An introduction to the geometry of homogeneous spaces
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1 Proceedings of The Thirteenth International Workshop on Diff. Geom. 13(2009) An introduction to the geometry of homogeneous spaces Takashi Koda Department of Mathematics, Faculty of Science, University of Toyama, Gofuku, Toyama , Japan 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, G G, (x, y) xy, x x 1 are C maps. Example 1 R n is an (abelian) Lie group under vector addition: (x, y) x + y, x 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) = R n2. The subset GL(n, R) = {A M(n, R) det A 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
2 122 Takashi Koda Example 3 SL(n, R) = {A GL(n, R) det A = 1}, dim = n 2 1 O(n) = {A GL(n, R) t AA = A t n(n 1) A = I n }, dim = 2 n(n 1) SO(n) = {A O(n) det A = 1}, dim = 2 U(n) = {A GL(n, C) A A = A A = I n }, dim = n 2 SU(n) = {A U(n) det A = 1}, dim = n 2 1 Sp(n) = {A GL(n, H) AA = A A = I n }, dim = 2n 2 + n where, H = {a + bi + cj + dk a, b, c, d R} denotes the set of all quaternions (i 2 = j 2 = k 2 = 1, ij = ji = k, jk = kj = i, ki = ik = j), and A = t A = 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 L a : G G defined by L a (x) = ax for x G is called the left translation. L a is a C map on G, and furthermore it is a diffeomorphism of G. L 1 a = L a 1. Similarly, the right translation R a by a G is a diffeomorphism defined by R a (x) = xa for x G. The inner automorphism i a : G G i a := L a R a 1, i a (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 (L a ) X = X, where ((L a ) X)f := X(f L a ) (f C (G)).
3 An introduction to the geometry of homogeneous spaces 123 More precisely, (L a ) x X x = X ax for x G ((L a ) x X x )f = X x (f L a ) 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) trace X = 0} M(n, R) o(n) = {X M(n, R) X + t X = 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)
4 124 Takashi Koda 1.4 Lie algebra of a Lie group Let X, Y be left invariant vector fields on a Lie group G, then hence we get (L a ) X = X, (L a ) Y = Y for a G, (L a ) [X, Y ] = [X, Y ]. In general, if ϕ : M N is a C map and vector fields X, Y on M and X, Y on N satisfy ϕ X = X, ϕ Y = Y, then we have 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 T e G at the identity e G. and conversely g X X e T e G (isomorphism between vector spaces) T e G X e X g defined by X x := (L x ) e X e (x G). Example An element x in GL(n, R) M(n, R) = R n2 may be expressed as x = (x i j), then x i j 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 Xj i x i i,j=1 j Then X e T e GL(n, R) may be identified with the n n matrix (Xj i (e)) M(n, R).. n gl(n, R) X = Xj i x i i,j=1 j (X i j(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), [, ])
5 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 ( ) d ϕ (0) = ϕ 0 T e G dt This is a one-to-one correspondense. For each left invariant vector field X g on G, we denote by ϕ X (t) the 1- parameter subgroup with the initial tangent vector ϕ X (0) = X e T e G. Definition 7. The map exp : g G defined by 0 exp X := ϕ X (1) is called the exponential map of a Lie group G. Example For each X M(n, R), the series of matrix e X = I + X + X2 2! converges in M(n, R) = R n2. The map exp : M(n, R) M(n, R), + + Xn n! is called the exponential map of matrices. If [X, Y ] = 0, exp(x + Y ) = exp X exp Y and we may see that Hence + X exp X = e X exp O = I, exp( X) = (exp X) 1. exp : M(n, R) GL(n, R). This coincides with the exponential map of the Lie group GL(n, R).
6 126 Takashi Koda 1.6 Adjoint representations Let G be a Lie group and i a denotes the inner automorphism by a G. We denote its differential by i a = L a R a 1 : G G, x axa 1 Ad(a) = Ad G (a) := (i a ). For any left invariant vector field X g, Ad(a)X = (i a ) X is also left invariant. satisfies Thus we have a homomorphism Ad(a) : g g (linear) [Ad(a)X, Ad(a)Y ] = Ad(a)[X, Y ]. Ad : G GL(g), which called the adjoint represenation of G. Proposition 1.6. For any a G and X g, satisfies a Ad(a) exp(ad(a)x) = i a (exp X). Let g be a Lie algebra with a bracket [, ]. For any X, the linear map (Jacobi identity). The homomorphism ad(x) : g g, Y ad(x)y := [X, Y ] ad(x)[y, Z] = [ad(x)y, Z] + [Y, ad(x)z] ad : g gl(g), is called the adjoint representation of g. Proposition 1.7. For any X, Y g, Ad(exp X)Y = e ad(x) Y = X ad(x) k=0 Definition 8. Let g be a Lie algebra. The bilinear form 1 k! ad(x)k Y. 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.
7 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 S 1 S }{{} 1 is called a toral subgroup of G, or simply a torus. 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 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 = xt x 1. x G (3) Any other maximal torus T of G is conjugate to T, i.e., a G s.t. T = at a 1 (4) Any maximal torus of G is a maximal abelian subgroup of G. The dimension l = dim T of T is called the rank of G. Example G = U(n). T := e iθ e iθn is a toral subgroup of U(n). θ 1,, θ n R = S 1 S 1 }{{} n
8 128 Takashi Koda Proposition Any unitary matrix A U(n) can be diagonalized by a unitary matrix P U(n), P 1 AP = e iθ e iθ n =: diag[eiθ 1,, e iθ n ] Example G = SO(n). We denote by R θ the rotation matrix ( cos θ sin θ R θ = sin θ cos θ ) and if n = 2m put T = R θ R θm θ 1,, θ m R = S 1 S 1 }{{} m and if n = 2m + 1 we put T = R θ Rθm 0 1 θ 1,, θ m R = S 1 S 1 }{{} m then T is a maximal torus of SO(n). Proposition Any orthogonal matrix A O(n) may be put into the following
9 An introduction to the geometry of homogeneous spaces 129 form by an orthogonal matrix Q O(n), t QAQ = 1 cos θ 1 sin θ 1 sin θ 1 cos θ 1... cos θ 0 r sin θ r sin θ r cos θ r Remark. R 0 = ( ) = I 2, R π = ( ) = I 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 Therefore, (ρ, V ) is completely reducible. (ρ(g)-invariant decomposition) (ρ, V ) = (ρ 1, V 1 ) (ρ k, V k ) where (ρ i, V i ) is irreducible. In particular, the adjoint representation (Ad, g) of G is completely reducible, and we have a decomposition where g = g 0 g 1 g m g 0 = {X g ad(y )X = 0 for Y g} (center of g) and g i is an ideal of g (i.e., [g, g i ] g i ) and g i is irreducible.
10 130 Takashi Koda Proposition Let A, B be two diagonalizable matrices. If [A, B] = 0, then A and B are simultaneously diagonalizable, i.e., x 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 completely reducible. If we consider the complexification g C = g R C and the representation (Ad, g C ) of S, the eigenvalues of Ad(s) are complex numbers with absolute value 1, and corresponding eigenspaces in g C are of dim C = 1. Then g may be decomposed into irreducible subspaces. g = V 0 V 1 V m where Ad(s) V0 = id on V 0 and for some linear form θ k : s R ( ) cos θk (v) sin θ Ad(exp v) Vk = k (v) sin θ k (v) cos θ k (v) on V k for v s w.r.t. an orthonormal basis on V k. These θ k are unique up to sign and order, and do not depent on the inner product and an orthonormal basis. Definition 11. Linear forms ±θ k (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 2π θ 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. n T = {diag[e ix 1,, e ix n ] x k R, x k = 0} k=1 The Lie algebra t of T is n t = {diag[ix 1,, ix n ] x k R, x k = 0} k=1
11 An introduction to the geometry of homogeneous spaces 131 x 1,, x n t are deined in natural way. linear form diag[ix 1,, ix n ] x k is denoted by x k We take the inner product (, ) on g defined by (X, Y ) := 1 tr (XY ) 2 and for 1 j < k n, 1 l n, put 1 U jk = 1, U jk = i i, 2El = 2 i which form an orthonormal basis of g. Then we may see that for any v = diag[ix 1,, ix n ] t ( ) cos (xj x Ad(exp v) = k ) sin (x j x k ) sin (x j x k ) cos (x j x k ) on span R {U jk, U jk }. Thus the root system of SU(n) is = {±(x j x k ) 1 j < k n} Definition 12. Let {λ 1,, λ l } be a basis of t. The lexicographic order λ 1 > λ 2 > > λ l > 0 is defined by l v = v i v 1 = = v k 1 = 0 λ i > 0 v k > 0 for some k k=1 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 {α 1,, α l } of all simple roots of G is called the simple root system of G (where l = rank G). Example G = SU(n) and T a maximal torus of all diagonal matrices. {x 1,, x n 1 } is a basis of t. Consider the lexicographic order x 1 > x 2 > > x n 1 > 0. Then
12 132 Takashi Koda + = {x j x k 1 j < k n} = { (x j x k ) 1 j < k n} and the simple root system is positive root system negative root system {α j = x j x j+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. g C := g R C, t C := t R C We extend [, ] complex-linearly on g C, then g C becomes a complex Lie algebra. t C becomes a maximal abelian Lie subalgebra of g C. The Killing form B C of g C 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 g C in natural way. (ad(v)x, Y ) + (X, ad(v)y ) = 0 for X, Y g C, v t C, that is, ad(v) is a skew-hermitian transformation. Thus for any v t C, ad(v) is diagonalizable. Hence, t C is a Cartan subalgebra of g C. 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 SU(n) G = SU(n), T is the maximal torus defined by T = {diag [e iθ 1,, e iθ n ] θ θ n = 0} su(n) = {X gl(n, C) X + X = O, tr X = 0} A basis of g is given by 1 U jk = 1, U i jk = i, E l = i i. Then g C is spanned (over R) by
13 An introduction to the geometry of homogeneous spaces 133 Hence g C = sl(n, C) and U jk, U jk, E l, iu jk, iu jk, ie l. t = {diag[iθ 1,, iθ n ] θ k R, θ θ n = 0}, t C = {diag[λ 1,, λ n ] λ k C, λ λ n = 0}. For v = diag[λ 1,, λ n ] t C, if we put E jk = 1 then we have ad(v)e jk = [v, E jk ] = (λ j λ k )E jk. Let x j : t C C be a linear form defined by x j (diag[λ 1,, λ n ]) = λ j, then we have ad(v)e jk = [v, E jk ] = (x j x k )(v)e jk. Thus if we put α = x j x k (t C ), ad(v)x = α(v)x for X g C α = CE jk. The root system of g = su(n) relative to t is = {±(x j x k ) 1 j < k n}, and we have the decomposition g C = t C + α, g C α, where g C α = {X g C ad(v)x = α(v)x for v t C } is the root space corresponding to a root α. (dim C g C α = 1). In usual, we denote by h a Cartan subalgebra, so in our case, h = t C. and then is just it. We take a basis {v 1,, v l } of h R as h R := {v h α(v) R ( α )} v j := diag[0,, 1,, 0] it
14 134 Takashi Koda and put {λ 1,, λ l } its dual basis of h R, λ j = x j. We consider the lexicographic order in h R. The root system is a union of the positive root system and the negative root system. If we put = + α j = x j x j+1 (1 j n 1 = l) 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 g C is nondegenerate, and we denote it by (, ). For λ h, we define v λ h by and we define an inner product on h by We put (v λ, v) = λ(v) for v h, (λ, µ) := (v λ, v µ ). (, ) R := (, ) hr, (, ) R := (, ) h R, which are positive definite. For convenience, we denote these simply by (, ). For α, β (α ±β), we consider the sequence β pα,, β α, β, β + α,, β + qα where β (p + 1)α and β + (q + 1)α. {β + mα p m q} is called the α-sequence containing β. We may see that In particular, if p = 0, i.e., β α, Therefore for simple roots α j, α k 2(β, α) (α, α) = p q Z. 2(β, α) (α, α) 0. 2(α j, α k ) (α j, α j ) 0.
15 An introduction to the geometry of homogeneous spaces 135 On the other hand where θ is the angle of α j and α k. 2(α j, α k ) 2(α j, α k ) (α j, α j ) (α k, α k ) = 4 cos2 θ, cos 2 θ = 0, 1 4, 1 2, 3 4 (or 1). Since π/2 θ < π, we have four cases. (1) θ = π 2, (2) θ = 2π 3, α j = α k (3) θ = 3π 4, α j : α k = 1 : 2 or 2 : 1 (4) θ = 5π 6, α j : α k = 1 : 3 or 3 : 1 Then we draw a diagram as follows: (1) we do not join α j and α k (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 {α 1,, α l } 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 = m 1 α m l α l in the Dynkin diagram, the integers m k are written on each α k. The highest root µ of su(n) is µ = x 1 x n = α α l, m 1 = = m l = 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) 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
16 136 Takashi Koda For an arbitraryly fixed point x 0 M, the closed subgroup H := {a G ax 0 = x 0 } is called an isotropy subgroup of G at x 0. Then M is diffeomorphic to the coset space G/H. M = G/H, x ah, where, a G is an element of G s.t. x = ax 0. 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, and the map τ g : G/H G/H, is called a left translation by g. (g, ah) (ga)h, ah (ga)h τ g π = π L g 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, on M is G-invariant if (τ g ) X, (τ g ) Y = X, Y for g G for any X, Y T M. Then τ g is an isometry of the Riemannian manifold (M,, ). Definition 15. A homogeneous space M = G/H is reductive if there exists a decomposition such that g = h + m (direct sum) 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 : T e G T o M onto linear
17 An introduction to the geometry of homogeneous spaces 137 induces an isomorphism For any X gh, Y gh T gh M T o M = T e G/ ker π e = g/h = m X gh, Y gh gh = (τ g 1) gh X gh, (τ g 1) gh Y gh eh = ((τ g 1) gh X gh, (τ g 1) gh Y gh ). Let G/H be a reductive homogeneous space, g = h m (Ad(H)-invariant decomposition). For any h H π(i h (g)) = τ h (π(g)) Hence for any X m g π e (Ad(h)X) = (τ h ) e X (g G). (π e : m = T o M). Then a G-invariant Riemannian metric, on M = G/H induces an inner product (, ) on m. (X, Y ) = X, Y o 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:, : G-inv. Riem. metric on G/H (, ) : Ad(H)-inv. inner product on m In the sequel, we use the same notation, for an Ad(H)-inv. inner product on m. Definition 16. A reductive homogeneous space M = G/H with a Riemannian metric, 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 ].
18 138 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 T o M, 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 (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) = τ exp tx (gh) is a curve in M through a point gh. The vector field X defined by XgH := γ (0) = d dt τ exp tx (gh) t=0 satisfies at the origin o = eh Xo = d dt π(exp tx) = π e (X). t=0 If G/H is reductive, the isomorphism π e : m = T o M may be expressed as m X Xo T o M. 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,. We denote by the Riemannian connection of (M/H,, ).
19 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 we have X Y, Z = X Y, Z + Y, X Z, Y X, Z + Y, Z X = 0. Proposition 2.3. Let (G/H,, ) be a naturally reductive homogeneous space. Then for v m = T o M. v X = [X, v] if X h 1 2 [X, v] m if X m 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. and The canonical connection D is given by for X, Y, Z m. { D v X [X, v] if X h = [X, v] m if X m D X Y = X Y + 1 T (X, Y ), 2 T (X, Y ) = [X, Y ] m, B(X, Y )Z = [[X, Y ] h, Z],
20 140 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] 1 2 [[X, Y ] m, Z] m for X, Y, Z m. Then we have 1 4 [[Y, Z] m, X] m 1 4 [[Z, X] m, Y ] m (R(X, Y )Y ) o = [[X, Y ] h, Y ] 1 4 [[X, Y ] m, Y ] m and the sectional curvature is given by R(X, Y )Y, X o = 1 4 [X, Y ] m, [X, Y ] m + [[X, Y ] h, X] m, Y. If (G/H,, ) is normal homogeneous R(X, Y )Y, X o = 1 4 [X, Y ] m, [X, Y ] m + [X, Y ] h, [X, Y ] h Symmetric spaces Definition 19. A connected Riemannian manifold (M, g) is called a Riemannian (globally) symmetric space if p M, s p I(M, g) s.t. s p 2 = id., p is an isolated fixed point of s p s p is the geodesic symmetry: where γ v (t) is the geodesic s.t. s p (γ v (t)) = γ v ( t), γ v (0) = p, γ v (0) = v T p M. The connected Lie group G = I 0 (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 p 0 M. The map σ : G G, a s p0 a s p0 is an involutive automorphism of G, and G σ 0 H G σ,
21 An introduction to the geometry of homogeneous spaces 141 where G σ = {a G σ(a) = a}, G σ 0 is the identity component of G σ. Such a pair (G, H) is called a symmetric pair. Examples M = G/H G H S n SO(n + 1) SO(n) sphere Gr k (R n ) O(n) O(k) O(n k) real Grassmann mfd. Gr k (C n ) 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 for X, Y, Z m. R = 0, R(X, Y )Z = [[X, Y ], Z] 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 h C of g C such that h C a, let be the root system of g C relative to h C. Then Σ coinsides with {α a α } (as well as their multiplicities).
22 142 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 Riemannian 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 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
23 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 φ 2 X = 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), [2] A. Arvanitoyeorgos, An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, Amer. Math. Sci., [3] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, [4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Interscience, New York, 1963 and [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, [6] K. Sekigawa, Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J., 7 (1978),
24 144 Takashi Koda [7] W. Ziller, The Jaacobi equation on naturally reductive compact Riemannian homogeneous spaces, Comment. Math. Helvetici, 52(1977),
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