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Symmetric Spaces Xinghua Gao May 5, 2014 Notations M N0 Np sp fΦ XΦ K(S) Drs I(M ) 1 Riemannian manifold. normal neighborhood of the origin in Tp M . normal neighbourhood of p, Np = exp N0 . geodesic symmetry with respect to p. dΦ f = f ◦ Φ. dΦ X. sectional curvature of M at p along the section S. set of tensor ﬁelds of type (r, s). the set of all isometries on M 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 Tp M , i.e. satisfying: (1)exp is a diﬀeomorphism of N0 onto an open neighborhood Np ; (2)if X ∈ N0 , 0 ≤ t ≤ 1, then tX ∈ N0 (star shaped). Deﬁnition 2 (geodesic symmetry) ∀q ∈ Np , consider the geodesic t → γ(t) ⊂ Np passing through p and q s.t. γ(0) = p, γ(1) = q. Then the mapping 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 symmetric if each point m ∈ M has an open neighborhood Nm on which the geodesic −1 symmetry sm is an aﬃne transformation. i.e. ∇X (Y ) = (∇X sm (Y sm ))sm , ∀X, Y ∈ X(M ). Deﬁnition 4 (Pseudo-Riemannian structure) Let M be a C ∞ -manifold. A pseudo-Riemannian structure on M is a symmetric nondegenerate (as bilinear form at each p ∈ M ) tensor ﬁeld g of type (0, 2). Remark: A pseudo-Riemannian manifold is a connected C ∞ -manifold with 1 a pseudo-Riemannian structure. g is called a Riemannian structure iﬀ gp is positive deﬁnite ∀p ∈ M . Deﬁnition 5 (Isometry) Let M and N be two C ∞ manifolds with pseudoRiemannian 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 ∈ 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 Riemannian locally symmetric space if for each p ∈ M ∃ 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 analytic Riemannian manifold, M is called Riemannnian globally symmetric if each p ∈ 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 ∀p ∈ M there is some sp ∈ 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 p ∈ Rn 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 at any x ∈ Sn is the reﬂection at the line Rx in Rn+1 , i.e. sx (y) = −y + 2⟨y, x⟩x (the component of y in x-direction, ⟨y, x⟩x, is preserved while the orthogonal complement y⟨y, x⟩x changes sign). In this case, the symmetries generate the full isometry group which is the orthogonal group O(n + 1). The isotropy group of the last standard unit vector en+1 = (0, ..., 0, 1)T 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 ∈ G. Then G is a symmetric space where the symmetry at the unit element e ∈ G is the inversion se (g) = g −1 . Then se (e) = e and dse v = −v for any v ∈ g = Te G, 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 ∈ G. This is certainly true for g = e, and for arbitrary g ∈ 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 Let S = Gk (Rn ) be the set of all k-dimensional linear subspaces of Rn . The group O(n) acts transitively on this set. The symmetry sE at any E ∈ Gk (Rn ) will be the reﬂection sE with ﬁxed space E, i.e. with eigenvalue 1 on E and −1 on E ⊥ . But what is the manifold structure and the Riemannian metric on Gk (Rn )? One way to see this is to embed Gk (Rn ) into the space S(n) of symmetric real n × n matrices: We assign to each k-dimensional subspace E ∈ Rn the orthogonal projection matrix pE with eigenvalues 1 on E and 0 on E ⊥ . Let P (n) = {p ∈ S(n)| p2 = p} 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 = {x ⊂ S(n)| trace x = k} Now we may identify Gk (Rn ) with P (n)k ⊂ S(n), using the embedding E 7→ pE which is equivariant in the sense gpE g T = pgE for any g ∈ 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 ( ) Ik 0 p0 = 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 with arbitrary L ∈ R(n−k)×k , thus P (n)k = Gk (Rn ) has dimension k(n − k). Deﬁne F : S(n) → S(n), F (p) = p2 − p, then P (n)k = Gk (Rn ) ⊂ F −1 (0), the kernel ker dFp is contained in Tp Gk (Rn ). But the subspace ker dFp = {v ∈ S(n)| vp + pv = v} is isomorphic to Hom(E, E ⊥ ) since it contains precisely the symmetric matrices mapping E = im p into E ⊥ and vice versa. Thus ker dFp = Tp Gk (Rn ). 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 isometrically on P (n)k . In particular, let sE ∈ O(n) be the reﬂection at the subspace 3 E and let ŝE be the corresponding conjugation, ŝE (x) = sE xsE . This is an isometry ﬁxing pE , and since sE ﬁxes E and reﬂects E ⊥ , the conjugation ŝE maps any x ∈ Tp Gk (Rn ) into −x (x is a linear map from E to E ⊥ and vice versa). Thus ŝE 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 ∈ 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 . If in addition, AdG (K) (the adjoint group of K in G) is compact, (G, K) is said to be a Riemannian symmetric pair. We can get symmetric pairs from symmetric spaces: Theorem 2 Let M be a symmetric space with a ﬁxed point p0 , G = I(M )0 be the identity component of the isometry group and let K be the isotropy group of G at p0 . Then the map G/K → M with K 7→ g(p0 ) is a bijection. The group G has an involutive automorphism σ given by σ : G → G, g 7→ sp0 ◦ g ◦ sp0 with stabilizer (Gσ )0 ⊂ K ⊂ Gσ . Ad(K) is compact since K is closed and bounded and Ad is a homeomorphism. So (G, K) is a Riemannian symmetric pair. 3.2 From Symmetric Pairs to Symmetric Space In fact symmetric pairs lead to symmetric spaces. Theorem 3 Let M be a Riemannian manifold and I(M) the set of all isometries of M. Then (1) The compact open topology of I(M ) turns I(M ) into a locally compact topological transformation group. 4 (2) Let p ∈ M and let K̃ denote the subgroup of I(M ) which leaves p ﬁxed. Then K̃ is compact. Theorem 4 Let (G, K) be a Riemannian symmetric pair. Let π denote the natural mapping of G onto G/K and put o = π(e). Let σ be any analytic, involutive automorphism of G on M = G/K s.t. (Gσ )0 ⊂ K ⊂ Gσ . Then there is a G-invariant Riemannian structure Q on M that makes (M, Q) a Riemannian globally symmetric space. The geodesic symmetry so satisﬁes so ◦ π = π ◦ σ, τ (σ(g)) = so τ (g)so , g∈G where τ is the parallel translation. In particular, so is independent of the choice of Q. So we get a Riemannian symmetric space M = G/K from a symmetric pair (G, K). Example 4: The Compact Grassmannian First consider the Grassmannian of oriented k-planes in Rk+l , denoted by M = G̃k (Rk+l ). Thus, each element in M is a k-dimensional subspace of Rk+l together with an orientation. We shall assume that we have the orthogonal splitting Rk+l = Rk ⊕ Rl , where the distinguished element p = Rk takes up the ﬁrst k coordinates in Rk+l and is endowed with its natural positive orientation. Let us ﬁrst identify M as a homogeneous space. Observe that O(k+1) acts on Rk+l . As such, it maps k-dimensional subspaces to k-dimensional subspaces, and does something uncertain to the orientations of these subspaces. We therefore get that O(k + 1) acts transitively on M. This is, however, not the isometry group as the matrix −I ∈ SO(k + l) acts trivially if k and l are even. The isotropy group consists of those elements that keep Rk ﬁxed as well as preserving the orientation. Clearly, the correct isotropy group is then: SO(k) × O(l) ⊂ O(k + 1). The tangent space at p = Rk is naturally identiﬁed with the space of k × l matrices M atk×l , or equivalently, with Rk ⊗ Rl . The isotropy action of SO(k) × O(l) on M atk×l now acts as follows: SO(k) × O(l) × M atk×l → M atk×l , (A, B, X) 7→ AXB −1 = AXB T The representation, when seen as acting on Rk ⊗Rl , is denoted by SO(k)⊗O(l). To see that M is a symmetric space, we have to show that the isotropy group contains the required involution. On the tangent space Tp M = M atk×l it is supposed to act by multiplication by −1. Thus, we have to ﬁnd (A, B) ∈ SO(k) × O(l) such that for all X, AXB T = −X. Clearly, we can just set: A = Ik , B = −Il . Depending on k and l, other choices are possible, but they will act in the same way. We have now exhibited M as a symmetric space, although we didn’t use the isometry group of the space. Instead, we used a ﬁnite covering of the isometry group and then had some extra elements that acted trivially. 5 Remark: If we deﬁne X to be the matrix that is 1 in the (1, 1) entry and otherwise zero, then AXB T = A1 (B 1 )T , where A1 is the ﬁrst column of A and B1 is the ﬁrst column of B. Thus, the orbit of X, under the isotropy action, generates a basis for M atk×l but does not cover all of the space. This is an example of an irreducible action on Euclidean space that is not transitive on the unit sphere. Note: Example 2(sphere) and 4(Grassmannian) arises as so are called extrinsic symmetric spaces: A submanifold S ⊂ RN is called extrinsic symmetric if it is preserved by the reﬂections ar all of its normal spaces. More precisely, let sp be the isometry of RN ﬁxing p whose linear part dsp acts as identity I on the normal space νp S and as −I on the tangent space Tp S, then S is extrinsic symmetric if sp (S) = S for all p ∈ S. Extrinsic symmetric spaces are classiﬁed. Figure 1: some classiﬁcation References [1] Sigurdur Helgason, Diﬀerential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978 [2] J.-H. Eschenburg, Lecture Notes on Symmetric Spaces [3] Peter Holmelin, Symetric spaces, 2005 [4] Peter Peterson, Riemannian Geometry, Springer, 2006 6