any such unit vector X we also have JZ = -JiZ for ali Z € C 1 X. Using Theorem 3 (ii) and a dimension argument, we then deduce that Mi x M2 is a complex, totally complex submanifold of CG2)m, each factor has Constant holomorphic sectional curvature eight, and there exists an almost Hermitian structure Ji e Jp such that JX = JiX for ali X e TPlMi and JY - -JXY for ali Y 6 TP2M2. Finally, using Theorem 8, we get that M is holomorphically congruent to some classical CPa(V) x CPb(VL) in CG2,m. •

18. The totally geodesie CG2>k

In this section we shall study not only CG2,m-i» but also CG2ik for any k e { 1 , . . . , m - 1}. If V is any (k + 2)-dimensional complex linear subspace of C m + 2 , we defìne a classical totally geodesie embedding Fv : CG2>k(V) -»• CG2tm , £&£?*-+£&£' and denote the image of this map also by CG2)k(V). So, let fc G { 1 , . . . ,rn - 1}, V a (ib + 2)-dimensional complex linear subspace of C m + 2 , and W :=V®VjC H m + 2 . Then MPk+1(W) is a (fc + l)-dimensional totally geodesie quaternionic projective subspace of

62 J. Berndt

HPm + 1 . Its intersection with CPm + 1 is the (k + 1)-dimensionai totally geodesie complex projective subspace CPk+1(V) of CPm+1. By restricting the construction of CG2,m

from H P m + 1 to WPk+1(W) we get a totally geodesie submanifold of CG2,m which is, by means of our identification according to Remark 4, precisely the classical totally geodesie submanifold CG2)k(V)' Further, both ip and V leave the horizontal subspaces of T(WPk+1(W)riQm+1) invariant. Therefore, CG2,k(v) i s a complex, quaternionic submanifold of CG2,m- If p E CG2,k(V) and A i , . . . , A* is an adapted quaternionic orthonormal basis of TpCG2,k(V), we may extend the basis to an adapted quaternionic orthonormal basis Ai , . . . ,X m of TpCG2,m so that

TpCG2tk(V) = HAi e • •. 0 HA* .

Conversely, suppose that p' £ CG2,m and Zi,..., Zm is an adapted quaternionic orthonormal basis of TpCG2>m with, say, JZV = J\ZV. Let q' E Qm+1 with Tr(q') = p' and Ji £ J / be the almost Hermitian structure such that ir*Ji\Hq> — Ji7r*\7iqi. Also, let z',v' £ €m+2 be the orthonormal vectors in Cm+2 so that qf = r(z'>') and 7^(7r/4) = JiC/?' with £ := hz>(v'j). Again, using Proposition 7 (iv) and Corollary 2, there exist complex orthonormal vectors Z[,..., Z'm E c m + 2 perpendicular to €z' 0 (CV so that the parallel translate of hzi(Z'vj) along 7$ from r(z') to q' is the horizontal lift of Zv at <?'. We now get

Tp/CG2,fc(v") = HZi e . . . e MZfc

when deflning

vr/:=c^ /eCi; /éCZ{.e...eGzi.

According to Theorém 7 there exists a holomorphic isometry / of CG .m with /(p) = V and f+X» = Zv for ali v. As /* maps WXV to HZ„, it maps TpCG2,k(V) to rpiCGa^CV), and thereforeCG2)k(V) to CG2)fc(V"). Thus CG2,fc(V0 and CG2)fc(V") are holomorphically congruent to each otner.

We summarize the preceding discussion in

THEOREM 10. Let k E {1, . . . , r a - 1}.

(i) Let V be a (k + 2)-dimensional complex linear subspace ofCm+2. Then CG2ik(V)

is a complex, quaternionic submanifold of CG2>m. For any p E CG2,À;(V) there

exists an adapted quaternionic orthonormal basis A i , . . . , Xm of TpCG2ìm such

that

Tp£G2,k(V) = HAi e . . . 0 HA* .

(ii) Conversely, let p E CG2,™ and X\,..., Am an adapted quaternionic orthonormal basis ofTpCG2tm- Then there exists a (k -\-T)-dimensional complex linear subspace V of£m+2 such that p E CG3|*(V) and TpCG2>k(V) = HAi 0 . . . 0 HA*.


(Hi) For any two (k + 2)-dimensionai complex linear subspaces V and V of Cm+2

the induced totally geodesie submanifolds CG2,k(V) and CG2,jb(V) ofCG2,m are holomorphically congruent to each other.

Theorem 10 implies

COROLLARY 7. Let V he a (k -f 2)-dimensionai complex linear subspace of C m + 2 for some k £ { 1 , . . . ,m — 1}. Then there exists for any p £ CG2,k(V) an (m — k + 2)-dimensionai complex linear subspace V ofCm+2 such that p £ CG2,m-k(V) and TpCG2,m-k(V) = ±PCG2tk(V).

Finally, as regards totally geodesie embeddings of CG jb m CG2,m» w e n a v e

THEOREM I I . Any totally geodesie submanifold M qfCG2,m which ìs isometric to the complex Grassmann manifold CG2 *, k > 2, with some symmetric metric, is equal to CG2,k(V) for some (k + 2)-dimensionai complex linear subspace V ofCm+2.

Proof. Suppose M is isometric to CG2,jb and embedded totally geodesically in (£G2,m> We denote the Kàhler structure of M by J and its quaternionic Kàhler structure by J. Let p be any point in M and X £ TpM a unit vector.

We first suppose that k > 3. As M is totally geodesie in CG2 „», we get by using Theorem 3, that if X is a singular tangent vector of M of type JX £ JX, then it is also a singular tangent vector of CG2,m of type JX £ JX. So Theorem 3 (ii) implies JX = ±JX. As we may choose a basis of TpM consisting of singular vectors of type JX £ JXy it follows that M is a complex submanifold of CG .m- Similarily, if X is a singular tangent vector of M of type JX _L JX, it is also a singular tangent vector of CG2,m of type JX _L JX. From Theorem 3 (i) and JX = ± JX we then induce »7X = JX. An analogous argument as for the other type of singular vectors proves that M is a quaternionic submanifold of CG2>m- Now, let X\,... ,Xk be an adapted quaternionic orthonormal basis of TPM. Then we have TPM = HXi © . . . 0 MXk. We extend Xi,..., Xk to an adapted quaternionic orthonormal basis of TpCG2,m and apply Theorem 10 to get that M is equal to CG2,k(V) for some (Ar + 2)-dimensional complex linear subspace ^ o f C m + 2 .

Next, suppose that k = 2. Then there are two possibilities for the behaviour of the singular tangent vectors X of M at p of type JX £ JX. First, if they are singular tangent vectors of CG2,m of type JX £ JX, we may conclude as in the previous case that M is equal to CGf

2,2(^) for some four-dimensional complex linear subspace V of Cm+2. Thus we assume from now on that singular tangent vectors of M at p of type JX £ JX are singular tangent vectors of CG2,m of type JX 1 JX. We shall now prove by induction that for each ra > 3 no such totally geodesie submanifold exists.

64 J. Berndt

To start with, let m == 3 and X\, X2 be an adapted quaternionic orthonormal basis of TPM. Then we have JXV L JXV for v = 1,2, and the table in Theorem 3 (i) shows that Xi 1 MCX2 and X2 1 MCXi, whence HCXi 1 W£X2. As IICXi, is of quaternionic dimension two, we get

3 = dimjCT Tp£G2>3 > dim^MCXi 0 U£X2) = 4 ,

which is a contradiction.

So let m > 4. According to the statements in Section 12, M is not a maximal totally geodesie submanifold in CG2)m and hence contained in some totally geodesie CG^.m-i» CP m , H P n (if rn = 2n), CP a x CP6 (a + 6 = m) or MG2,m in CG2,m. By means of the assumption for the induction, M is not contained in some CG2,m-i- As a rank-two symmetric space cannot be embedded totally geodesically in a rank-one symmetric space, M cannot be contained in some CP m or MPn . Next, suppose M is contained in some CPa x CP6 . Let F be any fiat in M with p e F. Then there exist orthonormal bases A"i,.X2 and Yi,Y2,of TJ,P such that X\, X2 and Yi, Y2 are singular tangent vectors of M of type JX £ JX and JX ± JX, respectively. The angle between the real lines MX„ and MYp is always TT/4. By comparing the Jacobi operators of M and CPa x CPÒ it follows that Xi,X.2, Y\, Y2 must be tangent to one factor of £Pa x CP6 . Therefore, TPF must be contained in, say, TPa£Pa, and hence F in CP a , which is impossible. The assertion then follows from the following statement, which we shall also prove by induction:

For each rn > 4 the real Grassmann manifold MG^ (« M) cannot be embedded in RG2ìm as a totally geodesie submanifold. For m = 4 this is clear. So let m > 5. It follows from [10] that the maximal totally geodesie submanifolds in MG2,m are MG2,m-i» (Sa x Sh)/{±ids« x ± id5b} (a + 6 = m) and CP n (if m = 2n). Thus MGJ"4 cannot be embedded in MG2iTn as a maximal totally geodesie submanifold. The assumption of the induction says that MG\A cannot be embedded totally geodesically in MG2)m-i' Finally, analogous arguments as above show that MG24 cannot be embedded totally geodesically in the sphere produets or CP n . •

For k = 1 the previous statement does not hold because of CG2>i — CP2 and the fact that CP 2 C CP m can be embedded totally geodesically in £G2>m as a totally complex submanifold (see Section 13).

19. The totally geodesie MG2tm

Let V be an (rn + 2)-dimensional real linear subspace of C m + 2 and MPm+1(V) the induced (m -f 1)-dimensionai real projective space embedded totally geodesically in CPm+1. We fix a point p0 € MPm+1(V) and put V0 := TPoMPm+1(V). Let J% G JPo

be an almost Hermitian structure leaving TPoCPm+l invariant and J2 e JPo be the almost


Hermitian structure orthogonal to J\ determined by

h2o(h;o1(x)j) = j2x ,

for ali X E V0, where z0 e S2m+3 0 V with r(z0) = p0. Then V0 0 J2V0 is a totally complex subspace of TJ,0HPm+1. Thus there exists a totally geodesie submanifold Mi in H P m + 1 congruent to C P m + 1 and determined by p0 G Mx and TPoMi = V0 © J2Vo. Recali that the focal set of ]RPm+1 in C P m + 1 is the set of ali points at distance 7r/4 from ffiPm+1 and isometric to the complex quadric in C P m + 1 . As TV : - Mx D Qm+1 is just the set of ali points at distance TT/4 from MPm + 1(7) in Mi (« CP m + 1 ) , it is isometric to the complex quadric in CP m + 1 , which we identify in the standard manner with the real Grassmann manifold M G ^ of ali oriented two-dimensional linear subspaces of ]Rm+2.

We choose (z,v) G CV2,m with p := r(z) e RPm+1(V) and £ := hz(vj) a unit normal vector of MPm + 1(F) in Mi at r(z). As £ is also a unit normal vector of C P m + 1

in MP m + 1 at T(Z), we may now adopt the notations as in Section 7. In addition, we put Vp := TpMPm+1(V) and denote by Ji,J2 almost Hermitian structures in Jp so that J iT p CP m + 1 = TpCPm+1 and J2VP is the normal space of RPm+1(V) in Mx atp.

The tangent space TqN of N at q = T(Z, V) is obtained by parallel translation of

(vp n Ti) © (j2vp n ±i) = (vp n Ti) © j2(vp n Ti)

along 7 = 7£ from p to <j. This implies that N is a horizontal submanifold of Qm+1, that is, each vector tangent to N belongs toTi.

The normal space ±.qN of N at q is the parallel translate of

MJ2£ © JiVp © 1£ © J2JiVp .

Thus _L?N fi TqQm+1 is the parallel translate of (JiVp fi Ti) © J2JiVp and is clearly

contained in the parallel translate of J{VP © JiJ2Vp, which is precisely '±qMi. We apply

Lemma 4 to obtain that N is totally geodesie in Qm+1.

Now we consider the integrai curve

t ^ r(é%(z, v))

of U passing through T(Z,V). As

r f e ^ s , t/)) = r(e"(s, t/)e"") = r(*, e 2 i \ ) ,

we have

hz(e2itvj) e ±pRPm+1(V)nTpMi <=> e2it £ {±1} <=> 2t e TTZ .

Thus we obtain

*-l(ir(T(zt v))) fi N = {T(Z, V), T(Z, -V)} .

66 J. Berndt

Now, z,v and z, — v span the same two-plane, but with different orientations. Therefore, 7r(JV) is the real Grassmann manifold RG2)m of ali two-dimensional linear subspaces in Mm + 2 , and N —• w(N) is the usuai twofold Riemannian covering map MG^m -* RG2>m.

By means of our identification, 7r(iV) is equal to the classical totally geodesie submanifold ^Gf2,m(Vr) as described in Section 12.

As N is horizontal and totally geodesie in Q m + 1 , also M.G2iTn(V) is totally geodesie

in CG2,m-

The description of TqN above shows that TqN is a horizontal, totally complex subspace of TqM.Pm+1. According to the construction of the quaternionic Kahler structure on CG2.ro > this implies that M.G2>m(V) is a totally complex submanifold of CG2,m-

Next, Vp is totally real with respect to Jp, and hence also Vpf)Ti has this property. Thus the parallel translate of Vp fi Ti along 7 fronti p to <j is totally real with respect to Jq.

Using Corollary 2, we get that > maps the parallel translate of Vp fi Ti into its orthogonal compìement in the parallel translate of Ti (which is just the eigenspace of A^) with respect to +1). Analogously, <p maps the parallel translate of J2VP fi ±1 = J2(VP C\ Ti) into its orthogonal compìement in the parallel translate of ±1 (which is just the eigenspace of Àj(£) with respect to -1 ) . Thus <p(TqN) is orthogonal to TqN, and by the construction of J we finally conclude, that M.G2,m(V) is a real submanifold of CG2,m.

Let X i , . . . , Xm G T7r(g)MG?2,m(Vr) be orthonormal vectors such that their horizontal lifts at q form an orthonormal basis of the parallel translate of Vp fi i-i along 7. Then Xi,:..,Xm is an adapted quaternionic orthonormal basis of T^^CG^m with, say, JXV — J\XU. The discussion above shows that

Tir{q)RG2ìm(v) = RX"! e . . . © i x m e MJ2Î ©... e m2xm

with some suitable almost Hermitian structure J2 G Jw(q) orthogonal to J\.

Conversely, let p' G CG^m» -X"ì, • • •, - m a n adapted quaternionic orthonormal basis of TpCG2,m with JX'V — J[X'V, and J'2 G i7p' an almost Hermitian structure orthogonal to J[. Let q' G Qm+1 with 7r(q') = p' and denote by Ji, J2 G J9' the almost Hermitian structures determined by Tr*Ju\Hqi = J'vn+\Hqi. Let z',v' G Cm+2 be the orthonormal vectors in Cmi~ with g' .= r(^ / ,v /) and j^(ir/4) = J\Uq<, where £' := hz/(v'j). Using Proposition 7 (iv) and Corollary 2 it follows that there exist complex orthonormal vectors Z,

1,...,Z,mm C m + 2 perpendicular to Cz' © Cv' so that the parallel translate of hz>(Z'„j)

along 7£ from r(z') to g' is the horizontal lift X'u of X£ at q'. There exists some t G M such that the parallel translate of each J2X'U along 7 from q' to r(z') is of the form h2i(Z'ue~2lt), and we have

hz>(Z'uj) = hz,en(Ke-itJ)

http://CG2.ro


and

hzl(Z,l/e-2it) = hzleii(Z

,ì/e-it).

It now follows that, when we defìne V to be the (m + 2)-dimensional real linear subspace of C m + 2 spanned by z'é\ v'e^, Z\eTi%,..., Z^c"»', we have

TpRG2,m {V) =RX[ e . . . e RX'm e m^x[ e . . . e n t ^ x ; m

Eventually, using Theorem 7 it follows that MG2>m(V) and MG2,m(V") are holomorphically congruent to each other. We summarize the preceding discussion in

THEOREM 12.

(i) Let V be an (ra + 2)-dimensional real linear subspace of£m+2. ThenRG2,m(V) is a totally geodesie, real, totally complex submanifold of CG .m- For any p E MG2,m(V) there exist an adapted quaternionic orthonormal basis Xi,... ,Xm

ofTpCG2tm with, say, JXV = 3\XV, and an almost Hermitian structure J2 E Jp

orthogonal to J\ such that *

TpMG2>m(v) = MXi e . . . e ^xm 0 MJ2X1 e . . . e Mj2xm .'

(ii) Conversely, let p E CG2,m, Xi,... ,Xm an adapted quaternionic orthonormal basis ofTpCG2,m with JXV = J\XV, and J2 E Jp an almost Hermitian structure orthogonal to J\. Then there exists an (m -f 2)-dimensional real linear subspace V of C m + 2 such that p E WLG2tm(v) and TpWLG2tm(V) = W 0 J2W with W:=MXi0. . .0MXm .

(Hi) Any two classical totally geodesie submanifolds MG2,m(^) and JRG?2,m(^/) in CG2,m are holomorphically congruent to each other.

As a consequence of Theorem 12 and Lemma 3 we obtain

COROLLARY 8. Let V be an (ra + 2)-dimensional real linear subspace ofCm+2. Then there exists for any p E M.G2,m(V) an (ra -f 2)-dimensionai real linear subspace V of C m + 2 such that p E MG2,m(^') and TpWLG2tm(V') = lpMG2)m(Vr).

Our next goal is to prove that the totally geodesie embeddings of MG2,m in CG2>m

are just the classical ones. For this we shall need some more information about the curvature of MCx2,m- The basic idea to get this information is to embed RG2,m in CG2,m

a s a totally geodesie submanifold and then to use the Gauss equation. To start with, let M.G2,m be one of the classical totally geodesie embeddings Mj2,m(V) in £G2,m- As RG .m is a totally complex submanifold of CG .m, there exists around each point in RG2,m a canonical locai basis Ji,J2,J3 of J such that J\ (resp. 32^3) maps the tangent space of ]RG2 m at any point q into the tangent (resp. normal space) of MG2,m at q. The locai

68 J. Berndt

almost Hermitian structure J\ on CG2]m induces a locai Kàhler structure Jo on RG2tm.

Furthermore, foJ, J3J induce a locai, parallel, rank-two vector subbundle of End(TJRG2,m)

consisting of symmetric endomorphisms. This locai subbundle can be extended in an

obvious manner to a global parallel rank-two vector subbundle A of End(TRG2tln). We

regard A as a Riemannian vector bundle such that locai sections of the form J2J, J3J

form an orthonormal basis everywhere. We cali such a basis a canonical locai basis of

A. By means of Proposition 15, if Ai,A2 is a canonical locai basis of A, then Av is a

symmetric endomorphism at each point satisfying Al = I and trAu = 0. In particular,

Av is diagonalizable and has the two real eigenvalues +1 and —1, both with the same

multiplicity m. Furthermore, since AVJQ = J^+iJJi = —J\JV+\J = —JQAU in the

above notation, these two eigenspaces are interchanged under Jo- Also, we have

Ai A2 + A2AX = J2JJ3J + J3JJ2J = -{J2J3 + J3J2) = 0

in the above notation, which means that the elements of a canonical locai basis of A

anticommute. Now, the Gauss equation and Theorem 2 yields the locai expression

R(X,Y)Z=:g(Y,Z)X-g(X,Z)Y

+ g(J0Y,Z)J0X - g(J0X, Z)JQY - 2g(J0X, Y)J0Z

2

+ ^2(g(AvYìZ)AuX-g(AvX1Z)AvY) v = l

for the Riemannian curvature tensor R of IRG .m- It is worthwhile to mention that this

expression can also be obtained by realizing MG%m as the complex quadric in C P m + 1 ;

the locai Kàhler structure Jo on MG2,m is obtained by projection with MGj m —*• MG2,m

of the Kàhler structure o n l G ^ induced from C P m + 1 . A canonical locai basis Ai,A2

is induced from the shape operators of MGj m with respect to a locai orthonormal frame

field of the normal bundle of the complex quadric in C P m + 1 .

Using Theorem 3 we see that RG2tm has two types of singular tangent vectors X,

namely those for which X J_ AX and those for which X E AX; and if X is of unit

length, the Jacobi operator Rx of MG2,m with respect to X has the following eigenvalues

K and eigenspaces TK :

(i) X LAX:

^ TK dimTK

0 RX&AX 3

1 (RX 0MJ o X e AX)L 2 m - 4

4 MJoX 1

Riemannian geometry ofcomplex two-plane Grassmannìans 69

(ii) X e AX, say AXX = X:

K TK

MX e {Y | Y 1 (RX 0 MJ0X), AXY = - Y }

MJoX e {Y | y i (MX e m0x), AXY = Y}

dimTK

m

m

We are now able to prove that there are no other totally geodesie embeddings of

MCr2,m m CG2,m m a n m e classical ones.

THEOREM 13. Any totally geodesie submanifold M tf/CG^.m which is isometric to the real Grassmann manifold MG ,™ wiih some symmetric metric is equal to WG2tm(V) for some (m-\- 2)-dimensionai real linear subspace V ofCm+2.

Proof. As M is isometric to MG .m with some symmetric metric, it is equipped with a locai Kàhler structure Jo and a parallel rank-two vector subbundle A of End(TM) such that the properties of J0 and A as stated above hold. Moreover, the Riemannian curvature tensor RM and the eigenvalues of the Jaeobi operators of M are as above up to some Constant factor.

Any singular tangent vector of M must be a singular tangent vector of CG2,m-Comparing the spectra of the Jacobi operators of M and CG2,m with respect to singular unit tangent vectors we get two possibilities:

(1) Kmax(M) = Sanarne {3,4},

(2) Kmax(M) = 4,

where Kmax(M) denotes the value of the maximal sectional curvature of M.

First, suppose that (1) holds. Then J0X = ±JX holds for any X € TM with X J_ AX. As each tangent space of M is spanned by singular tangent vectors of type X X AX, it follows that M is a complex submanifold of CG .m- This implies that M, equipped with the Kàhler structure induced from CG2,m» is a Hermitian symmetric space. But this is a contradiction, because every Hermitian symmetric space of compact type is simply connected.

Therefore, only (2) remains as a possibility. We fìx some point p £ M and choose a canonical locai basis Ai,A2 of Ap. Moreover, let Xi,..., Xm be a complex (with respect to Jo) orthonormal basis of TpM with AiX„ = Xv. Then we hàve A\JQXV — -JQXV

for ali v. Each Xv and JQXU is a singular tangent vector of M of type X E AX and of CG2,m of type JX e JX. Comparing the spectra of the corresponding Jacobi operators it follows that JXV and JJ0XU are perpendicular to TPM for ali v. Hence M is a real submanifold of CG2 m- Now, for each v E { 1 , . . . , m} there exists some almost Hermitian

70 J. Berndt

structure J$' £ Jp such that JXV = -J\V*XU. Comparing the zero eigenspaces of the

Jacobi operators with respect to Xv we get JQX^, _L WXU and JJQX^ = J$\j0XM for ali

fi ^ v. As ra > 3, the last equation yields J$' = . . . — J2 = : ^2- So we have

JXU = —J2XV and JJ$XU = J2J$XV

for ali ^, or equivalently,

J2JXV = X„ and J2J J$XV = —JQXV .

Às 'Xi,..:ìXmi JQXI ,.-..,. JoXm is a basis of TPM, and since J4IÀ\, = X„ and

AIJQXV = —JQXU for ali ^, we eventually get

Ai = J2^ •

In an analogous way we may deduce that

A2 = J3J

with some almost Hermitian structure J3 £ Jp. As

0 = AiA 2 + A2Ai = J2JJ3J + Js/^2J = -(foJs + J3/2) ,

we conclude that J2 and J3 are perpendicular.

Now, using the explicit expressions for the curvature tensors RM and R of M

( « MG2,m) and CC?2,m» and using the facts that M is real in CG2>m, ^1 = J2J and

^42 = J3J, the Gauss equation yields

0 = RM(X, Y)Z - R(X, Y)Z

= g(J0Y, Z)J0X - g(J0X, Z)JQY - 2g(J0X, Y)J0Z

- 9(hhY,Z)hhX + g(J2J3X,Z)J2J3Y + 2g(J2J3X,Y)J2J3Z

for ali X,Y, Z £TpM. Therefore, we have J0 = ±J2J3. In particular, at the point p, Jo

is an almost Hermitian structure in Jp orthogonal to J2 and J3. As X^ is perpendicular to

XUÌJQXUÌJ2XV,J3XV for /i ^ v (the last two vectors are perpendicular to M anyway),

we get XM 1 MXV and hence MX^ 1 MXV for ali \i ± v. Thus Xi,...,Xm is a

quaternionic orthonormal basis of TpCG2im. Moreover, since JXV = —32XV for ali v, it

is an adapted one. As

T P M = MXi e . . . e MXm © MJ0X1 e . . . e J 0xm ,

we may apply now Theorem 12 (with J\,J2 there replaced by - J2, Jo here) and eventually conclude that M is equal to some classical totally geodesie MG2)m(V) in CG2im. •


20. Complex-quaternionic orthonormal bases

Let A' G TpCG2,m be a singular unit vector of type JX _L JX. Then X determines the 8-dimensional subspace HCA of Tp£G2>m> Suppose that m is even, say m = 2n. A complex-quaternionic orthonormal basis of TPCG2,211 consists of unit vectors A i , . . . , Xn G Tp€G2,2n, for which the 8-dimensional spaces HCATi,... ,HCAn

are mutuaily perpendicular. The following useful lemma may be proved in a straightforward manner.

LEMMA 5. Let Xi,...,Xn be a complex-quaternionic orthonormal basis of TpCG2t2n, m ~ 2?i, J i , J2 G Jp orthonormal almost Hermitian structures, and J3 := Ji J2. Then A { , . . . , X'm, defined by

Xv := —j=(Xv — J\JXU) , A~n+;, := -^=(—J2AV + J3 JA„) , z/ = 1 , . . . , n.,

is fl« adapted quaternionic orthonormal basis of TpCG2,m with JX'U = JiA£ /or a// */ = 1 , . . . , m.

Conversely, let X[,..., X'm be an adapted quaternionic orthonormal basis of TpCG?2,m» m — 2n, vw7/i JA^ = 3\X'V, and J2 G Jp an almost Hermitian structure orthogonal to J\. Then A i , . . . , Xn, defined by

1 Xv : = ~7K(X'U + J2xli+») > ^ = 1, • • •, n ,

« « complex-quaternionic orthonormal basis of'TPCG'2,2^

Using this transformation from a complex-quaternionic orthonormal basis to an adapted quaternionic orthonormal basis, and vice versa, Theorem 7 may be reformulated in terms of complex-quaternionic orthonormal bases as follows.

THEOREM 14. Let f be a holomorphic isometry of ' CG^n- Furthermore, let p G CG2,2n» Xi,...,Xn a complex-quaternionic orthonormal basis of TpC(j2,2n» omd J\,J2 G Jp two orthonormal almost Hermitian structures. Then f*X\,..., f*Xn is a complex-quaternionic orthonormal basis of Tf^CG2,2n, and there exist orthonormal almost Hermitian structures J[, J'2 G Jf(P) such that f*J1 = J[f* and /* J2 — J'2f*.

Conversely, let pi,P2 G CG^m Ai, . . . ,Xn a complex-quaternionic orthonormal basis of TPlCG2t2n and Yi, . . . ,Yn a complex-quaternionic orthonormal basis of TP2C6?2)2n» < i) 2 £ Jpi orthonormal almost Hermitian structures, and J[,J2 G JP2

orthonormal almost Hermitian structures. Then there exists a unique holomorphic isometry f o/CG'2|2n with f(pi) = p2> f*Xv - Yu for allv, /* J\ = J[f* and /* J2 = J'2f*-

72 J. Berndt

21. The totally geodesie HP n

Let j be a quaternionic structure on C2 n + 2 compatible with the Hermitian structure on C2 n + 2 , that is,.; is an orthogonal transformation of C2 n + 2 and i,j, ij makes C2 n + 2 into a right quaternionic vector space. It is easy to prove that

F : CP 2 n + 1 -+ Q2n+1 , T(Z) hH. T(Z, ZJ) , z E 5 4 n + 3 C €2n+2

is a well-defined smooth map. Let z, v E £2n+2 be unit vectors with v _L Cz := Mz0M2Ì. Then ^ (u ) is a unit tangent vector of CP 2 n + 1 at r(z) and

j v : R -* C P 2 n + 1 , * •-+ r(costf 2 + sint v)

is the geodesie in CP2n+1 with 7^(0) = r(2) and 7^(0) = hz(v). By computing the tangent vector of F o j v at 0, we obtain for the differential F*z of F at z the expression

l o , ìf v E Czj.

Hence F is a submersion and for each £ E CP 2 n + 1 there exists an open neighborhood Ut of l in C P 2 n + 1 such that F\Ut is a submersion onto a 4n-dimensional submanifold Ni C P(CP 2 n + 1 ) of Q2n+1. Suppose that r(z) E C and v 1 Czj. The geodesie 7 in HLP2n+1 with initial values 7(0) = r(z,zj) and 7(0) = hT^tZ^(v,vj) E TT^ZyZ3)Ni is the curve

ti-». r(costf (z, z?) + sin/ (v, vj)) = F(yv(t))

and lies entirely in P(CP2 n + 1) and, for small i, in Ni. Therefore, each Ni is a totally geodesie submanifold of HP 2 n + 1 . By means of rigidity of totally geodesie submanifolds the union of ali pieces Ni forms a totally geodesie submanifold of ELP2n+1. Hence N := P(CP 2 n + 1 ) is an open part of a totally geodesie CP2 n or UPn in MP2 n + 1 . A straightforward calculation shows that r(zrzj) — T(Z', Z'J) if and only if z' E Cz 0 Czj. So the fibre of F over F(T(Z)) is the CP1 in C P 2 n + 1 determined by ali complex linear lines in Cz 0 Czj. Hence F : C P 2 n + 1 -+ N is a fibre bundle with typical fibre CP 1 and so N is diffeomorphic to CP2n+1 /CP1 « MPn. In fact, one might view F as the twistor map CP 2 n + 1 -» HP" assigning to each complex linear line Cz in C 2 n + 2 the quaternionic linear line Cz © Czj in the right quaternionic vector space (C2n+2,j). Summing up, we have shown so far that AT is a totally geodesie MPn in HP 2 n + 1 . As N is contained in Q2n+l, it is also totally geodesie in Q2n+1.

The above formula for F*z shows that

TT(ZlZj)N = {/»(,,„)(*, vj) I 1; E C2 n + 2 , « 1 Ù 0 C*j} , .

which is a subspace of 7iT'(*,*.?)• Hence AT is a horizontal submanifold of Q 2 n + 1 . Furthermore, a simple calculation shows that ir(F(z)) = 7r(F(z')) if and only if F(z) =


F(z'), whence ir\N is injective. Consequently, ir(N) is a totally geodesie submanifold of C6'2,2n diffeomorphic to HLPn. By construction, n(N) coincides with the classical totally geodesie submanifold MPn(j) in CG^n as described in Section 12.

As any totally geodesie MPn in WP2n+1 is invariant under the quaternionic Kàhler structure of HP 2 n + 1 , MPn(j) is a quaternionic submanifold of €G2,2n.

Now let z G Cn+2 be an arbitrary unit vector. We put q := r(z,zj), p := 7r(q) and £ := hz(zjj). Let z,zj,X\,... ,À'2n be a complex orthonormal basis of C2 n + 2

with Xn+V — Xvj for ali v — 1,. ; . ,n. Then hz(zj),hz(X\),... ,hz(X2n) is a complex orthonormal basis of TT^CP2n+1. We denote by X„ e 7ig the parallel translate of hz(Xv) along 7^ from r(z) to q. Let Ji G Jq be the almost Hermitian structure determined by 7^(7r/4) = -J\Uq. According to Corollary 2 and Proposition 7 (iv) we then have

<pXv = J\XV , v - l , . . . , 2n .

Defìne XL, . . . ,X2 n G TpCG2,2n and Ji € Jp by TT*^ = X„ and 7r*7i|ft? = «7ifl\„|Wg. Then X i , . . . , X2n is an adapted quaternionic orthonormal basis of TpCG2,2n with JXV — J\XV. Let J2 G *7? be the almost Hermitian structure for which the parallel translate of hz(Xuj) along 7 from r(z) to q is J2XV for ali v — 1 , . . . , 2rc, and denote by J2 G Jp the almost Hermitian structure with TT* J 2 | ^ ? = ^ff*!^. The above formula for the tangent space of N at q implies that

1 Zv := -7=(Xv + Ì2^n+/ / ) , Z/ = 1, . . . , fi

are tangent vectors of MPn(j) at p. As they are quaternionic orthonormal, and since M.Pn(j) is a quaternionic submanifold, we get

T p M P n O ) = M Z i 0 . ' . . e H Z n . .

According to Lemma 5, Zi,...,Zn is a complex-quaternionic orthonormal basis of TpCG2,2n- This implies that JWZV = M.JZ„, is the orthogonal cómplement of WZU

in WCZ„, which is also perpendicular to MZ^ for ali p-i^v. Therefore, J maps rpHPn(.?) into J-pWPn(j), which means that MPn(j) is a real submanifold of CG2)2n- Each Zv is a singular tangent vector of CG2,2n of type JX _L JX. According to Lemma 3, each non-zero vector in MZV is also singular of type JX _L JX. As the linear combination of any two complex-quaternionic orthonormal singular vectors of type JX _L JX is also singular of that type, we eventually conclude that each non-zero tangent vector to MPn(j) is singular of type JX _L JX.

Conversely, let Z[,..., Z'n be a complex-quaternionic orthonormal basis of Tp/CG2,2n at some point p' G CG2,2n and J[, J'2 G Jp> two orthonormal almost Hermitian

74 J. Berndt

structures. According to Lemma 5, X[,..., X'2n defined by

1 1 Xv := ~7~(^v ~ J\JZV) » Xn+v : = —7=(—J2%v + J1J2JZI) } i> = 1,. . . ,n

is an adapted quaternionic orthonormal basis of TpCG2,2n with JX'U = J { ^ for ali 1/ = 1,...., 2n. We choose g' G Q2 n + 1 with 7r(<jf') = p' and denote by X£ the horizontal lift of X'v at q' and by J{, J'2 G *7?' the almost Hermitian structures with T* J'U \Hq> = J'v TT* \Hq>. There exist uniquely determined complex orthonormal vectors z',?/ G C2 n + 2 such that 5' = 7£'(7r/4) and —J[Uq' = 7£/(7r/4), where £' := hz>(v'j). For */ = 1,. . . ,n, denote by-X£ the unit vector in C2 n + 2 for which the parai lei translate oihzi(X'u) along y^ from r(z') to q' is X£. We now defìne a quaternionic structure / on C2 n + 2 , compatible with the Hermitian structure on C2n+2 , by ij' + / i = 0, z'J = v', and for each v — 1 , . . . , n the parallel translate of hz>(X'ì/j'j) along 7^ from r(z') to q' is just J'2X'n+v. Then * , / ,* / is a quaternionic structure on C2 n + 2 and the induced totally geodesie HP n ( / ) in CG2,2n satisfìes p' G HP n ( / ) and

•TpiHPn(/) = mz[ e . . . e mz'n.

By means of Theorem 14, there exists a holomorphic isometry / of CG^^n with f(p) = p' and f+Z„ = Z'u. This isometry maps WPn(j) onto HP n ( / ) , whence HPn(j) and HP n ( / ) are holomorphically congruent to each other.

We summarize the preceding discussion in

THEOREM 15.

(i) Let j be a quaternionic structure on C2n+2 compatible with its Hermitian structure. Then WPn(j) is a totally geodesie, real, quaternionic submanifold o/CG^n- Any non-zero vector tangent to MPn(j) is a singular tangent vector ofCG2t2n of type JX -L JX. For each p G MPn(j) there exists a complex-quaternionic orthonormal basis Xi,... ,Xn ofTpCG2,2n such that

Tpwpn(j) = mxx e . . . e mxn.

(ii) Conversely, let p G CG2,2n and X\ì...ìXn a complex-quaterniónic orthonormal basis ofTpCG2,2n> Then there exists a quaternionic structure jon C2 n + 2 compatible with its Hermitian structure such that p G MPn(j) and TpWPn(j) = MXi 0 . . . 0 MXn.

(Hi) Any two classical totally geodesie submanifolds I?"(j) and MPn(f) of CG2,2n are holomorphically congruent to each other.

If Xi,...,Xn is a complex-quaternionic orthonormal basis of TpCG2,2n> then JXi,... ,JXn is also a complex-quaternionic orthonormal basis of TpCG2,2n> and


MJXÌ 0 . . . 9 MJXn is the orthogonal complement of MXi 0 . . . 0 MXn in TpCCr2,2n. Theorems 14 and 15 therefore imply

COROLLARY 9. For any p e HPn(j) there exists a MPn(j') such that p € UPn(j') and TpMPn(f) = ±pMPn(j).

We shall now prove that any totally geodesie embedding of some MPn into CG2t2n is in fact a classical one.

THEOREM 16. Any totally geodesie submanifold M ofCG2,2n which is isometric to an n-dimensional quaternionic projective space with some Fubini Study metric, is equal to some classical totally geodesie submanifold I ? " ( j ) in CG2,2n-

Proof. We denote by Ac the value of the quaternionic sectional curvature of M, fìx a point p E M and choose a unit vector X £ TPM. Then Rx\TpM has the eigenvalues 0,c, 4c with corresponding multiplicities l,4ra - 4,3, respectively. The eigenspace with respect to Ac is JX, where J denotes the quaternionic Kàhler structure of M. Theorem 3 implies the following possibilities:

d ) c = l ;

(2) e = 1 — cos a and Ac = 1 + cos a;

(3) e = 1 — cosa and 2(1 — sino;) = 4(1 — cosa);

where a = l(JX,JX).

First, suppose that (2) or (3) holds. Then Theorem 3 (iii) implies that there exists a unit vector Y in the eigenspace of Rx\TpM with respect to e satisfying JY e JY. This gives a contradiction, because the spectrum of Ry does not contain the value e.

Therefore (1) holds for ali unit vectors X £TPM. Let X i , . . . , Xn be a quaternionic orthonormal basis of TPM. The table in Theorem 3 (i) then implies Xv L H O ^ for ali v -fi p, and hence also IHICX^ J_ WCX^ for ali v ^ p. Therefore, X\,.•.., Xn is a complex-quaternionic orthonormal basis of TpCG2,2n- Now, let Y £TPM be a unit vector vector in the eigenspace of Rx\TpM with respect to 4. Then, according to Theorem 3 (i), Y can be written in the form

Y = cost JX + sint JiX

with some t £ R and some almost Hermitian structure J\ € Jp. Since also RyX = AX, we get, by using Theorem 2,

AX - cos21 RJXX + sin21 RjlXX

+ sin* cos/ (R(X,JX)JlX + R(X, JXX)JX)

= 4 X - 8 s i n / cosi JXJX .

76 J. Berndt

As 3\ JX J_ X, this rmplies either sin t = 0 or cos i = 0. In the first case we get, by varying with Y in the 3-dimensional eigenspace of Rx \TPM with respect to 4, JX C MJX, which is clearly impossible. Thus the second possibility holds, which implies JX = JX, and hence TPM = MX1 0 . . . 0 MXn. The assertion then follows from part (ii) of Theorem 15. •

22. Isometric reflections in submanifolds

We will now study the reflections of CG2,m in the above totally geodesie submanifolds.

THEOREM 17.

(i) The reflection of CG2,m in any of the classical totally geodesie submanifolds CPm(£L), CPa(V) x CPb(VL) (a + b = m), and CG2,k(V) (0 < k < m - 1) (where k = 0 givesjust a single point) is a well-defined global holomorphic isometry of CG .m- Conversely, if N is a connected submanifold of CG2,m far which the locai reflections in itare holomorphic isometries, then N is an open part ofone ofthe classical totally geodesie submanifolds CPm(t1), CPa(V)xCPb(VL) (a+b = m), orCG2lk(V)(0<k<m-l).

(ii) The reflection of CG ,™ in any of the classical totally geodesie submanifolds ^G2>m(V) and MPn(j) (m = 2n) is a well-defined global anti-holomorphic isometry of CG^m- Conversely, if N is a connected submanifold of CG2,m> far which the locai reflections in it are anti-holomorphic isometries, then N is an open part of one of the classical totally geodesie submanifolds MG2,m(V) or MPn(j) (m = 2n).

Proofi The reflection in a single point of CG2ìm is'-a holomorphic isometry, because (£G2,m is a Hermitian symmetric space. Let M be any other of the above mentioned classical totally geodesie submanifolds. As M is compact, one can define the reflection / in M in some open neighborhood of M. By means of Corollaries 5-9 there exists for any p £ M a totally geodesie submanifold M of CG2,m such that p e M and TpM = LpM. It now follows from Theorem 3 in [12] that / is an isometry. Since CG2,m is complete, simply connected, and real analytic, / can be extended to a global isometry / of CG2,m-Clearly, / is the reflection of CG2,m in M.

First, suppose that M is some CPm{lL), CPa(V) xCPb{VL), or CG2,k(V). Then M is complex and we obtain

UJX = JX = Jf*X


for ali X e TM, and

f,JY = -JY = JUY

for ali Y G i-M. As any isometry of CG2,m is either holomorphic or anti-holomorphic, we conclude that / is holomorphic. Next, if M is some RG2,m(V) or HPn(j) , then

fifJX — —3 X =• —Jj+X

for ali X e TM, and

f,JY = JY = -Jf*Y

for ali Y G LM. Thus, in this case, / is anti-holomorphic.

Next, suppose that a locai reflection f in N (as in the assertion) is an isometry. Let p G N be a point where / is defined. According to [12], N is totally geodesie in CG2,m» and there exists a totally geodesie submanifold N of CG2,m with p € N and TP7V = -LpAT. Let À' be any tangent vector of N at p. Then /*X = X, and hence /*JX = Jf*X = JX if / is holomorphic, and f*JX — —Jf*X — —JX if / is anti-holomorphic. Therefore, N and N are complex (resp. real) submanifolds in case the locai reflections in it are holomorphic (resp. anti-holomorphic). Clearly, the real dimension of Àr or N must be at least 2ra. According to the existence results of maximal totally geodesie submanifolds in CG2,m (see Section 12), N or N must be an open part of some CP™^ 1) , CPa(V) x CPb(VL) (a+ b = m), CG2,k(V) (m/2 < k < m - 1), MPn(j) (m = 2n), or M-r2,m(V)- The assertion then follows by applying Corollaries 5-9 and Theorems 4,6,8-13,15 and 16. •

23. Isoparametric systems of codimension one

Let V be an (m + 1)-dimensionai complex linear subspace of C m + 2 and put CG2,m-i := CG2>m-i(V). Consider the subgroup G of the group of holomorphic isometries of CG2,m leaving CG^.m-i invariant. Let pi,P2 £ CG2,m-i and £i G LPlCG2,m-i, £2 € -LP2CG2,m-i be unit vectors. As _LPlCG2,m-i is both a complex and quaternionic subspace of TPlCG2,m with real dimension four, there exists an almost Hermitian structure J\ G JPl such that J£i = Ji£i. Similarily, there exists an almost Hermitian structure J[ G JP2 such that J& = J\£,2> We exténd £i to an adapted quaternionic orthonormal basis X\,... ,Xm_i,£i of TPlCG2,m with JXU = 3\XV for ali v, and £2 to an adapted quaternionic orthonormal basis Y\ì... , ym-i,£2 of TP2CG2,m

with jy„ = J[YU for ali z/. According to Theorem 7, there exists a holomorphic isometry / of CG2,m with /(pi) = p2, f*Xu = y„ for ali ^, and /*£i = &• As X i , . . . , X m _ i is an adapted quaternionic orthonormal basis of TPlCG2,m-i> and y i , . . . , y m - i is an adapted quaternionic orthonormal basis of TP2CG2)m-i, we conclude that /*TPlCG2,m_i = TP2CG2,m-i, and hence /(CG2 ,m- i) = CG2,m_i. This proves

78 J. Berndt

that / G G and G acts transitively on the unit normal bundle of CG2,m-i- Therefore, the principal orbits of the action of G on CG2,m are of codimension one and are the tubes around the singular orbit CG2,m-i-

Now let p G CG2,m-i» £ a unit normal vector of CG2,m-i at p, and J\ the almost Hermitian structure in Jp so that J£ = Ji£. We decompose TpCG2,m orthogonally into

TpCG2,m = io e U e i-s 0 T0 e T2 ,

where ±o := M ,

1 8 := MJ£ ,

Tb := {X € TpCG2,m-i | JX — — JiX} ,

T2 := {X G TpCG2,m-i | J * = J iX} . Note that these spaees are the normal and tangential components of the eigenspaces of the Jacobi operator R{. The index refers to the corresponding eigenvalue.

Next, let 7 : M -»• CG2)m be the geodesie in CG2,m with T(0) - p and 7(0) = £. By means of Corollaries 4 and 7, 7 lies in the totally geodesie CG2,i « CP2(8) tangent to the normal space of CG2,m-i at p. Thus 7 is closed with period length 7r/\/2. Denote by Zx the Jacobi field along 7 with initial values Zx(0) = 0 and Z'X (0) = X if X G ±pCG2,m-i» and with Z*(0) = X and Z'x(0) = 0 if X G rpCG2,m-i. Further, by Px we denote the parallel vector field along 7 with Px(0) = X. Then we have

Zx{t) = j-sm(y/8t)Px(t), if X G ±8 ,

^ W = 7 2 S Ì n ( \ / 2 ^ x W , i f X € l 2 ,

•^jrW = ^ ( < ) , ifXGTo ,

Zx(t) = cos(V2t)Px(t), ifXeT2.

From this we deduce that the first focal point of CC?2,m-i along 7 is at distance 7T/A/8,

which is just the antipodal point q of p along 7. Furthermore, the tangent space of the focal set F (which is a singular orbit of the action of G and hence a submanifold) at q is obtained by parallel translation of To © ±2 along 7 from p to q. Note (see also Lemma 3) that JX = — J\X for ali X G Tb©±2, whence TgF is a complex, totally complex subspace of TqCG2,m with half real dimension and the property JY = J[Y for ali Y G TqF and some almost Hermitian structure J[ € Jq. If A denotes the shape operator of F at q with respect to 7(7r/\/8), then, see also the proof of Proposition 7,

AZx{*lyM) = (-Z'x(ir/VS))T ,


where the index denotes the tangential component to TqF, and X e T0 © 1 2 - Using the above explicit expressions for Zx we get A — 0. Therefore, F is totally geodesie in £G2,m> Using Theorem 5 we conclude now, that F is a classica! totally geodesie submanifold CPm(£L) in CG ,™- ft ÌS n o t n a r^ t o prove that £ is the complex linear line in C m + 2 perpendicular to V.

Eventually, denote by Mr the tube of radius 0 < r < ir/y/S around CG2,m-i ar|d by Ar the shape operator of Mr at y(r) with respect to y(r). Again, we use the formula

ArZx(r) = (-Zx(r)f .

A straightforward computation yields, that the principal curvatures of Mr at y(r) are

-V/8cot(V/8r) , -V2cot(V2r) , V/2tan(V2r) , 0 ,

and the corresponding eigenspaces are obtained by parallel translation of

along 7 from p to j(r). We summarize the results in the following

THEOREM 18. Let V be an (m+\)-dimensional complex linear subspace ofCm+2

and t the complex linear line in C m + 2 perpendicular to V. The focal set ofCG2,m-i{V) in CG2,m is CPm (£ 1 ) and consists precisely of ali points in CG2,m at distance 7r/\/8 from CCr2.ro-i(V"). or equivalently, of ali antipodal points ofCG2,m-i{V) in CG2,m- Any tube Mr of radius 0 < r < w/y/S around CG2>m-i(V) is an embedded isoparametric hypersurface ofCG2,m withfour (resp. threefor r — w/y/32) distinct principal curvatures. More precisely, denote by £ the "outward" unit normal field of Mr for somefixed r. Then there exists an almost Hermitian stmcture J\; which is uniquely defined at each point of Mr, such that J£ = Ji£. The principal curvatures of Mr with respect to £ are

, Ai := ->/8cot(>/8r) , \2 := -v /2cot(v /2r) , A3 := \/2tan(V2r) , A4 := 0 ,

with corresponding multiplicities

m(Ai) = 1 , m(\2) — 2 , ra(A3) = m(A4) = 2m — 2 .

The corresponding eigenspaces are

E(X2) := C 1 * ,

JE7(A3) := {X e TMr | X J_ H? , JX = J iX} ,

£(A4) := {X <E TMr \ X J_ H£ , JX = - J iX} .

http://CCr2.ro-

80 J. Berndt

We now construct the second homogeneous isoparametric system. Consider the subgroup G of ali holomorphic isometries of CG2,2n leaving some totally geodesie MPn. := MPn(j) invariant. Choose any two points pi,P2 G HP n and unit normal vectors Ì i G J-PlMPn, (2 e l P 2 H P n . As HP n is a real submanifold of CG2|2n, we have J& e TPlMPn and J£2 G TP2MPn. We extend «7£i to a quaternionic orthonormal basis Xiì...}Xnî,J(i of TPlMPn and J ^ to a quaternionic orthonormal basis Yi,. ..,yn_i,«7^2 of Tp2HPn. Then both bases are complex-quaternionic orthonormal bases of the respective tangent spaces of CG2,2n- By means of Theorem 14 there exists a holomorphic isometry / of CG2)2n with /(pi) = />2> f*Xu = Yv for ali v, and /*J£i = J 6 - Using Theorem 15 we see that / (HP n ) = MPn , and hence f e G. Furthermore, as / is holomorphic, we get

f*d — —J f*£i — —Jf*J£i = —J £2 = £2 •

This implies that G acts transitively on the unit normal bundle of H P n . Thus, the principal orbits of the action of G on CG^n are of codimension one and arise as tubes around the singular orbit MPn .

We proceed now as in the previous case. Let p G HP n and £ a unit normal vector of MPn at p. From Corollary 9 and Theorem 15 we get J£ 1 Jf . We decompose TpCG2,2n orthogonally into

TpCG2,2n = io e l i e i 4 e To e Ti e r 4 ,

where

l o := M ,

l i := (ECO 1 n l p M P n ,

1 4 : = J { ,

To := JJÌ ,

Ti :=(HCf)- L nr p l IP n ,

T4 := MJ£ .

Also here, these spaces are the normal and tangential components of the eigenspaces of the Jacobi operator R{, and the index refers to the corresponding eigenvalue.

Let 7 : M —> CG2,2n be the geodesie in CG2,2n with 7(0) = p and 7(0) = £. According to Corollary 9, 7 lies in the totally geodesie MPn(f) tangent to the normal space of HP n at p. As HP n ( / ) has Constant quaternionic sectional curvature four, 7 is closed with period length ir. For Zx, as defined analogously to the preceding case, we


have

Zx(t) = smtPx(t), i f X € ± i ,

Zx(t) = \ sin % Px(t), if X e ±4 ,

Zx(t) = Px(t), ifXeT0ì

Zx(t) = cosi Px(t), ifXeTi ,

Zx(t)- cos2* Px(t), if XeTA.

This implies that the focal points of HPn along 7 are at y(t), where t is an integrai multiple of 7r/4. SO the focal set Fn := Fn(j) of MPn consists of ali points at distance w/4 from HPn , and the set of antipodal points of HPn is HPn itself. Furthermore, the normal space of Fn at q = 7(7r/4) is obtained by parallel translation of ±0 © T4 = C£ along 7 from p to q. Hence, Fn is a complex hypersurface of CG^n- The principal curvatures of Fn

with respect to 7(7r/4) can be computed with the same methods as before. It turns out that there are precisely three of them, namely — 1,0, +1, and the corresponding eigenspaces are obtained by parallel translation of _l_i, ±4 ©T0, Tu respectively. Às 1 4 $ T 0 = JC£, the zero eigenspace is J±qF

n and hence independent of the choice of the unit normal at q.

Now let Mr be the tube of radius 0 < r < 7r/4 around MPn. Using the same methods as in the previous example we get, that the principal curvatures of Mr with respect to y(r) are

— cotr , —2cot2r , tanr , 2tan2r , 0 ;

the corresponding eigenspaces are obtained by parallel translation of

-Li j -1-4 ; T\ , T4 , To ,

along 7 from p to j(r), respectively. Summing up the preceding discussion we obtain

THEOREM 19. The focal set Fn(j) of WPn(j) in CG2,2n is a complex hypersurface of£G2,2n cind consists of ali points in CG2,2n at distance ir/'4 from MPn(j). The principal curvatures of Fn(j) with respect to any unit normal vector are + 1 , 0 , - 1 . At each point q G Fn(j), the nuli space ofthe shape operator is JLqF

n(j) and independent of the choice of the unit normal at q. The other two eigenspaces are real subspaces and mapped into each other by J.

Any tube Mr of radius 0 < r < w/i around MPn (j) is an embedded isoparametric hypersurface ofCG2,2n with five distinct principal curvatures. More precisely, the principal curvatures with respect to the "outward" unit normal field £ of Mr are

Ai := — cot r , X2 := - 2 cot 2r , A3 := tan r , A4 := 2 tan 2r , A5 := 0 ,

82 J. Berndt

and the respective multiplicities are

m(\i) = m(A3) — 4n - 4 , ra(A2) = ra(A5) = 3 , ra(A4) = 1 .

For the corresponding spaces of principal curvature vectors we have

E(X4) = mt , E(X2) = Ji , £(A5) = JJÌ ,

and the eigenspaces E(\\) and E(\3) are mapped info each other by J.

REMARK 7. The two isoparametric systems discussed above are the only ones arising in this manner from a totally geodesie submanifold of CG^m- This might, for example, be deduced from the fact that in ali the other cases the Jacobi operators have different spectrum when varying the normal vectors.

24. A remark on the dual symmetric space CG*2 m .

Denote by CGJm the non-compact Riemannian symmetric space dual to CG .m-The Riemannian curvature tensor of CG% m is obtained from the one of CG2,m by just adding a minus sign in front of it. By duality, the above discussion of CG2,m may be transferred to CGJ m by using the corresponding dual objects. The only exception arises for the isoparametric systems in the preceding section. Here, CG*2 m_x and the m-dimensional complex hyperbolic space CHm are not focal sets of each other, but determine different homogeneous isoparametric systems on CGJ m with just one singular orbit. The principal curvatures of a tube of any radius r e M+ around CGJ m _i or CHm are the same, namely

-V8coth(V8r) , -\/2coth(\/2r) , - \ /2tanh(\/2r) , 0 ,

but the multiplicities are different in their order, namely

1 , 2 , 2ra - 2 , 2ra - 2

for the tube around CGJ m _ i , and

1 , 2m - 2 , 2 , 2m - 2

for the tube around CHm. Similarily, the quaternionic hyperbolic space MHn determines a homogeneous isoparametric system on CGJ 2n

w*tn Just o n e singular orbit. The principal curvatures of the tube with radius r E M+ around MHn are

—2coth2?1 , — coth?' , — 2tanh2r , — tanhr , 0 ,

with respective multiplicities

3 , 4n - 4 , 1 , 4?i - 4 , 3 .


REFERENCES

[1] ALEKSEEVSKY D.V., M ARCHI AFA VA S., Quaternionic-like structures on a manifold: Note IL Automorphism groups and their interrelations, Rend. Mat. Acc. Lincei (9) 4 (1993), 53-61.

[2] BERNDT J., Real hypersurfaces with Constant principal curvatures in complex space forms, Geometry and Topology of Submanifolds li, Proc. Conf. Avignon/France 1988 (Eds. M. Boyom, J.-M. Morvan and L. Verstraelen), World Scientific, Singapore, 1990, 10-19.

[3] BERNDT J., VANHECKE L., Geometry ofweakly symmetric spaces, J. Math. Soc. Japan 48 (1996), 745-760.

[4] BESSE A.L., Manifolds ali of whose geodesics are closed, Springer-Verlag, Berlin, Heidelberg, 1978.

[5] BESSE A.L., Einstein manifolds, Springer-Verlag, Berlin, Heidelberg, 1987. [6] BLAIR D.E., Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics 509,

Springer-Verlag, Berlin, Heidelberg, New York, 1976. [7] BLAIR D.E., VANHECKE L., Symmetries and (p-symmetric spaces, Tóhoku Math. J. 39

(1987), 373-383. [8] BORISENKO A.A., NIKOLAEVSKII Y.A., Grassmann manifolds and the Grassmann image

of submanifolds, Uspeki Mat. Nauk 46 (1991), 41-83, translation in Russian Math. Surveys 46 (1991), 45-94.

[9] CARTAN E., Sur certaines formes riemanniennes remarquables des géométries à groupe fondamental simple, Ann. Sci. École Norm. Sup. 44 (1927), 345-467.

[10] CHEN B.Y., NAGANO T., Totally geodesie submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977), 745-755.

[11] CHEN B.Y., NAGANO T., Totally geodesie submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405-425.

[12] CHEN B.Y., VANHECKE L., Isometric, holomorphic and symplectic reflections, Geom. Dedicata 29 (1989), 259-277.

[13] FERUS D., KARCHER H., MUNZNER H.-F., Cliffordalgebren und neue isoparametrische Hyperflachen, Math. Z. 177 (1981), 479-502.

[14] HELGASON S., Differential geometry and symmetric spaces, Academic Press, New York, London, 1962.

[15] MARCHIAFAVA S., Curvature sezionali critiche e curvature caratteristiche di una varietà Kàhleriana quaternionale, Rend. Mat. Ser. VII 13 (1993), 467-497.

[16] OGIUE K., On fiberings of almost contact manifolds, Kodai Math. Sem. Rep. 17 (1965), 53-62.

[17] O'NEILL B., The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469.

[18] TAKAHASHI T.,Sasakian (p-symmetric spaces, Tdhoku Math. J. 29 (1977), 91-113. [19] TONDEUR P., Foliations on Riemannian manifolds, Springer-Verlag, New York, 1988.

Jurgen BERNDT

Universitàt zu Kòln, Mathematisches Institut

Weyertal 86-90, 50931 Kòln, Germany.

e-mail: [email protected]

mailto:[email protected]

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