An Introduction to the Geometry of Homogeneous Spaces

Home , Homogeneous space

Proceedings of The Thirteenth International Workshop on Diﬀ. Geom. 13(2009) 121-144 An introduction to the geometry of homogeneous spaces

Takashi Koda Department of Mathematics, Faculty of Science, University of Toyama, Gofuku, Toyama 930-8555, Japan e-mail : [email protected]

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 2 M(n, R) =∼ Rn . The subset GL(n, R) = {A ∈ M(n, R) | det A 6= 0} is an open submanifold of M(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 n(n − 1) O(n) = {A ∈ GL(n, R) | tAA = A tA = I }, dim = n 2 n(n − 1) SO(n) = {A ∈ O(n) | det A = 1}, dim = 2 ∗ ∗ 2 U(n) = {A ∈ GL(n, C) | A A = A A = In}, dim = n SU(n) = {A ∈ U(n) | det A = 1}, dim = n2 − 1 ∗ ∗ 2 Sp(n) = {A ∈ GL(n, H) | AA = A A = In}, dim = 2n + 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 A is 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 a diﬀeomorphism of G. −1 La = 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

−1 ia := La ◦ Ra−1 , ia(x) = axa (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 − YX 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) | trace X = 0} ⊂ M(n, R) o(n) = {X ∈ M(n, R) | X + tX = 0} ⊂ M(n, R) so(n) = {X ∈ o(n) | trace X = 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 X0,Y 0 on N satisfy 0 0 ϕ∗X = X , ϕ∗Y = Y , then we have 0 0 ϕ∗[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).

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

i x = (xj),

i then xj are global coordinates of GL(n, R). A left invariant vector ﬁeld X ∈ gl(n, R) on GL(n, R) may be written as

Xn ∂ X = Xi . j ∂xi i,j=1 j

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

Xn ∂ gl(n, R) 3 X = Xi ←→ (Xi(e)) ∈ M(n, R) j ∂xi j i,j=1 j 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 µ ¶ 0 d ϕ (0) = ϕ∗0 ∈ TeG dt 0 This is a one-to-one correspondense. For each left invariant vector ﬁeld X ∈ g on G, we denote by ϕX (t) the 1- 0 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 X2 Xn eX = I + X + + ··· + + ··· 2! n!

2 converges in M(n, R) =∼ Rn . 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 exp O = I, exp(−X) = (exp X)−1. Hence exp : 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. −1 ia = La ◦ Ra−1 : G → G, x 7→ axa 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(exp X). 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 ] satisﬁes ad(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, X∞ 1 Ad(exp X)Y = ead(X)Y = ad(X)kY. k! k=0 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

Example sl(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 isomorphic to S1 × · · · × S1 is called a toral subgroup of G, or simply a torus. | {z } k

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 0 is any other torus with T ⊂ T 0 ⊂ G =⇒ T 0 = 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 = xT x−1. x∈G (3) Any other maximal torus T 0 of G is conjugate to T , i.e.,

∃a ∈ G s.t. T 0 = aT a−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).   ¯   eiθ1 ¯   0 ¯   .  ¯ T :=  ..  ¯ θ , ··· , θ ∈ R =∼ S1 × · · · × S1   ¯ 1 n  | {z }  ¯  0 eiθn ¯ 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),

  eiθ1  0   .  P −1AP =  ..  =: diag[eiθ1 , ··· , eiθn ]   0 eiθn

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

µ ¶ cos θ sin θ R = θ − sin θ cos θ and if n = 2m put

  ¯  ¯  Rθ1 ¯   0  ¯   ..  ¯ ∼ 1 1 T =  .  ¯ θ1, ··· , θm ∈ R = S × · · · × S  ¯  | {z }  ¯  m 0 Rθm and if n = 2m + 1 we put

  ¯  ¯  Rθ1 ¯   0  ¯   ..  ¯  . ∼ 1 1 T =   ¯ θ1, ··· , θm ∈ R = S × · · · × S  R  ¯  | {z }  θm  ¯   ¯  m 0 1 ¯ 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),   1    0   ..   .       1       −1     ..   .      t  −1  QAQ =        cos θ1 sin θ1       − sin θ1 cos θ1       ..   .       cos θ sin θ   0 r r  − sin θr cos θr

Remark. µ ¶ µ ¶ 1 0 −1 0 R = = I ,R = = −I , 0 0 1 2 π 0 −1 2

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 product on 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 where g0 = {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 choose an Ad(S)-invariant inner product on g. Then the representation (Ad, g) of S is C completely reducible. If we consider the complexiﬁcation g = g ⊗R C and the representation (Ad, gC) of S, the eigenvalues of Ad(s) are complex numbers with C absolute value 1, and corresponding eigenspaces in g are of dimC = 1. Then g may be decomposed into irreducible subspaces.

g = V0 ⊕ V1 ⊕ · · · ⊕ Vm where

Ad(s)|V0 = id on V0 and for some linear form θk : s → R µ ¶ cos θk(v) − sin θk(v) Ad(exp v)|Vk = on Vk for v ∈ s sin θk(v) cos θk(v) w.r.t. an orthonormal basis on Vk. These θk are unique up to sign and order, and do 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 1 ± θ 2π k 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.

Xn ix1 ixn T = {diag[e , ··· , e ] | xk ∈ R, xk = 0} k=1

The Lie algebra t of T is

Xn t = {diag[ix1, ··· , ixn] | xk ∈ R, xk = 0} k=1 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 1 (X,Y ) := − tr (XY ) 2 and for 1 ≤ j < k ≤ n, 1 ≤ ` ≤ n, put            1   i  √ √   0     Ujk =   ,Ujk =   , 2E` = 2 i  −1   i  which form an orthonormal basis of g. Then we may see that for any v = diag[ix1, ··· , ixn] ∈ t µ ¶ cos (x − x ) − sin (x − x ) Ad(exp v) = j k j k sin (xj − xk) cos (xj − xk)

0 on spanR{Ujk,Ujk}. 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  X`  1 k−1 i v = ··· = v = 0 v = v λi > 0 ⇐⇒  k k=1 v > 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 ` = rank G).

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 .

C C g := g ⊗R C, t := t ⊗R C We 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 for any 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

iθ1 iθn T = {diag [e , ··· , e ] | θ1 + ··· + θn = 0} su(n) = {X ∈ gl(n, C) | X + X∗ = O, tr X = 0} A basis of g is given by            1   i    0     Ujk =   ,Ujk =   ,E` = i .  −1   i  −i

Then gC is spanned (over R) by An introduction to the geometry of homogeneous spaces 133

0 Ujk, Ujk, E`, 0 iUjk, iUjk, iE`. Hence gC = sl(n, C) and

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

C For v = diag[λ1, ··· , λn] ∈ t , if we put      1    Ejk =     then we have ad(v)Ejk = [v, Ejk] = (λj − λk)Ejk. C Let xj : t → C be a linear form deﬁned by

xj(diag[λ1, ··· , λn]) = λj, then we have ad(v)Ejk = [v, Ejk] = (xj − xk)(v)Ejk. C ∗ Thus if we put α = xj − xk ∈ (t ) ,

C ad(v)X = α(v)X for X ∈ gα = CEjk. The root system of g = su(n) relative to t is

∆ = {±(xj − xk) | 1 ≤ j < k ≤ n}, and we have the decomposition X C C C g = t + , gα, α∈∆ where C C C gα = {X ∈ g | ad(v)X = α(v)X for v ∈ t } C is the root space corresponding to a root α.(dimC gα = 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 put αj = xj − xj+1 (1 ≤ j ≤ n − 1 = `) + then they are the simple roots and α ∈ ∆ may be written as a Z≥0-coeﬃcients linear combination of simple roots. The restriction to h of the Killing form B of gC is nondegenerate, and we denote it 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 hR 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) = p − q ∈ Z. (α, α)

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

2(β, α) ≤ 0. (α, α)

Therefore for simple roots αj, αk

2(α , α ) j k ≤ 0. (αj, αj) An introduction to the geometry of homogeneous spaces 135

On the other hand 2(α , α ) 2(α , α ) j k j k = 4 cos2 θ, (αj, αj) (αk, αk) where θ is the angle of αj and αk. 1 1 3 cos2 θ = 0, , , (or 1). 4 2 4 Since π/2 ≤ θ < π, we have four cases. π (1) θ = , 2 2π (2) θ = , kα k = kα k 3 j k 3π √ √ (3) θ = , kα k : kα k = 1 : 2 or 2 : 1 4 j k 5π √ √ (4) θ = , kα k : kα k = 1 : 3 or 3 : 1 6 j k 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). The highest root µ (that is, µ is a positive root such that µ ≥ α for any α ∈ ∆) can be 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 transitively 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 map τg : 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 subgroup H. A Riemannian metric h , i on M is G-invariant if

h(τg)∗X, (τg)∗Y i = hX,Y i for ∀g ∈ G for any X,Y ∈ TM. Then τg is an isometry of the Riemannian manifold (M, h , i).

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 complement m := 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/ ker π∗e = g/h = m

For any XgH ,YgH ∈ TgH M

hXgH ,YgH igH = h(τg−1 )∗gH XgH , (τg−1 )∗gH YgH ieH

= ((τg−1 )∗gH XgH , (τg−1 )∗gH YgH ).

Let G/H be a reductive homogeneous space,

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

For any h ∈ H π(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 h , i on M = G/H induces an inner product ( , ) on m. (X,Y ) = hX,Y io Then ( , ) 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:

h , i : G-inv. Riem. metric on G/H l ( , ) : Ad(H)-inv. inner product on m

In the sequel, we use the same notation h , i for an Ad(H)-inv. inner product on m.

Deﬁnition 16. A reductive homogeneous space M = G/H with a Riemannian metric h , i is naturally reductive if there exists an Ad(H)-invariant decomposition

g = h + m (direct sum), Ad(H)m ⊂ m such that

h[X,Y ]m,Zi + hY, [X,Z]mi = 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 h , i 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

h , i = ( , )|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 (then G 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 ¯ ¯ ∗ 0 d ¯ XgH := γ (0) = ¯ τexp tX (gH) dt t=0 satisfies at the origin o = eH ¯ ¯ ∗ d ¯ Xo = ¯ π(exp tX) = π∗e(X). dt t=0 ∼ If G/H is reductive, the isomorphism π∗e : m = ToM may be expressed as

∗ m 3 X ↔ Xo ∈ ToM.

Proposition 2.1. For X,Y ∈ g,

[X∗,Y ∗] = −[X,Y ]∗.

Let G/H be a reductive homogeneous space with a G-invariant Riemannian metric h , i. We denote by ∇ the Riemannian connection of (M/H, h , i). 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∗hY,Zi = h[X∗,Y ],Zi + hY, [X∗,Z]i for any vector ﬁelds Y,Z ∈ X(G/H).

Since ∗ X hY,Zi = h∇X∗ Y,Zi + hY, ∇X∗ Zi, we have ∗ ∗ h∇Y X ,Zi + hY, ∇Z X i = 0.

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

Then   ∗ [X, v] if X ∈ h ∇vX =  1 2 [X, v]m if X ∈ m ∼ for v ∈ m = ToM.

Deﬁnition 18. Let (G/H, h , i) 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 ½ ∗ [X, v] if X ∈ h DvX = [X, v]m if X ∈ m and 1 D Y = ∇ Y + T (X,Y ), X X 2 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, h , i) be a naturally reductive homogeneous space. Then the Riemannian curvature tensor R is given by 1 (R(X∗,Y ∗)Z∗) = −[[X,Y ] ,Z] − [[X,Y ] ,Z] o h 2 m m 1 1 − [[Y,Z] ,X] − [[Z,X] ,Y ] 4 m m 4 m m for X,Y,Z ∈ m. Then we have 1 (R(X∗,Y ∗)Y ∗) = −[[X,Y ] ,Y ] − [[X,Y ] ,Y ] o h 4 m m and the sectional curvature is given by 1 hR(X∗,Y ∗)Y ∗,X∗i = h[X,Y ] , [X,Y ] i + h[[X,Y ] ,X] ,Y i. o 4 m m h m If (G/H, h , i) is normal homogeneous 1 hR(X∗,Y ∗)Y ∗,X∗i = h[X,Y ] , [X,Y ] i + h[X,Y ] , [X,Y ] i ≥ 0. o 4 m m h h

2.5 Symmetric spaces

Deﬁnition 19. A connected Riemannian manifold (M, g) is called a Riemannian (globally) symmetric space if

2 ∀p ∈ M, ∃sp ∈ I(M, g) s.t. sp = 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.

0 γv(0) = p, γv (0) = v ∈ TpM.

The connected Lie group G = I0(M, g) (identity component) acts transitively on M, and M becomes a Riemannian homogeneous space G/H, where a closed subgroup 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

σ σ G0 ⊂ H ⊂ G , An introduction to the geometry of homogeneous spaces 141 where

σ σ σ G = {a ∈ G | σ(a) = a},G0 is the identity component of G .

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

Examples

M = G/H G H Sn SO(n + 1) SO(n) sphere n Grk(R ) O(n) O(k) × O(n − k) real Grassmann mfd. n Grk(C ) U(n) U(k) × U(n − k) complex Grassmann mfd.

The Lie algebra g may be decomposed into ±1-engenspaces of the diﬀerential dσ : 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 the Ad(H)-invariant decomposition

g = h ⊕ m.

We take a maximal abelian subspace a ⊂ m. For any H ∈ a, we consider the linear maps ad(H): g → g, which are simultaneously diagonalizable. If there exists X ∈ g such 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 relative to a. The set Σ of all roots of M relative to a is called a restricted root system of M. Σ can be obtained as follows: take a Cartan subalgebra hC of gC such that hC ⊃ 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 homogeneous manifold? W. Ambrose and I. M. Singer characterized it by the existense of some tensor ﬁeld 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) ∇X R = T (X) · R

(3) ∇X T = T (X) · T where ∇ 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, ∇˜ R˜ = 0, ∇˜ T˜ = 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 almost complex structure; J 2 = −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) ∇X R = T (X) · R

(3) ∇X T = T (X) · T

(4) ∇X J = 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 with an 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 metric satisfying 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 manifold 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) ∇X R = T (X) · R

(3) ∇X T = T (X) · T

(4) ∇X η = −η ◦ T (X)

(5) ∇X φ = T (X) · φ

References [1] W. Ambrose and I. M. Singer, On homogeneous Riemannian manifolds, Duke Math. 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. I and II, Interscience, New York, 1963 and 1969. [5] T. Koda and Y. Watanabe, Homogeneous almost contact Riemannian manifolds 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, Hokkaido Math. J., 7 (1978), 206–213. 144 Takashi Koda

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