Vol. 51 (2003) REPORTS ON MATHEMATICAL PHYSICS No. 2i3

DIFFERENTIAL INVARIANTS OF IMMERSIONS OF WITH METRIC FIELDS*

PAVLA MUSILOVA and DEMETER KRUFKA

MANIFOLDS

Institute of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, KoWska 2, 611 37 Bmo, Czech Republic

(e-mails: pavla@physics.muni.cz, krupka@physics.muni.cz)

(Received August 31, 2002 - Revised March 12. 2003)

The problem of finding all (higher order) differential invariants of immersions f : X -+ Y, where X and Y are manifolds endowed with metric fields, is investigated. The underlying jet manifolds are introduced, and the action of the corresponding differential groups on them are analysed. It is shown that the differential invariants can be described by means of the factorization of the group action with respect to a distinguished subgroup. By the orbit reduction method a basis of first-order invariants is found. In particular, this basis is used for a characterization of all first-order invariant Lagrangians depending on two metrics and an immersion.

Keywords: Differential invariant, immersion, orbit reduction method, variational principle, invariant Lagrangians. MS &ssMcation: 53A55, 58A20, 58E30, 83E30.

1. Introduction The problem of constructing differential invariants of a given field with values in

some manifold Y, or a collection of fields, defined on a smooth manifold, belongs to the fundamental problems of the theory of invariants and its applications. In this paper, the orbit reduction method developed in [8] is applied to the problem of finding all differential invariants of immersions of manifolds endowed with metric fields.

One well known differential invariant of this type is constructed as follows. Let X (resp. Y) be a manifold, and let h (resp. g) be a metric field on X (resp. Y), Every immersion f : X + Y induces another metric on X, f *g, and we have a differential invariant e(f) = 1/2(f*g)jjh’j. This differential invariant, the energy density, defines the energy functional

E(f) = J

e(f)(x)Jdx’ A.. . A dx”,

X

*This research was supported by grants GACR 201/00/0724 and MSM 143100006.

[3071

308 P. MUSILOVti and D. KRUPKA

whose critical points are the harmonic mappings. A particular question arises as to what extent the general covariance condition determines the function e(f).

Let us recall basic concepts of the differential invariant theory in the context of the present work, see [4-61. Let L:; be the r-th differential group of the Euclidean space IF?“. Let IV* 0 R”* be the symmetric tensor product, where KY* is the dual. We denote by Met W” c (IF* 0 W*) the subset of regular tensors, and by T,: Met R” the manifold of r-jets with source 0 E R” and target in Met IV. Further we denote by Imm J&,(&P, II??) the manifold of regular r-jets with source 0 E R” and target at OER’“.

Then we set for fixed indices n and m, n 5 m,

Qr = T,’ Met R” x TL Met Rm x Imm J;otd, (R” , IV”).

Qr is endowed with a natural left action of the Lie group LL+’ x L>+’ given by

(L;+’ x L;+‘) x Q” 3 <<j,‘+‘cx, j;+‘y), (j,‘h, jo’g, j;+‘f)) +

+ (j;(j;o . h), j,‘(j,‘y . g>, jor+‘(y 0 f ~a-‘)) E Qr,

where j&r . h : W” + Met&!? is defined by (j:cz - h)(x) = j~(t,~~t__~-~~~$ - h(cr-l(x)), with multiplication on the right-hand side denoting the tensor action of the group GL,(R). Analogously for j,’ y . g. Note that this formula represents the transforma- tions, induced on two metric fields and their derivatives as well as derivatives of an immersion by the coordinate transformations (at a point of R” x Rm), see e.g. [6].

Now let P be any left (LS, x LL)-manifold, where 1 5 s 5 r + 1 (later the case P = IR will be considered). Recall that a diferential invariant on Qr with values in P is any smooth mapping x : Qr + P such that

for all 4 E Qr, $+‘a! E LL+‘, ji+‘y E Lz’. Denote by K,‘+l,S the kernel of the canonical group morphism LL+’ -+ Li;

Kr+lvs is a nilpotent normal subgroup of L, ofn the quotient Q’/(K,‘+‘*”

“+l. In this paper, we give a description x KL+1.S) for r = 1, s = 1. This quotient has the

structure of an (Li x Lk)-manifold. Any set of coordinates on this quotient manifold can be viewed as a basis of differential invariants. In particular, every differential invariant x : Q’ -+ P factorizes through an (LA x Li)-equivariant mapping ~0 : Q1/(K,2v1 x K2s1) + P.

In [ll], themconstruction of Q’/(K,‘+‘.” x KL+l,‘) is given for arbitrary r, and s = 1; the geometrical meaning of the quotient projections Qr + Q’/(K,‘+‘q’ x K$‘,‘), or, which is the same, of bases of differential invariants, is also discussed.

For the first-order case, treated in this paper, the basis of differential invariants gives us a criterion for an immersion to be &fine. In the Riemannian case, a relation with the second fnndamental form appear.

DIFFERENTIAL INVARIANTS OF IMMERSIONS OF hfANlFOLDS WITH METRIC FIELDS 309

As an application, we also discuss the meaning of the differential invariants of immersions for the theory of natural Lagrange structures. In particular, the result of the factorization gives us a description of all first-order natural Lagrangians depend- ing on immersion of manifolds with metric fields.

2. Preliminaries In this section we recall two basic theorems on orbit spaces which we apply

in the orbit reduction method for computing the differential invariants. The proof of the first one can be found e.g. in [ 11, the proof of the second is immediate, see [5].

THEOREM 1. Let Q be a connected left G-manifold, n : Q + Q/G the quotient projection. The following two conditions are equivalent:

1. There exists a smooth structure on Q/G such that n is a submersion. 2. The set 2 = I(p,q) E Q x Q I n(p) = n(q)} is a closed submanifold of

Q x Q.

If these two equivalent conditions are satisfied, then the smooth structure on Q/G is unique. The set Q/G endowed with this smooth structure is the orbit manifold.

Let p : G + H be a surjective Lie group homomorphism with kernel K, Q be a left G-manifold, P’ be a left H-manifold and rr : Q + P’ be a p-equivariant surjective submersion, i.e. n(gq) = p(g)n(q) for all 4 E Q, g E G. Having p, we can consider every left H-manifold P as a left G-manifold by gy = p(g)y, g E G, y E P.

THEOREM 2. i’f each n-‘(x), x E P’ is a K-orbit in Q, then there is a bijection between the G-equivariant maps x : Q + P and the H-equivariant maps ~0 : P’ + P given by x = ~0 o n.

3. First-order difFerenthI invariants

We now compute the differential invariants on Q’ with values in left (Lf, x LL)-manifold. If q E Q’ and (A, C) E Lz x Li, we denote q = (A-‘, C-l) .q. Let hijv gavv hij,k, &rv,q, f{, fiT, where 1 5 i ( j 5 n, 1 5 k 5 n, 1 ( CT 5 u 5 m, 1 5 q 5 m, be the canonical coordinates on Qt. and let ($, +) (resp. (cz, c&)) be the canonical coordinates on Lz (resp. Li). Let A-’ = (bj, bj,), C-l = (d,“, d$). If we denote hij ($ = xij, gov(@ = Zo”, hij.k(a = xij,k, &v,,(a = Egv,)l, .fL(@ = x, fiy (4) = xj, we get for the action of Lt x L$ on Q’ the expressions

310

We introduce a new global chart on Q’ adapted to this action, with coordinates (hij, g,“, f;a, J;!k, A:,,, K;), with the help of the transformation equations

r;!k = 2 ‘h”(h9j.k + hqk,j - hjk,y), hij,k = hqi I$ + hqj q(lkt

A& = ~P(gav.r, + garl,v - gvr,,,), guv,, = ga,A:, + ga&,,

Kj”k = rjl&’ - A;,fjv ff - fjak, qk=q,fi”-A&fj”f;-K;.

The coordinates rjik,A$ KS are symmetric in the subscripts. We put h’jhjk = Sl, gU”gv,, = 8;. The action of the group Li x Lk on Q’ is expressed as follows:

The action of K,f*’ x K,$ ’ on Q’ is expressed by

With the help of expressions (1) we can describe the quotient space Q’/(K,fs’ x K;‘). The following theorem is the direct consequence of Theorem 1.

THEOREM 3. The set Q’/(K,f” x K;‘) has the orbit manifold structure. In global coordinates (hij, g,,“, fi”, qi, A&, KS) on Q1 and global coordinates (hij , gOv , fy, KS) on Q’/(Kz,’ x K$’ ), the quotient projection x is expressed by

(hii, guu, fi”+ ci, AZ,,, KG) + (hii, guv, fi”t KS)-

To study the structure of the orbit manifold Q’/(Kz,’ x K,$‘), consider the manifold

Qo= Met W” x Met Wm x reg (IP* @I IR”) x (IV* 0 IV* @ W”)

DIFFERENTZAL INVARIANTS OF IMMEXSIONS OF MANIFOLDS WITH METRIC FIELDS 3 11

with the tensor action of the group Lf, x LL. The set Q’/(Ki,l x K:‘) is endowed with the structure of a left (LA x Lk)-manifold given by

(Lf, x L;) x Q’/(K$’ x K;‘) 3 ((j,+, J,y). Eql~) +-

+ [(i,‘~2(j,‘a), iL2(j,‘y)) . qlK E Q’/(K,2S1 x K:‘),

where i’,2 : Lf, + Li is canonical inclusion. The following assertion is an imme- diate cokequence of the existence of global coordinates on both Q’/(Ki,l x Ki ’ ) and Qo.

THEOREM 4. The Zef (Li x L!,,)-manifold Q’/(Kj$l xK2’) is isomorphic with Q,-,.

The following theorem is a consequence of expressions (1) and Theorem 2.

THEOREM 5. Every differential invariant x : Q’ -+ P has a unique expression in the form x = ~0 on, where ~0 : Q’/(K$l x X2’) + P is (LA x Lk)-equivarianr mapping.

COROLLARY. Theorem 5 means that the coordinate functions hij , g,, , fi” , KG on Q’/(K,2,’ x K:‘), form a basis of the first-order differential invariants.

4. Geometrical interpretation of adapted coordinates In this section we consider an injective immersion f : X + Y. Denote by V(X),

respectively V(Y), the modules of smooth vector fields on X, respectively Y. Let

Vh : V(X) x V(X) 3 (u, u) + v;v E V(X),

respectively vg : V(Y) x V(Y) 3 (Q, fi) --f v,g/? E V(Y),

be the covariant derivative on the manifold X resp. Y given by the Levi-Civita connection rjk = (l/Z)h’q(hqj,k + hqk,j - hjk,q)r resp. A& = (1/2)gaa(g,,,, + gafl,v - g”wi).

Recall that any vector field u on X, defines a vector field along f, f*v. Choose a point x E X. There exists a neighbourhood V of y = f(x) and a vector field i7 on V such that G = f*u on V fl f(X). Consider a vector field $ ij on V and its restriction to V n f(X). One can easily verify then this constructron gives us a well-defined vector field along f.

To every x E X we assign a bilinear mapping Kf (x) : T,X x T,X + Tfcsj Y such that

K#)(u(x), u(x)) = f*(V,htW) - (V,gW(x)),

where ii and 5 are any extensions of f*u and f*u on a neighbourhood of f(x). This formula defines a bilinear mapping Kf : V(X) x V(X) + V(Y) along f .

312 t? MLJSILOVA and D. KRUPKA

Let (U, q~p), q = (xi>, respectively (V, *), @ = (y”), Let f(u) c V. Denote u = uja/axj, u = vka/axk. We nents. We have for every x E U

be a chart on X resp. Y. express K,f (x) in compo-

Kf(X)(U(X), v(x)) = K~(n)u’(x)uj(n) ( a \ - aY" 1fC.r).

THEOREM 6. The components of Kf are

K_S = rjl& - A;, fj”f; - 4;.

RJZMARK. The bilinear field Kf along f can also be interpreted within the theory of connections on vector bundles and their pull-back bundles, see e.g. [5].

Recall that a mapping f : X + Y is said to be u&e if for every geodesic a! : (-8, E) + X the curve f o Q : (--E, E) + Y is a geodesic. Using this definition, we get by a direct computation:

THEOREM 7. The following conditions are equivalent: 1. K;=O. 2. The mapping f is ajine.

REMARK. In the special case of Riemannian metric g on Y and induced metric h = f*g on X, the mapping Kf corresponds to the second fundamental form (associated with different normal vector fields to f(x)).

5. Natural Lagrangian structures In this section we show that the basis of the first-order differential invariants,

given by Theorem 5, can be used for a description of all possible, first-order generally covariant Lagrangians depending on immersions of manifolds with metric fields. To this purpose we have to extend the standard theory of natural Lagrange structures, see [9], to fibrations over pairs of manifolds.

In general, one should consider for a Lagrangian a differential invariant .C : Q + I!$ where R is the real line endowed with the action (A, t) + 1 det Al-’ . t of GL, (R). However, if Q includes a metric jleld, then the corresponding metric volume element allows us to consider the functions L instead of C, that is, the Lagrange functions instead of odd base form on underlying jet manifolds. For the categorical aspects of the theory of natural Lagrange structures, and for the theory of odd base forms the reader is refered to [9] and [7]. Thus, for our purposes it is sufficient to recall a simplified version of the definition, which can be used on manifolds with metric fields.

We define a natural Lagrange structure to be a pair (Q, L), where Q is a left LL-manifold, and L : Q + R is an LL-invariant function (i.e. a real-valued differential invariant).

Using this definition, we can paraphrase Theorem 5 as follows.

DIFFERENTIAL INVARIANTS OF IMMERSIONS OF MANIFOLDS WlTH METRIC FIELDS 3 13

THEOREM 8. Every generally covariant first-order Lagrange function depending on an immersion f : X + Y and metric fields on X and Y, respectively, is a function of coordinates hij, g,,, Jr, K,$ only.

In particular, for the induced metrics we have the following result:

REMARK. Let f : X + Y be an injective immersion, let g be a Riemannian metric on Y and let h = f*g be th e induced metric on X. Then every first-order invariant Lagrange function depends on the metric tensor components, the Jacobian matrix (f”), and the second fundamental form components only.

In accordance with the results of the classical theory of invariants [2, 131, every algebraic invariant function on spaces endowed with a tensor action can by constructed by operations of the tensor product, alternation and contraction. The typical example of a zeroth order Lagrange function is the energy density, mentioned in the introduction, and used e.g. in the string theory. An example of the first-order Lagrange function is given by L1 = h’jh”‘K& K&g,,; this Lagrangian usually serves as the geometric part of a higher order Lagrangian in the string theory, see e.g. [3].

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