Proceedings of The Thirteenth International

Workshop on Diff. Geom. 13(2009) 121-144

An introduction to the geometry of homogeneous spaces

Takashi KodaDepartment of Mathematics, Faculty of Science, University of Toyama, Gofuku,Toyama 930-8555, Japane-mail : koda@sci.u-toyama.ac.jp

Abstract. This article is a survey of the geometry of homogeneous spaces for students. In

particular, we are concerned with classical Lie groups and their root systems. Then we

describe curvature tensors of the Riemannian connection and the canonical connection of

a homogeneous space G/H in terms of the Lie algebra g.

1 Lie groups

1.1 Definition and examples

Definition 1. A set G is called a Lie group if

(1) G is an (abstract) group,

(2) G is a C∞ manifold,

(3) the group operations

G×G → G, (x, y) 7→ xy,

G → G, x 7→ x−1

are C∞ maps.

Example 1 Rn is an (abelian) Lie group under vector addition:

(x, y) 7→ x + y, x 7→ −x.

Example 2 (Classical Lie groups) Let M(n,R) be the set of all real n×n matrices.We may identify it as

M(n,R) ∼= Rn2.

The subset GL(n,R) = {A ∈ M(n,R) | det A 6= 0} is an open submanifold ofM(n,R), and becomes a Lie group under the multiplication of matrices. GL(n,R)is called the general linear group.

Definition 2. Let G be a Lie group. A Lie group H is a Lie subgroup of G if

(1) H is a submanifold of the manifold G (H need not have the relative topology),

(2) H is a subgroup of the (abstract) group G.

121

122 Takashi Koda

Example 3

SL(n,R) = {A ∈ GL(n,R) | det A = 1}, dim = n2 − 1

O(n) = {A ∈ GL(n,R) | tAA = A tA = In}, dim =n(n− 1)

2

SO(n) = {A ∈ O(n) | det A = 1}, dim =n(n− 1)

2U(n) = {A ∈ GL(n,C) | A∗A = A∗A = In}, dim = n2

SU(n) = {A ∈ U(n) | detA = 1}, dim = n2 − 1Sp(n) = {A ∈ GL(n,H) | AA∗ = A∗A = In}, dim = 2n2 + n

where, H = {a + bi + cj + dk | a, b, c, d ∈ R} denotes the set of all quaternions(i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j), and A∗ = tA =t Ais the conjugate transpose of a matrix A.

Theorem 1.1. (Cartan) Let H be a closed subgroup of a Lie group G. Then H

has a structure of Lie subgroup of G. (The topology of H coincides with the relative

topology and the structure of Lie subgroup is unique).

1.2 Left translation

For any fixed a ∈ G, the map La : G → G defined by

La(x) = ax for x ∈ G

is called the left translation. La is a C∞ map on G, and furthermore it is adiffeomorphism of G.

La−1 = La−1 .

Similarly, the right translation Ra by a ∈ G is a diffeomorphism defined by

Ra(x) = xa for x ∈ G.

The inner automorphism ia : G → G

ia := La ◦Ra−1 , ia(x) = axa−1 (x ∈ G)

is C∞ map on G.

Definition 3. Let G be a Lie group. A vector field X on G is left invariant if

(La)∗X = X,

where

((La)∗X)f := X(f ◦ La) (f ∈ C∞(G)).

An introduction to the geometry of homogeneous spaces 123

More precisely,

(La)∗xXx = Xax for x ∈ G

((La)∗xXx)f = Xx(f ◦ La) for f ∈ C∞(G)

We denote by g the set of all left invariant vector fields on G.

Proposition 1.2. (1) For X, Y ∈ g and λ, µ ∈ R, we have λX + µY ∈ g.

(2) For X,Y ∈ g, we have [X, Y ] ∈ g.

Remark. For vector fields X,Y on a manifold, bracket [X, Y ] is defined by

[X, Y ]f := X(Y f)− Y (Xf) (f : C∞ function).

1.3 Lie algebras

Definition 4. An n-dimensional vector space g over a field K = R or C with a

bracket [ , ] : g× g → g is called a Lie algebra if it satisfies

(1) [ , ] is bilinear,

(2) [X, X] = 0 for ∀X ∈ g and

(3) (Jacobi identity) [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0 for ∀X,Y, Z ∈ g.

Example M(n,R), M(n,C) and M(n,H) with a commutator [ , ] defined by

[X, Y ] = XY − Y X

are Lie algebras over R.

Definition 5. A vector subspace h of a Lie algebra g is called a Lie subalgebra if

[h, h] ⊂ h.

Example

sl(n,R) = {X ∈ M(n,R) | traceX = 0} ⊂ M(n,R)o(n) = {X ∈ M(n,R) | X + tX = 0} ⊂ M(n,R)

so(n) = {X ∈ o(n) | traceX = 0} ⊂ o(n)u(n) = {X ∈ M(n,C) | X + X∗ = O} ⊂ M(n,C)

su(n) = {X ∈ u(n) | trace X = 0} ⊂ u(n)

124 Takashi Koda

1.4 Lie algebra of a Lie group

Let X, Y be left invariant vector fields on a Lie group G, then

(La)∗X = X, (La)∗Y = Y for a ∈ G,

hence we get(La)∗[X, Y ] = [X, Y ].

In general, if ϕ : M → N is a C∞ map and vector fields X, Y on M and X ′, Y ′ onN satisfy

ϕ∗X = X ′, ϕ∗Y = Y ′,

then we haveϕ∗[X, Y ] = [X ′, Y ′].

Proposition 1.3. The set g of all left invariant vector fields on a Lie group G is

a Lie algebra with respect to the Lie bracket [X,Y ] of vector fields X,Y on G.

g is called the Lie algebra of a Lie group G.

Proposition 1.4. The Lie algebra g of a Lie group G may be identified with the

tangent space TeG at the identity e ∈ G.

g 3 X 7→ Xe ∈ TeG (isomorphism between vector spaces)

and conversely

TeG 3 Xe 7→ X ∈ g defined by Xx := (Lx)∗eXe (x ∈ G).

Example An element x in GL(n,R) ⊂ M(n,R) ∼= Rn2may be expressed as

x = (xij),

then xij are global coordinates of GL(n,R). A left invariant vector field X ∈ gl(n,R)

on GL(n,R) may be written as

X =n∑

i,j=1

Xij

∂

∂xij

.

Then Xe ∈ TeGL(n,R) may be identified with the n×n matrix (Xij(e)) ∈ M(n,R).

gl(n,R) 3 X =n∑

i,j=1

Xij

∂

∂xij

←→ (Xij(e)) ∈ M(n,R)

Then we may see that the Lie algebra gl(n,R) of GL(n,R) is isomorphic to M(n,R)as Lie algebras:

(gl(n,R), [ , ]) ∼= (M(n,R), [ , ])

An introduction to the geometry of homogeneous spaces 125

Theorem 1.5. Let G be a Lie group and g its Lie algebra.

(1) If H is a Lie subgroup of G, h is a Lie subalgebra of g

(2) If h is a Lie subalgebra, there exists a unique Lie subgroup H of G such that the

Lie algebra of H is isomorphic to h.

1.5 Exponential maps

Definition 6. A homomorphism ϕ : R→ G is called a 1-parameter subgroup of

a Lie group G. ϕ(t) is a curve in G such that

ϕ(0) = e, ϕ(s + t) = ϕ(s)ϕ(t).

For any 1-parameter subgroup ϕ : R→ G, we consider its initial tangent vector

ϕ′(0) = ϕ∗0

(d

dt

)

0

∈ TeG

This is a one-to-one correspondense.For each left invariant vector field X ∈ g on G, we denote by ϕX(t) the 1-

parameter subgroup with the initial tangent vector ϕX′(0) = Xe ∈ TeG.

Definition 7. The map exp : g → G defined by

exp X := ϕX(1)

is called the exponential map of a Lie group G.

Example For each X ∈ M(n,R), the series of matrix

eX = I + X +X2

2!+ · · ·+ Xn

n!+ · · ·

converges in M(n,R) ∼= Rn2. The map

exp : M(n,R) → M(n,R), X 7→ exp X = eX

is called the exponential map of matrices.If [X,Y ] = 0,

exp(X + Y ) = exp X exp Y

and we may see that

expO = I, exp(−X) = (exp X)−1.

Henceexp : M(n,R) → GL(n,R).

This coincides with the exponential map of the Lie group GL(n,R).

126 Takashi Koda

1.6 Adjoint representations

Let G be a Lie group and ia denotes the inner automorphism by a ∈ G.

ia = La ◦Ra−1 : G → G, x 7→ axa−1

We denote its differential by

Ad(a) = AdG(a) := (ia)∗.

For any left invariant vector field X ∈ g, Ad(a)X = (ia)∗X is also left invariant.

Ad(a) : g → g (linear)

satisfies[Ad(a)X, Ad(a)Y ] = Ad(a)[X, Y ].

Thus we have a homomorphism

Ad : G → GL(g), a 7→ Ad(a)

which called the adjoint represenation of G.

Proposition 1.6. For any a ∈ G and X ∈ g,

exp(Ad(a)X) = ia(expX).

Let g be a Lie algebra with a bracket [ , ]. For any X, the linear map

ad(X) : g → g, Y 7→ ad(X)Y := [X,Y ]

satisfiesad(X)[Y, Z] = [ad(X)Y, Z] + [Y, ad(X)Z]

(Jacobi identity). The homomorphism

ad : g → gl(g), X 7→ ad(X)

is called the adjoint representation of g.

Proposition 1.7. For any X,Y ∈ g,

Ad(expX)Y = ead(X)Y =∞∑

k=0

1k!

ad(X)kY.

Definition 8. Let g be a Lie algebra. The bilinear form

B(X, Y ) := tr(ad(X) ◦ ad(Y )) (X,Y ∈ g)

is called the Killing form of g, where the right-hand-side means the trace of the

linear map

ad(X) ◦ ad(Y ) : g → g.

An introduction to the geometry of homogeneous spaces 127

Examplesl(n,R) B(X, Y ) = 2n tr(XY )o(n) B(X, Y ) = (n− 2) tr(XY )su(n) B(X, Y ) = 2n tr(XY )sp(n) B(X, Y ) = (2n + 2) tr(XY )

1.7 Toral subgroups and maximal tori

Definition 9. Let G be a connected Lie group. A subgroup S of G which is iso-

morphic to S1 × · · · × S1

︸ ︷︷ ︸k

is called a toral subgroup of G, or simply a torus.

A toral subgroup S is an arcwise-connected compact abelian Lie group.

Definition 10. A torus T is a maximal torus of G if

T ′ is any other torus with T ⊂ T ′ ⊂ G =⇒ T ′ = T

Theorem 1.8. Any torus S of G is contained in a maximal torus T .

Theorem 1.9. (Cartan–Weil–Hopf) Let G be a compact connected Lie group.

Then

(1) G has a maximal torus T .

(2) G =⋃

x∈G

xTx−1.

(3) Any other maximal torus T ′ of G is conjugate to T , i.e.,

∃a ∈ G s.t. T ′ = aTa−1

(4) Any maximal torus of G is a maximal abelian subgroup of G.

The dimension ` = dim T of T is called the rank of G.Example G = U(n).

T :=

eiθ1 0. . .

0 eiθn

∣∣∣∣∣∣∣∣θ1, · · · , θn ∈ R

∼= S1 × · · · × S1

︸ ︷︷ ︸n

is a toral subgroup of U(n).

128 Takashi Koda

Proposition 1.10. Any unitary matrix A ∈ U(n) can be diagonalized by a unitary

matrix P ∈ U(n),

P−1AP =

eiθ1 0. . .

0 eiθn

=: diag[eiθ1 , · · · , eiθn ]

Example G = SO(n). We denote by Rθ the rotation matrix

Rθ =(

cos θ sin θ− sin θ cos θ

)

and if n = 2m put

T =

Rθ1 0. . .

0 Rθm

∣∣∣∣∣∣∣∣θ1, · · · , θm ∈ R

∼= S1 × · · · × S1

︸ ︷︷ ︸m

and if n = 2m + 1 we put

T =

Rθ1 0. . .

Rθm

0 1

∣∣∣∣∣∣∣∣∣∣

θ1, · · · , θm ∈ R

∼= S1 × · · · × S1

︸ ︷︷ ︸m

then T is a maximal torus of SO(n).

Proposition 1.11. Any orthogonal matrix A ∈ O(n) may be put into the following

An introduction to the geometry of homogeneous spaces 129

form by an orthogonal matrix Q ∈ O(n),

tQAQ =

1 0. . .

1

−1. . .

−1

cos θ1 sin θ1

− sin θ1 cos θ1

. . .

0 cos θr sin θr

− sin θr cos θr

Remark.

R0 =(

1 00 1

)= I2, Rπ =

( −1 00 −1

)= −I2,

1.8 Root systems

Let G be a compact connected Lie group.Let (ρ, V ) be any representation of G.

ρ : G → GL(V ) (homomorphism)

(V is a vector space over R or C). We may choose a ρ(G)-invariant inner producton V . For any ρ(G)-invariant subspace W ⊂ V , we have

V = W ⊕W⊥ (ρ(G)-invariant decomposition)

Therefore, (ρ, V ) is completely reducible.

(ρ, V ) = (ρ1, V1)⊕ · · · ⊕ (ρk, Vk)

where (ρi, Vi) is irreducible.In particular, the adjoint representation (Ad, g) of G is completely reducible,

and we have a decomposition

g = g0 ⊕ g1 ⊕ · · · ⊕ gm

whereg0 = {X ∈ g | ad(Y )X = 0 for ∀Y ∈ g} (center of g)

and gi is an ideal of g (i.e., [g, gi] ⊂ gi) and gi is irreducible.

130 Takashi Koda

Proposition 1.12. Let A, B be two diagonalizable matrices. If [A,B] = 0, then A

and B are simultaneously diagonalizable, i.e.,

∃x 6= 0 s.t. Ax = λx, Bx = µx (λ, µ : eigenvalues)

Let G be a compact connected Lie group, and S a torus of G. We choosean Ad(S)-invariant inner product on g. Then the representation (Ad, g) of S iscompletely reducible. If we consider the complexification gC = g ⊗R C and therepresentation (Ad, gC) of S, the eigenvalues of Ad(s) are complex numbers withabsolute value 1, and corresponding eigenspaces in gC are of dimC = 1. Then g maybe decomposed into irreducible subspaces.

g = V0 ⊕ V1 ⊕ · · · ⊕ Vm

whereAd(s)|V0 = id on V0

and for some linear form θk : s → R

Ad(exp v)|Vk=

(cos θk(v) − sin θk(v)sin θk(v) cos θk(v)

)on Vk for v ∈ s

w.r.t. an orthonormal basis on Vk. These θk are unique up to sign and order, anddo not depent on the inner product and an orthonormal basis.

Definition 11. Linear forms ±θk (k = 1, · · · ,m) are called roots of G relative to

S. If S is a maximal torus T of G, they are called simply the roots of G. The set

∆ of all roots of G is called the root system of G.

Remark. Sometimes we define roots as

± 12π

θk

so as to take Z-values on the unit lattice exp−1(e) ⊂ t.

Example G = SU(n) and T is the maximal torus defined as before.

T = {diag[eix1 , · · · , eixn ] | xk ∈ R,

n∑

k=1

xk = 0}

The Lie algebra t of T is

t = {diag[ix1, · · · , ixn] | xk ∈ R,

n∑

k=1

xk = 0}

An introduction to the geometry of homogeneous spaces 131

x1, · · · , xn ∈ t∗ are deined in natural way.

linear form diag[ix1, · · · , ixn] 7→ xk is denoted by xk

We take the inner product ( , ) on g defined by

(X, Y ) := −12tr (XY )

and for 1 ≤ j < k ≤ n, 1 ≤ ` ≤ n, put

Ujk =

1

−1

, U ′jk =

i

i

,√

2E` =√

2

i

which form an orthonormal basis of g. Then we may see that for any v =diag[ix1, · · · , ixn] ∈ t

Ad(exp v) =(

cos (xj − xk) − sin (xj − xk)sin (xj − xk) cos (xj − xk)

)

on spanR{Ujk, U ′jk}.

Thus the root system ∆ of SU(n) is

∆ = {±(xj − xk) | 1 ≤ j < k ≤ n}

Definition 12. Let {λ1, · · · , λ`} be a basis of t∗. The lexicographic order λ1 >

λ2 > · · · > λ` > 0 is defined by

v =∑̀

k=1

viλi > 0 ⇐⇒

v1 = · · · = vk−1 = 0

vk > 0 for some k

A root α ∈ ∆ is positive (resp. negative) if α > 0 (resp. α < 0) relative to the

lexicographic order w.r.t. the basis.

∆ = ∆+ ∪∆−

A positive root α is called a simple root if it can not be expressed as a sum of two

positive roots. The set {α1, · · · , α`} of all simple roots of G is called the simple

root system of G (where ` = rankG).

Example G = SU(n) and T a maximal torus of all diagonal matrices. {x1, · · · , xn−1}is a basis of t∗. Consider the lexicographic order x1 > x2 > · · · > xn−1 > 0. Then

132 Takashi Koda

∆+ = {xj − xk | 1 ≤ j < k ≤ n} positive root system∆− = {−(xj − xk) | 1 ≤ j < k ≤ n} negative root system

and the simple root system is

{αj = xj − xj+1 | 1 ≤ j ≤ n− 1}

1.9 Root systems – again –

Here, we shall calculate the root system in terms of Lie algebra.Let G be a compact connected semisimple Lie group, T a maximal torus of G,

g the Lie algebra of G, t the Lie algebra of T .

gC := g⊗R C, tC := t⊗R CWe extend [ , ] complex-linearly on gC, then gC becomes a complex Lie algebra.

tC becomes a maximal abelian Lie subalgebra of gC.The Killing form BC of gC is the natural extension of the Killing form B of g.If we take an Ad(G)-invariant inner product ( , ) on g, we can make an Ad(G)-

invariant Hermitian inner product ( , ) on gC in natural way.

(ad(v)X, Y ) + (X, ad(v)Y ) = 0

for X, Y ∈ gC, v ∈ tC, that is, ad(v) is a skew-Hermitian transformation. Thus forany v ∈ tC, ad(v) is diagonalizable.

Hence, tC is a Cartan subalgebra of gC.

Definition 13. A Lie aubalgebra h is called a Cartan subalgebra of a complex

Lie algebra g if

(1) h is maximal abelian subalgebra,

(2) ad(H) is diagonalizable for any H ∈ h.

1.10 SU(n)

G = SU(n), T is the maximal torus defined by

T = {diag [eiθ1 , · · · , eiθn ] | θ1 + · · ·+ θn = 0}su(n) = {X ∈ gl(n,C) | X + X∗ = O, tr X = 0}A basis of g is given by

Ujk =

1

−1

, U ′jk =

i

i

, E` =

i

−i

.

Then gC is spanned (over R) by

An introduction to the geometry of homogeneous spaces 133

Ujk, U ′jk, E`,

iUjk, iU ′jk, iE`.

Hence gC = sl(n,C) and

t = {diag[iθ1, · · · , iθn] | θk ∈ R, θ1 + · · ·+ θn = 0},tC = {diag[λ1, · · · , λn] | λk ∈ C, λ1 + · · ·+ λn = 0}.

For v = diag[λ1, · · · , λn] ∈ tC, if we put

Ejk =

1

then we havead(v)Ejk = [v,Ejk] = (λj − λk)Ejk.

Let xj : tC → C be a linear form defined by

xj(diag[λ1, · · · , λn]) = λj ,

then we havead(v)Ejk = [v, Ejk] = (xj − xk)(v)Ejk.

Thus if we put α = xj − xk ∈ (tC)∗,

ad(v)X = α(v)X for X ∈ gCα = CEjk.

The root system of g = su(n) relative to t is

∆ = {±(xj − xk) | 1 ≤ j < k ≤ n},

and we have the decomposition

gC = tC +∑

α∈∆

, gCα,

wheregCα = {X ∈ gC | ad(v)X = α(v)X for v ∈ tC}

is the root space corresponding to a root α. (dimC gCα = 1).In usual, we denote by h a Cartan subalgebra, so in our case, h = tC. and then

hR := {v ∈ h | α(v) ∈ R (∀α ∈ ∆)}

is just it.We take a basis {v1, · · · , v`} of hR as

vj := diag[0, · · · , 1, · · · , 0] ∈ it

134 Takashi Koda

and put {λ1, · · · , λ`} its dual basis of hR∗,

λj = xj .

We consider the lexicographic order in hR∗. The root system is a union of the

positive root system and the negative root system.

∆ = ∆+ ∪∆−

If we putαj = xj − xj+1 (1 ≤ j ≤ n− 1 = `)

then they are the simple roots and α ∈ ∆+ may be written as a Z≥0-coefficientslinear combination of simple roots.

The restriction to h of the Killing form B of gC is nondegenerate, and we denoteit by ( , ).

For λ ∈ h∗, we define vλ ∈ h by

(vλ, v) = λ(v) for v ∈ h,

and we define an inner product on h∗ by

(λ, µ) := (vλ, vµ).

We put( , )R := ( , )|hR , ( , )R := ( , )|h∗R ,

which are positive definite. For convenience, we denote these simply by ( , ).For α, β ∈ ∆ (α 6= ±β), we consider the sequence

β − pα, · · · , β − α, β, β + α, · · · , β + qα ∈ ∆

where β − (p + 1)α 6∈ ∆ and β + (q + 1)α 6∈ ∆.

{β + mα | − p ≤ m ≤ q}

is called the α-sequence containing β. We may see that

2(β, α)(α, α)

= p− q ∈ Z.

In particular, if p = 0, i.e., β − α 6∈ ∆,

2(β, α)(α, α)

≤ 0.

Therefore for simple roots αj , αk

2(αj , αk)(αj , αj)

≤ 0.

An introduction to the geometry of homogeneous spaces 135

On the other hand2(αj , αk)(αj , αj)

2(αj , αk)(αk, αk)

= 4 cos2 θ,

where θ is the angle of αj and αk.

cos2 θ = 0,14,12,34

(or 1).

Since π/2 ≤ θ < π, we have four cases.

(1) θ =π

2,

(2) θ =2π

3, ‖αj‖ = ‖αk‖

(3) θ =3π

4, ‖αj‖ : ‖αk‖ = 1 :

√2 or

√2 : 1

(4) θ =5π

6, ‖αj‖ : ‖αk‖ = 1 :

√3 or

√3 : 1

Then we draw a diagram as follows:(1) we do not join αj and αk ◦ ◦(2) join by single line ◦−−◦(3) join by double line ◦==>◦ (arrow from longer root to shorter one)(4) join by triple line ◦≡≡>◦ (arrow from longer root to shorter one)

Thus the Dynkin diagram has vertices {α1, · · · , α`} and edges (1) – (4). Thehighest root µ (that is, µ is a positive root such that µ ≥ α for any α ∈ ∆) canbe written as

µ = m1α1 + · · ·+ m`α`

in the Dynkin diagram, the integers mk are written on each αk.The highest root µ of su(n) is

µ = x1 − xn = α1 + · · ·+ α`, m1 = · · · = m` = 1

2 Geometry of homogeneous spaces

2.1 Homogeneous spaces

Definition 14. A C∞ manifold M is homogeneous if a Lie group G acts transi-

tively on M , i.e.,

(1) There exists a C∞ map : G×M → M, (a, x) 7→ ax such that

(i) a(bx) = (ab)x

(ii) ex = x for any x ∈ M

(2) For any x, y ∈ M , there exists an element a ∈ G such that y = ax

136 Takashi Koda

For an arbitraryly fixed point x0 ∈ M , the closed subgroup

H := {a ∈ G | ax0 = x0}

is called an isotropy subgroup of G at x0.Then M is diffeomorphic to the coset space G/H.

M ∼= G/H, x 7→ aH,

where, a ∈ G is an element of G s.t. x = ax0.Conversely, if H is a closed subgroup of a Lie group G, then the coset space

G/H becomes a C∞ manifold, and G acts on M by

G×G/H → G/H, (g, aH) 7→ (ga)H,

and the mapτg : G/H → G/H, aH 7→ (ga)H

is called a left translation by g.

τg ◦ π = π ◦ Lg

2.2 G-invariant Riemannian metrics

Let M = G/H be a homogeneous space of a Lie group G and a closed subgroupH. A Riemannian metric 〈 , 〉 on M is G-invariant if

〈(τg)∗X, (τg)∗Y 〉 = 〈X,Y 〉 for ∀g ∈ G

for any X,Y ∈ TM . Then τg is an isometry of the Riemannian manifold (M, 〈 , 〉).

Definition 15. A homogeneous space M = G/H is reductive if there exists a

decomposition

g = h + m (direct sum)

such that

Ad(H)m ⊂ m, i.e., Ad(h)m ⊂ m (∀h ∈ H).

If g has an Ad(H)-invariant inner product ( , ), then the orthogonal complementm := h⊥ satisfies the above property.

π : G → M = G/H

is the natural projection, o = eH ∈ G/H.

π∗e : TeG → ToM onto linear

An introduction to the geometry of homogeneous spaces 137

induces an isomorphism

ToM ∼= TeG/ kerπ∗e ∼= g/h ∼= m

For any XgH , YgH ∈ TgHM

〈XgH , YgH〉gH = 〈(τg−1)∗gHXgH , (τg−1)∗gHYgH〉eH

= ((τg−1)∗gHXgH , (τg−1)∗gHYgH).

Let G/H be a reductive homogeneous space,

g = h⊕m (Ad(H)-invariant decomposition).

For any h ∈ Hπ(ih(g)) = τh(π(g)) (g ∈ G).

Hence for any X ∈ m ⊂ g

π∗e(Ad(h)X) = (τh)∗eX (π∗e : m ∼= ToM).

Then a G-invariant Riemannian metric 〈 , 〉 on M = G/H induces an inner product( , ) on m.

(X,Y ) = 〈X,Y 〉oThen ( , ) is Ad(H)-invariant.

(Ad(h)X, Ad(h)Y ) = (X, Y ), for ∀h ∈ H, X, Y ∈ m.

Hence, we have the one-to-one correspondence:

〈 , 〉 : G-inv. Riem. metric on G/Hl

( , ) : Ad(H)-inv. inner product on m

In the sequel, we use the same notation 〈 , 〉 for an Ad(H)-inv. inner product onm.

Definition 16. A reductive homogeneous space M = G/H with a Riemannian met-

ric 〈 , 〉 is naturally reductive if there exists an Ad(H)-invariant decomposition

g = h + m (direct sum), Ad(H)m ⊂ m

such that

〈[X, Y ]m, Z〉+ 〈Y, [X,Z]m〉 = 0

for any X,Y, Z ∈ m, where [X,Y ]m denotes the m-component of [X,Y ].

138 Takashi Koda

Remark. Accoring to the direct sum decomposition g = h⊕m,

[X, Y ] = [X,Y ]h + [X, Y ]m

Definition 17. A homogeneous space M = G/H with a Riemannian metric 〈 , 〉is normal homogeneous if there exists a bi-invariant Riemannian metric ( , ) on

G such that, if we identify

m := h⊥ (with respect to ( , ))

with ToM , then

〈 , 〉 = ( , )|m×m.

Remark. If M = G/H is normal homogeneous, M is naturally reductive.

Example If the Kiling form B of a compact Lie group G is negative definite (thenG is semisimple), using the inner product −B on g, G/H is normal homogeneous.

2.3 Riemannian connection and the canonical connection

Let G/H be a homogeneous space. For X ∈ g,

γ(t) = τexp tX(gH)

is a curve in M through a point gH. The vector field X∗ defined by

X∗gH := γ′(0) =

d

dt

∣∣∣∣t=0

τexp tX(gH)

satisfies at the origin o = eH

X∗o =

d

dt

∣∣∣∣t=0

π(exp tX) = π∗e(X).

If G/H is reductive, the isomorphism π∗e : m ∼= ToM may be expressed as

m 3 X ↔ X∗o ∈ ToM.

Proposition 2.1. For X,Y ∈ g,

[X∗, Y ∗] = −[X,Y ]∗.

Let G/H be a reductive homogeneous space with a G-invariant Riemannianmetric 〈 , 〉. We denote by ∇ the Riemannian connection of (M/H, 〈 , 〉).

An introduction to the geometry of homogeneous spaces 139

Proposition 2.2. For any X ∈ g, the vector field X∗ is a Killing vector field on

G/H, i.e.,

X∗〈Y,Z〉 = 〈[X∗, Y ], Z〉+ 〈Y, [X∗, Z]〉

for any vector fields Y, Z ∈ X(G/H).

Since

X∗〈Y,Z〉 = 〈∇X∗Y, Z〉+ 〈Y,∇X∗Z〉,

we have

〈∇Y X∗, Z〉+ 〈Y,∇ZX∗〉 = 0.

Proposition 2.3. Let (G/H, 〈 , 〉) be a naturally reductive homogeneous space.

Then

∇vX∗ =

[X, v] if X ∈ h

12 [X, v]m if X ∈ m

for v ∈ m ∼= ToM .

Definition 18. Let (G/H, 〈 , 〉) be a naturally reductive homogeneous space. Then

there exists a metric connection D with torsion tensor T and the curvature tensor

B such that

DT = 0, DB = 0.

D is called the canonical connection.

The canonical connection D is given by

DvX∗ ={

[X, v] if X ∈ h[X, v]m if X ∈ m

and

DXY = ∇XY +12T (X, Y ),

T (X, Y ) = −[X,Y ]m,

B(X, Y )Z = −[[X, Y ]h, Z],

for X, Y, Z ∈ m.

140 Takashi Koda

2.4 Curvature tensor

Theorem 2.4. Let (G/H, 〈 , 〉) be a naturally reductive homogeneous space. Then

the Riemannian curvature tensor R is given by

(R(X∗, Y ∗)Z∗)o = −[[X, Y ]h, Z]− 12[[X, Y ]m, Z]m

−14[[Y, Z]m, X]m − 1

4[[Z, X]m, Y ]m

for X, Y, Z ∈ m.

Then we have

(R(X∗, Y ∗)Y ∗)o = −[[X,Y ]h, Y ]− 14[[X,Y ]m, Y ]m

and the sectional curvature is given by

〈R(X∗, Y ∗)Y ∗, X∗〉o =14〈[X, Y ]m, [X, Y ]m〉+ 〈[[X, Y ]h, X]m, Y 〉.

If (G/H, 〈 , 〉) is normal homogeneous

〈R(X∗, Y ∗)Y ∗, X∗〉o =14〈[X,Y ]m, [X,Y ]m〉+ 〈[X,Y ]h, [X,Y ]h〉 ≥ 0.

2.5 Symmetric spaces

Definition 19. A connected Riemannian manifold (M, g) is called a Riemannian

(globally) symmetric space if

∀p ∈ M, ∃sp ∈ I(M, g) s.t. sp2 = id., p is an isolated fixed point of sp

sp is the geodesic symmetry:

sp(γv(t)) = γv(−t),

where γv(t) is the geodesic s.t.

γv(0) = p, γv′(0) = v ∈ TpM.

The connected Lie group G = I0(M, g) (identity component) acts transitivelyon M , and M becomes a Riemannian homogeneous space G/H, where a closedsubgroup H is an isotropy subgroup at p0 ∈ M . The map

σ : G → G, a 7→ sp0 ◦ a ◦ sp0

is an involutive automorphism of G, and

Gσ0 ⊂ H ⊂ Gσ,

An introduction to the geometry of homogeneous spaces 141

where

Gσ = {a ∈ G | σ(a) = a}, Gσ0 is the identity component of Gσ.

Such a pair (G,H) is called a symmetric pair.

Examples

M = G/H G HSn SO(n + 1) SO(n) sphere

Grk(Rn) O(n) O(k)×O(n− k) real Grassmann mfd.Grk(Cn) U(n) U(k)× U(n− k) complex Grassmann mfd.

The Lie algebra g may be decomposed into ±1-engenspaces of the differentialdσ : g → g.

g = h⊕m, dσ|h = id, dσ|m = −id

Then we see that[h, h] ⊂ h, [h, m] ⊂ m, [m, m] ⊂ h

and hence the canonical connection D coincides with the Riemannian connection∇, thus

∇R = 0,

R(X, Y )Z = −[[X, Y ], Z]

for X, Y, Z ∈ m.

2.6 The root system of a symmetric space

Let M = G/H be a Riemannian symmetric space. Then as before, we have theAd(H)-invariant decomposition

g = h⊕m.

We take a maximal abelian subspace a ⊂ m. For any H ∈ a, we consider the linearmaps ad(H) : g → g, which are simultaneously diagonalizable. If there exists X ∈ gsuch that

ad(H)X = α(H)X

for all H ∈ a for some nonzero linear form α ∈ a∗, α is called a root of M relativeto a. The set Σ of all roots of M relative to a is called a restricted root system ofM .

Σ can be obtained as follows: take a Cartan subalgebra hC of gC such thathC ⊃ a, let ∆ be the root system of gC relative to hC. Then Σ coinsides with{α|a | α ∈ ∆} (as well as their multiplicities).

142 Takashi Koda

3 Homogeneous structures

When does a Riemmanian manifold (M, g) become a Riemannian homogeneousmanifold?

W. Ambrose and I. M. Singer characterized it by the existense of some tensorfield T of type (1, 2) on M , which is called a homogeneous structure.

Theorem 3.1. ([1]) Let (M, g) be a connected, simply connected, complete Rie-

mannian manifold. (M, g) is a Riemannian homogeneous space if and only if there

exists a tensor field T of type (1, 2) on M satisfying

(1) g(T (X)Y, Z) + g(Y, T (X)Z) = 0

(2) ∇XR = T (X) ·R(3) ∇XT = T (X) · Twhere ∇ and R denotes the Riemannian connection and the Riemannian curvature

tensor of (M, g), respectively.

∇̃ := ∇− T

is called a A–S connection. (1)–(3) are equivalent to ∇̃g = 0, ∇̃R̃ = 0, ∇̃T̃ = 0.Remark. Here, we consider the tensor field T : X(M)×X(M) → X(M) of type

(1, 2) as, for X ∈ X(M),

T (X) : X(M) → X(M), Y 7→ T (X)Y

and

(T (X) ·R)(Y, Z)W := T (X)(R(Y, Z)W )−R(T (X)Y,Z)W−R(Y, T (X)Z)W −R(Y, Z)T (X)W

(T (X) · T )(Y )Z := T (X)T (Y )Z − T (T (X)Y )Z − T (Y )T (X)Z

About the homogeneity of almost Hermitian manifold (M, g, J), (J is an almostcomplex structure; J2 = −I), the following result was shown by K. Sekigawa.

Theorem 3.2. ([6]) Let (M, g, J) be a connected, simply connected, complete

almost Hermitian manifold. M is a homogeneous almost Hermitian manifold if and

only if there exists a tensor field T of type (1, 2) on M satisfying

(1) g(T (X)Y, Z) + g(Y, T (X)Z) = 0

(2) ∇XR = T (X) ·R(3) ∇XT = T (X) · T(4) ∇XJ = T (X) · J

An introduction to the geometry of homogeneous spaces 143

An almost contact metric manifold (M, φ, ξ, η, g) is a C∞ manifold M withan almost contact metric structure (φ, ξ, η, g) such that(1) φ is a tensor field of type (1, 1),(2) ξ ∈ X(M) is called a characteristic vector field,(3) η is a 1-form,(4) g is a Riemannian metricsatisfying

φ2X = −X + η(X)ξ,η(ξ) = 1,

g(X, ξ) = η(X),g(φX, φY ) = g(X, Y )− η(X)η(Y )

About the homogeneity of (M, φ, ξ, η, g), the following result was shown.

Theorem 3.3. ([5]) Let (M, φ, ξ, η, g) be a connected, simply connected, complete

almost contact metric manifold. M is a homogeneous almost contact metric mani-

fold if and only if there exists a skew-symmetric tensor field T of type (1, 2) on M

satisfying

(1) g(T (X)Y,Z) + g(Y, T (X)Z) = 0

(2) ∇XR = T (X) ·R(3) ∇XT = T (X) · T(4) ∇Xη = −η ◦ T (X)

(5) ∇Xφ = T (X) · φ

References

[1] W. Ambrose and I. M. Singer, On homogeneous Riemannian manifolds, DukeMath. J., 25 (1958), 647–669.

[2] A. Arvanitoyeorgos, An Introduction to Lie Groups and the Geometry of Ho-mogeneous Spaces, Amer. Math. Sci., 2003.

[3] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Aca-demic Press, New York, 1978.

[4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. Iand II, Interscience, New York, 1963 and 1969.

[5] T. Koda and Y. Watanabe, Homogeneous almost contact Riemannian mani-folds and infinitesimal models, Bollettino U. M. I. (7) 11-B (1997), Suppl. fasc.2, 11–24.

[6] K. Sekigawa, Notes on homogeneous almost Hermitian manifolds, HokkaidoMath. J., 7 (1978), 206–213.

144 Takashi Koda

[7] W. Ziller, The Jaacobi equation on naturally reductive compact Riemannianhomogeneous spaces, Comment. Math. Helvetici, 52(1977), 573–590.