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Symmetric Spaces Symmetric Spaces Xinghua Gao May 5, 2014 Notations M Riemannian manifold. N0 normal neighborhood of the origin in TpM. Np normal neighbourhood of p, Np = exp N0. sp geodesic symmetry with respect to p. Φ f dΦf = f ◦ Φ. Φ X dΦX. K(S) sectional curvature of M at p along the section S. r Ds set of tensor fields of type (r; s). I(M) the set of all isometries on M 1 Affine Locally Symmetric Space, Isometry Group and Others Definition 1 (normal neighborhood) A neighborhood Np of p in M is called a normal neighbourhood if Np = exp N0, where N0 is a normal neighborhood of the origin in TpM, i.e. satisfying: (1)exp is a diffeomorphism of N0 onto an open neighborhood Np; (2)if X 2 N0, 0 ≤ t ≤ 1, then tX 2 N0(star shaped). Definition 2 (geodesic symmetry) 8q 2 Np, consider the geodesic t ! γ(t) ⊂ Np passing through p and q s.t. γ(0) = p, γ(1) = q. Then the mapping 0 q ! q = γ(−1) of Np onto itself is called geodesic symmetry w.r.t p, denoted by sp. Remark: sp is a diffeomorphism of Np onto itself and (dsp)p = −I. Definition 3 (Affine locally symmetric) M is called affine locally symmet- ric if each point m 2 M has an open neighborhood Nm on which the geodesic −1 sm sm symmetry sm is an affine transformation. i.e. rX (Y ) = (rXsm (Y )) , 8X; Y 2 X(M). Definition 4 (Pseudo-Riemannian structure) Let M be a C1-manifold. A pseudo-Riemannian structure on M is a symmetric nondegenerate (as bilinear form at each p 2 M) tensor field g of type (0; 2). Remark: A pseudo-Riemannian manifold is a connected C1-manifold with 1 a pseudo-Riemannian structure. g is called a Riemannian structure iff gp is positive definite 8p 2 M. Definition 5 (Isometry) Let M and N be two C1 manifolds with pseudo- Riemannian structures g and h, respectively. Let ϕ be a mapping of M into N. Then ϕ is called an isometry if ϕ is a diffeomorphism of M onto N and ϕ∗h = g. ϕ is called a local isometry if for each p 2 M there exist open neighborhoods U of p and V of ϕ(p) s.t. ϕ is an isometry of U onto V . Definition 6 (Riemannian locally symmetric space) M is called a Rie- mannian locally symmetric space if for each p 2 M 9 a normal neighborhood Np of p on which the geodesic symmetry sp is an isometry, i.e. sp(g) = g, where g is the pseudo-Riemannian structure on Np ⊂ M.(sp is a local isometry from M to itself). 2 Definition of Symmetric Space Definition 7 (Riemannian Globally Symmetric Space) Let M be an an- alytic Riemannian manifold, M is called Riemannnian globally symmetric if each p 2 M is an isolated fixed point of an involutive (its square but not the mapping itself is the identity) isometry sp of M. Or equivalently 8p 2 M there is some sp 2 I(M) with the properties: sp(p) = p, (dsp)p = −I. Example 1: Euclidean Space Let M = Rn with the Euclidean metric. The geodesic symmetry at any point n p 2 R is the point reflection sp(p + v) = p − v. The isometry group is the Euclidean group E(n) generated by translations and orthogonal linear maps; the isotropy group of the origin is the orthogonal group O(n). Note that the symmetries do not generate the full isometry group E(n) but only a subgroup which is an order-two extension of the translation group. Example 2: The Sphere Let M = Sn be the unit sphere with the standard scalar product. The symmetry n n+1 at any x 2 S is the reflection at the line Rx in R , i.e. sx(y) = −y + 2hy; xix (the component of y in x-direction, hy; xix, is preserved while the orthogonal complement yhy; xix changes sign). In this case, the symmetries generate the full isometry group which is the orthogonal group O(n + 1). The isotropy group T of the last standard unit vector en+1 = (0; :::; 0; 1) is O(n) ⊂ O(n + 1). Example 3: Compact Lie groups Let M = G be a compact Lie group with biinvariant Riemannian metric, i.e. left and right translations Lg, Rg : G ! G acts as isometries for any g 2 G. Then G is a symmetric space where the symmetry at the unit element e 2 G is −1 the inversion se(g) = g . Then se(e) = e and dsev = −v for any v 2 g = TeG, so the involutive condition is satisfied. We have to check that se is an isometry, 2 i.e. (dse)g preserves the metric for any g 2 G. This is certainly true for g = e, and for arbitrary g 2 G we have the relation se ◦ Lg = Rg−1 ◦ se which shows (dse)g ◦ (dLg)e = (dRg−1 )e ◦ (dse)e. Thus (dse)g preserves the metric since so do the other three maps in the above relation. Example 4: Projection model of the Grassmannians n n Let S = Gk(R ) be the set of all k-dimensional linear subspaces of R . The n group O(n) acts transitively on this set. The symmetry sE at any E 2 Gk(R ) will be the reflection sE with fixed space E, i.e. with eigenvalue 1 on E and −1 on E?. n But what is the manifold structure and the Riemannian metric on Gk(R )? n One way to see this is to embed Gk(R ) into the space S(n) of symmetric real n × n matrices: We assign to each k-dimensional subspace E 2 Rn the ? orthogonal projection matrix pE with eigenvalues 1 on E and 0 on E . Let P (n) = fp 2 S(n)j p2 = pg denote the set of all orthogonal projections. This set has several mutually disconnected subsets, corresponding to the trace of the elements which here is the same as the rank: P (n)k = P (n) ⊂ S(n)k;S(n)k = fx ⊂ S(n)j trace x = kg n Now we may identify Gk(R ) with P (n)k ⊂ S(n), using the embedding E 7! pE T which is equivariant in the sense gpEg = pgE for any g 2 O(n). In fact, each pE lies in this set, and vice versa, a symmetric matrix p satisfying p2 = p has only ? eigenvalues 1 and 0 with eigenspaces E = im p and E = ker p, hence p = pE, and the trace condition says that E has dimension k. P (n)k is a submanifold of the affine space S(n)k since it is the conjugacy class of the matrix ( ) I 0 p = k 0 0 0 i.e. the orbit of p0 under the action of the group O(n) on S(n) by conjugation. The isotropy group of p0 is O(k)×O(n−k) ⊂ O(n). A complement of TI (O(k)× O(n − k)) in TI O(n) is the space of matrices of the type ( ) 0 −LT L 0 (n−k)×k n with arbitrary L 2 R , thus P (n)k = Gk(R ) has dimension k(n − k). 2 n −1 Define F : S(n) ! S(n), F (p) = p − p, then P (n)k = Gk(R ) ⊂ F (0), the n kernel ker dFp is contained in TpGk(R ). But the subspace ker dFp = fv 2 S(n)j vp + pv = vg is isomorphic to Hom(E; E?) since it contains precisely the symmetric matrices mapping E = im p into E? and vice versa. Thus n ker dFp = TpGk(R ). Now we equip P (n)k ⊂ S(n) with the metric induced from the trace scalar product < x; y >= trace(xT y) = trace(xy) on S(n). The group O(n) acts isometrically on S(n) by conjugation and preserves P (n)k, hence it acts isomet- rically on P (n)k. In particular, let sE 2 O(n) be the reflection at the subspace 3 E and lets ^E be the corresponding conjugation,s ^E(x) = sExsE. This is an ? isometry fixing pE, and since sE fixes E and reflects E , the conjugations ^E n ? maps any x 2 TpGk(R ) into −x (x is a linear map from E to E and vice versa). Thuss ^E is the symmetry at pE. 3 Homogeneous description For a symmetric space, we have theorem: Theorem 1 Let M be a Riemannian symmetric space and p0 any point in M. Let G = I(M)0 be the identity component of the isometry group and K be the isotropy group of G at p0. Then K is a compact subgroup of the connected group G and G=K is analytically diffeomorphic to M under the mapping gK ! g(p0), g 2 G. So we can get think of symmetric spaces as the homogeneous space of the isometry group G. A natural question to ask is what group G and subgroup K will lead to a symmetric space. To answer this question, we need the definition of Riemannian symmetric pairs. 3.1 From Symmetric space to Symmetric Pairs Definition 8 (symmetric pair) Let G be a connected Lie group and K a closed subgroup. The pair (G; K) is called a symmetric pair if there exists an involutive analytic automorphism σ of G s.t. (Gσ)0 ⊂ K ⊂ Gσ, where Gσ is the set of fixed points of σ in G and (Gσ)0 is the identity component of Gσ.
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