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Home , Symmetric space

Symmetric Spaces

Xinghua Gao May 5, 2014

Notations

M . 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) of M at p along the section S. r Ds set of tensor fields of type (r, s). I(M) the set of all on M

1 Affine Locally Symmetric Space, 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 ∈ 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 symmet- ric if each point m ∈ M has an open neighborhood Nm on which the geodesic −1 sm sm symmetry sm is an affine transformation. i.e. ∇X (Y ) = (∇Xsm (Y )) , ∀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 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 ∈ 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 ∈ 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 an- alytic 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: Let M = Rn with the Euclidean metric. The geodesic symmetry at any point n p ∈ R is the point reflection sp(p + v) = p − v. The is the E(n) generated by translations and orthogonal linear maps; the isotropy group of the origin is the 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 ∈ S is the reflection at the line Rx in R , 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 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 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 −1 the inversion se(g) = g . Then se(e) = e and dsev = −v for any v ∈ 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 ∈ 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 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 ∈ 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 ∈ 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}

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 ∈ 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 ∈ 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 = {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 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 ∈ O(n) be the reflection at the subspace

3 E and lets ˆE be the corresponding conjugation,s ˆE(x) = sExsE. This is an ⊥ isometry fixing pE, and since sE fixes E and reflects E , the conjugations ˆE n ⊥ maps any x ∈ TpGk(R ) into −x (x is a linear map from E to E and vice versa). Thuss ˆE 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 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 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 → 7→ ◦ ◦ G has an involutive automorphism σ given by σ : G G, g 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 isome- tries of M. Then (1) The compact open topology of I(M) turns I(M) into a locally compact topo- logical transformation group.

4 (2) Let p ∈ M and let K˜ denote the subgroup of I(M) which leaves p fixed. Then K˜ 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)) = soτ(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 First consider the Grassmannian of oriented k-planes in Rk+l, denoted by M = k+l k+l G˜k(R ). Thus, each element in M is a k-dimensional subspace of R togeth- er 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 at p = Rk is naturally identified with the space of k × l k l matrices Matk×l, or equivalently, with R ⊗R . The isotropy action of SO(k)× O(l) on Matk×l now acts as follows:

−1 T SO(k) × O(l) × Matk×l → Matk×l, (A, B, X) 7→ AXB = AXB

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 TpM = Matk×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, AXBT = −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 other- T 1 T wise zero, then AXB = A1(B ) , 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 Matk×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 N sp be the isometry of R fixing p whose linear part dsp acts as identity I on the normal space νpS and as −I on the tangent space TpS, 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 S- paces, Academic Press, 1978 [2] J.-H. Eschenburg, Lecture Notes on Symmetric Spaces [3] Peter Holmelin, Symetric spaces, 2005 [4] Peter Peterson, , Springer, 2006

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