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An Introduction to the Geometry of Homogeneous Spaces Proceedings of The Thirteenth International Workshop on Di®. 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 De¯nition and examples De¯nition 1. A set G is called a Lie group if (1) G is an (abstract) group, (2) G is a C1 manifold, (3) the group operations G £ G ! G; (x; y) 7! xy; G ! G; x 7! x¡1 are C1 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) = fA 2 M(n; R) j det A 6= 0g 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. De¯nition 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) = fA 2 GL(n; R) j det A = 1g; dim = n2 ¡ 1 n(n ¡ 1) O(n) = fA 2 GL(n; R) j tAA = A tA = I g; dim = n 2 n(n ¡ 1) SO(n) = fA 2 O(n) j det A = 1g; dim = 2 ¤ ¤ 2 U(n) = fA 2 GL(n; C) j A A = A A = Ing; dim = n SU(n) = fA 2 U(n) j det A = 1g; dim = n2 ¡ 1 ¤ ¤ 2 Sp(n) = fA 2 GL(n; H) j AA = A A = Ing; dim = 2n + n where, H = fa + bi + cj + dk j a; b; c; d 2 Rg 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 ¯xed a 2 G, the map La : G ! G de¯ned by La(x) = ax for x 2 G 1 is called the left translation. La is a C map on G, and furthermore it is a di®eomorphism of G. ¡1 La = La¡1 : Similarly, the right translation Ra by a 2 G is a di®eomorphism de¯ned by Ra(x) = xa for x 2 G: The inner automorphism ia : G ! G ¡1 ia := La ± Ra¡1 ; ia(x) = axa (x 2 G) is C1 map on G. De¯nition 3. Let G be a Lie group. A vector ¯eld X on G is left invariant if (La)¤X = X; where 1 ((La)¤X)f := X(f ± La)(f 2 C (G)): An introduction to the geometry of homogeneous spaces 123 More precisely, (La)¤xXx = Xax for x 2 G 1 ((La)¤xXx)f = Xx(f ± La) for f 2 C (G) We denote by g the set of all left invariant vector ¯elds on G. Proposition 1.2. (1) For X; Y 2 g and ¸; ¹ 2 R, we have ¸X + ¹Y 2 g. (2) For X; Y 2 g, we have [X; Y ] 2 g. Remark. For vector ¯elds X; Y on a manifold, bracket [X; Y ] is de¯ned by [X; Y ]f := X(Y f) ¡ Y (Xf)(f : C1 function): 1.3 Lie algebras De¯nition 4. An n-dimensional vector space g over a ¯eld K = R or C with a bracket [ ; ]: g £ g ! g is called a Lie algebra if it satis¯es (1) [ ; ] is bilinear, (2) [X; X] = 0 for 8X 2 g and (3) (Jacobi identity) [X; [Y; Z]] + [Y; [Z; X]] + [Z; [X; Y ]] = 0 for 8X; Y; Z 2 g. Example M(n; R);M(n; C) and M(n; H) with a commutator [ ; ] de¯ned by [X; Y ] = XY ¡ YX are Lie algebras over R. De¯nition 5. A vector subspace h of a Lie algebra g is called a Lie subalgebra if [h; h] ½ h. Example sl(n; R) = fX 2 M(n; R) j trace X = 0g ½ M(n; R) o(n) = fX 2 M(n; R) j X + tX = 0g ½ M(n; R) so(n) = fX 2 o(n) j trace X = 0g ½ o(n) u(n) = fX 2 M(n; C) j X + X¤ = Og ½ M(n; C) su(n) = fX 2 u(n) j trace X = 0g ½ u(n) 124 Takashi Koda 1.4 Lie algebra of a Lie group Let X; Y be left invariant vector ¯elds on a Lie group G, then (La)¤X = X; (La)¤Y = Y for a 2 G; hence we get (La)¤[X; Y ] = [X; Y ]: In general, if ' : M ! N is a C1 map and vector ¯elds X; Y on M and X0;Y 0 on N satisfy 0 0 '¤X = X ;'¤Y = Y ; then we have 0 0 '¤[X; Y ] = [X ;Y ]: Proposition 1.3. The set g of all left invariant vector ¯elds on a Lie group G is a Lie algebra with respect to the Lie bracket [X; Y ] of vector ¯elds 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 identi¯ed with the tangent space TeG at the identity e 2 G. g 3 X 7! Xe 2 TeG (isomorphism between vector spaces) and conversely TeG 3 Xe 7! X 2 g de¯ned by Xx := (Lx)¤eXe (x 2 G): 2 Example An element x in GL(n; R) ½ 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 ¯eld X 2 gl(n; R) on GL(n; R) may be written as Xn @ X = Xi : j @xi i;j=1 j i Then Xe 2 TeGL(n; R) may be identi¯ed with the n£n matrix (Xj(e)) 2 M(n; R). Xn @ gl(n; R) 3 X = Xi Ã! (Xi(e)) 2 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 De¯nition 6. A homomorphism ' : R ! G is called a 1-parameter subgroup of a Lie group G. '(t) is a curve in G such that '(0) = e; '(s + t) = '(s)'(t): For any 1-parameter subgroup ' : R ! G, we consider its initial tangent vector µ ¶ 0 d ' (0) = '¤0 2 TeG dt 0 This is a one-to-one correspondense. For each left invariant vector ¯eld X 2 g on G, we denote by 'X (t) the 1- 0 parameter subgroup with the initial tangent vector 'X (0) = Xe 2 TeG. De¯nition 7. The map exp : g ! G de¯ned by exp X := 'X (1) is called the exponential map of a Lie group G. Example For each X 2 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 2 G. ¡1 ia = La ± Ra¡1 : G ! G; x 7! axa We denote its di®erential by Ad(a) = AdG(a) := (ia)¤: For any left invariant vector ¯eld X 2 g, Ad(a)X = (ia)¤X is also left invariant. Ad(a): g ! g (linear) satis¯es [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 2 G and X 2 g, exp(Ad(a)X) = ia(exp X): Let g be a Lie algebra with a bracket [ ; ].
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