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1 1. Riemannian symmetric spaces Definition 1.1. A (globally, Riemannian) symmetric space is a Riemannian manifold (X, g) such that for all x X, there exists an isometry s x Iso(X, g) such that s x (x) = x and T x s x = id TxX. Remark 1.1. When it exists, an isometry satisfying s x (x) = x and T x s x = id TxX must be unique, as for any f Iso(X, g), f is completely determined by the data of its 1- jet at some given point x 0, that is by the data of f(x 0 ) and T x0 f (provided that X is connected). In fact, this simply comes from the local linearizability of isometries. And in the present case, the action of s x (a priori near x) is given by: s x (exp x (v)) = exp x ( v), for any small enough v T x X. Such isometries are often called geodesic symmetries. Note that by the same arguments as above, s 2 x = id for all x. The first natural instances of symmetric spaces are (simply connected) spaces of constant sectional curvature. Indeed, for instance if x = (1, 0,..., 0) S n, then 1 1 s x =... O(n + 1) = Iso(Sn ) 1 is a geodesic symmetry at x. By homogeneity of S n, we conclude that it is symmetric. For the Euclidean space, if x = 0, then id is a geodesic symmetry at x. For the hyperbolic space, in the hyperboloid model, if x = (1, 0,..., 0) then 1 1 s x =... O(1, n) 1 is a geodesic symmetry at x (which again is enough by homogeneity). But there are of course other examples, for instance the complex hyperbolic space H n C (plus the quaternionic hyperbolic space and the octonionic hyperbolic plane). Fact 1.1. Let (X, g) be a symmetric space. Then, it is geodesically complete and homogeneous. In fact, the identity component Iso(X, g) 0 acts transitively on X. Proof. If a geodesic c is defined on an interval ]a, b[, since the geodesic symmetry at some c(b ε) will always preserve c near the symmetry point, we can always extend c across c(b). If x, y M, consider m the middle of some geodesic segment between x and y and let f = s m s x. Immediately, f(x) = y and f Iso(X) 0 (consider the path f t = s c(t) s x ). Recall that by Myers-Steenrod theorem, the isometry group of any Riemannian manifold is a Lie group. Precisely, it admits a unique differential structure (compatible with the compact-open topology) which makes it a Lie group acting differentiably on the manifold. 1

2 In particular, if X is a symmetric space, G := Iso(X) 0 is a connected Lie group acting differentiably and transitively on X. Let us fix once for all an origin o X, and let K := {g G : go = o} denote the isotropy group at o. Fact 1.2. K is compact Lie subgroup of G. Remark 1.2. In fact, having compact isotropy is a necessary and sufficient condition for a faithful Riemannian homogeneous space to admit a left-invariant Riemannian metric. That is, if G is any connected Lie group and H a closed Lie subgroup of G such that (1) G G/H faithfully ; (2) the corresponding map G Diff(G/H) is proper, then, there exists a G-invariant Riemannian metric on G/H if and only if H is compact. Here, the condition are immediately satisfied because G = Iso(X) 0 and by construction of the Lie group structure of G. Counter-examples are given in general with G = R 2 and H = R < G (it is possible to construct G-invariant metrics on G/H because the action of G is not faithful), or with {φ t } < O(n) a one-parameter subgroup injectively immersed (but non-properly) (like for instance a line of irrational slope in some torus in O(n)), then take G = R φ t R n. It acts transitively on R n by Euclidean transformations, with isotropy R = {φ t } at 0. Proof of the remark. Note M = G/H and let x 0 = H M. Consider the isotropy representation ρ x0 : h H T x0 h GL(T x0 M). If H is compact, then by classic averaging arguments, there exists a non-trivial, ρ x0 (H)-invariant, (Euclidean) scalar product <.,. > on T x0 M. Then, for any x = g.x 0 M, and for any u, v T x M, we can coherently define g x (u, v) =< g 1 u, g 1 v >, which will be G-invariant by construction. Conversely, if M is endowed with a G-invariant metric g, then ρ x0 takes values in O(T x0 M, g x0 ) which is compact. An element h Ker ρ x0 has trivial 1-jet at x 0, and is thus trivial by faithfullness of the action. Moreover, the assumption implies that ρ x0 is a proper homomorphism of Lie group (it is clear once we have seen the proof of Myers- Steenrod s theorem), which then concludes because O(T x0 M, g x0 ) is compact. Thus, if X is a symmetric space and G = Iso(X) 0, then X G/K as differentiable manifolds. The question is now to see what is special with symmetric spaces, among Riemannian homogeneous space. Of course, we have symmetries. Let s denote the geodesic symmetry at o and σ Aut(G) the involution Note G σ = {g G : σ(g) = g}. Fact 1.3. We have (G σ ) 0 < K < G σ. σ : G G g sgs. Remark 1.3. In fact, this condition on K characterizes symmetric pairs of Lie groups, in the sense that a couple (G, K) with G connected, and K < G compact is such that X = G/K can be endowed with a symmetric Riemannian metric such that G = Iso(X) 0 if and only if there exists an involutive σ Aut(G) such that K is stuck between (G σ ) 0 and G σ. For instance, it can be easily proved that for G = SO(n), the possible K s are centralizers of diagonal matrices with 1 s and 1 s on the diagonal, which are thus locally 2

3 isomorphic to SO(p) SO(q), with p + q = n. These, groups are maximal (at least at the Lie algebra level) among proper Lie subgroups of SO(n), but they are not the only ones. For instance, SO(7) admits Lie subgroups isomorphic to the exceptional Lie group G 2, which is proper, maximal for the inclusion, but not of the form SO(p) SO(q). Consequently, (SO(7), G 2 ) is not a symmetric pair in the above sense. Proof of Fact 3. By uniqueness of geodesic symmetries, for any g G, gs x g 1 = s gx. Consequently, for any k K, we have sks = k. Conversely, any element of G σ will preserve the set of fixed points of s. Since the fixed points of a geodesic symmetry are isolated, if {φ t } is any one parameter subgroup of G σ, we deduce that φ t o = o, and then (G σ ) 0 K. Let us now define the Cartan involution θ Aut(g) by θ = T e σ, which, as the above remarks indicates, is a very important object in our symmetric space. Because θ 2 = id, we can decompose g with respect to the eigenspaces of θ: g = {θ = id} {θ = id}. }{{}}{{} =k=lie(k) :=p This decomposition g = k p is called the Cartan decomposition of g (with respect to X and its fixed origin o). Fact 1.3 indicates that k is exactly the Lie algebra of K. Proposition 1.1. (1) The Cartan decomposition is Ad(K)-invariant. (2) [k, k] k, [k, p] p and [p, p] k. (3) The restriction of the Killing for to k is negative definite: B k k < 0. And the spaces k and p are orthogonal with respect to the Killing form. From now on, we identify the tangent space T o X p in the following way: X p v = d dt e tx.o T o X. t=0 Under this identification, the isotropy representation at o is conjugate to Ad p : K GL(p). (recall that the Cartan decomposition is Ad(K) invariant). The geometric interpretation of the +1 eigenspace of θ is clear (Lie algebra of the isotropy). For the other (p), we introduce the notion of transvection. Definition 1.2. A transvection of X is an isometry φ such that there exists a geodesic c(t) and a number t 0 such that (1) φ(c(t)) = c(t + t 0 ) ; (2) T φ induces the parallel transport along c. Remark 1.4. c may not be unique: for instance, if X is the Euclidean space, then its transvections are translations, and any straight line is convenient. Proposition 1.2. If X p, then {e tx } is a one-parameter groups of transvections of X along the geodesic {t exp o (tx)} (in the second case, X is seen as an element of T o X). In fact, any one parameter group of transvections of X is of the form {ge tx g 1 }, with g G and X p. 3

4 The idea that the parallel transport is induced by the differential action of one parameters subgroups selected by the Cartan involution will finish the implementation of the geometry of X into algebraic data in G. Indeed : Proposition 1.3. Let R be the (3, 1)-curvature tensor of X, and let X, Y, Z p T o X. Then, R o (X, Y, Z) = [[X, Y ], Z]. }{{} k Proof. We use another identification: to any X g corresponds a Killing vector field of X, say X Kill(X), naturally given by X x = d dt e tx.x, t=0 i.e. X is the infinitesimal generator of the one-parameter group of isometries {e tx }. Then, it is a general fact that [X, Y ] = [X, Y ] (the bracket on the left is the one of vector fields). Now, if X p and T is any vector field on X, then the key point is that ( X T ) o = d dt o φ t (o) T φ t=0 X X (o) = d ( ) dt φ t T X φ t=0 X (o) = [X, T ](o). In particular, if T is of the form T = Y, with Y p, then ( X Y ) o = 0 since [X, Y ] = [X, Y ] and [X, Y ] k. Then, since R is tensorial, we have that for any X, Y, Z p T o X R o (X, Y, Z) = ( X Y Z Y X Z [X,Y ] Z) o. Note that the last term is 0 since [X, Y ] o = 0. For the others, we have X Y Z = [X, Y Z] = [X,Y ] Z + Y [X, Z] - by some sort of Leibniz rule for the Lie derivation in directions given by Killing fields. Again, the first term is 0, and we obtain that R o (X, Y, Z) = ( Y [X, Z]) o ( X [Y, Z]) o 4 = [Y, [X, Z]] o [X, [Y, Z]] o = [[Y, X], Z] o (Jacobi) = [[X, Y ], Z] o = [[X, Y ], Z] o. The last thing which remains unknown is the link between the metric on X and the induced quadratic form on p T o X. This will depend on the type of the symmetric space, which will be defined in the setting of irrecudible, simply-connected symmetric spaces. From now on, X is assumed simply-connected.

5 Recall that we say that a Riemannian manifold is irreducible if it cannot be written as a non-trivial Riemannian product. In the simply connected case, irreducibility can be read on the holonomy: Fact 1.4. Let N be a complete, simply-connected Riemannian manifold and x 0 N. Then, it is irreducible if and only if its holonomy group Hol(x 0 ) acts irreducibly on T x0 N. Starting from an a priori non-irreducible (simply connected) X, we can consider its De Rham decomposition into a product of irreducible manifolds. Each factor will be symmetric (they must be complete and recall that s x (exp x (v)) = exp x ( v), so the geodesic symmetries will preserve each factor). The next proposition characterizes the type of each irreducible factor. Proposition 1.4. Assume X irreducible (and simply-connected). Then: (1) There exists α R such that B p p = αg o (.,.) (where g denotes the metric of X). (2) The sectional curvature κ of X is given by: { 1 κ(p ) = αb([x, Y ], [X, Y ]) if α 0; 0 else, where P T o X is a 2-plane, and X, Y p T o X form an orthonormal basis of P. In particular, κ has constant sign (but may vanish, and often does), which is the opposite of the sign of α. (3) (a) If α = 0, then X is isometric to R. (b) If α < 0, then X and G = Iso(X) 0 are compact, and G has finite center. (c) If α > 0, then G has trivial center. Proof. The key point is to see that the holonomy Hol(o) < O(T o X) is contained in the isotropy K. This follows from the fact that for any loop γ based at o, the parallel transport along γ can be approximated by the one along piecewise-geodesic loops based at o, and the parallel transport along such loops belongs to K (because it is realised by a finite product of transvections), and K is closed. Therefore, by irreducibility, Ad(K) must act irreducibly on p because it is conjugate to the isotropy representation of K on T o X, which contains the holonomy. (1) We have two Ad(K)-invariant quadratic forms on p: first the restriction of the Killing form (this is general and easy to see on the definition), second the inner product corresponding to the metric g o. By irreducibility of Ad(K), they must be proportional. (2) This follows from the formula on R o (X, Y, Z) and the previous point. (3) (a) If α = 0, then κ 0 by homogeneity and X is isometric to some Euclidean space, which must be 1-dimensional by irreducibility. (b) If α < 0, we prove that the Ricci curvature is positive and conclude with Myers theorem. (c) If α > 0, then κ 0 and X is CAT(0). An element g in Z(G) is such that x d(x, gx) is constant (Clifford map). If this constant is non-zero, g is hyperbolic and since the minimum is realized everywhere, X splits into the product of all axes of translation of g, contradicting irreducibility. 5

6 Definition 1.3. Let X be a simply connected symmetric space, and let X = R k X 1 X s be its De Rham decomposition. (1) X is said to be of compact type if k = 0 and all X is are in the case α < 0 of the last proposition. (2) X is said to be of non-compact type if k = 0 and all X is are in the case α > 0 in the last proposition. Recall that we have a finite group F and a short exact sequence 1 Iso(R k ) Iso(X 1 ) Iso(X s ) Iso(X) F 1 Fact 1.5. If X is of non-compact type, then it has non-positive sectional curvature κ 0 and G = Iso(X) 0 is semi-simple without compact factor. In particular, X is diffeomorphic to R dim X. Proof. The sign of the sectional curvature is clear by the last proposition. For the group, its Lie algebra splits into g = g 1 g s where g i is the Lie algebra of Iso(X i ), and each has a Cartan decomposition g i = k i p i and the Killing form is < 0 on k i and > 0 on p i. This proves that the Killing form of g is non-degenerate, which is enough by Cartan s criterion of semi-simplicity. Remark 1.5 (Concerning simple-connectedness). It is possible but not immediate to prove that if X is symmetric, then any cover of it is also symmetric. I am not sure that it is standard, but it seems natural to declare that a general X is irreducible when its universal cover is (to exclude cases like (S n S n )/{±1}) and to assign it the same type as X. Under this convention, any irreducible symmetric space of non-compact type is simplyconnected. Otherwise stated, if X is simply connected, irreducible of non-compact type, there does not exist non-trivial Γ < Iso(X) acting freely properly discontinuously and such that Γ is normalized by all symmetries s x, x X. This is not true in the compact type as RP n = S n /{±1} is still symmetric. Anyway, this justifies our assumption on the topology of X because we are interested in the non-compact type. For instance, some authors (like Eberlein) start studying symmetric spaces of non-compact type by assuming simple-connectedness. 2. Symmetric spaces of non-compact type We fix X a symmetric space of non-compact type, G = Iso(X) 0, an origin o X, K = Stab G (o), θ the Cartan involution at o and g = k p the corresponding Cartan involution. We start with a proposition in fact valid for general symmetric spaces. Proposition 2.1. Complete, totally geodesic submanifolds of X passing through o are in 1 1 correspondence with vector subspaces V p such that [V, [V, V ]] V, the correspondence being V N := exp x0 (V ). Proof. This condition is clearly an obstruction, as for any X, Y, Z V T o N, we must have R o (X, Y, Z) = [[X, Y ], Z] V since N is assumed totally geodesic. To see that it is sufficient, consider h := V [V, V ]. Then, the condition gives that h < g is a Lie subalgebra. If H < G is the corresponding connected Lie subgroup, then it is easy to see that V coincides with the H-orbit H.o. If x N and v T x N, there is h H 6

7 and v V = T o N such that h.o = x and h v = v. Because H acts isometrically it sends γ (t) = exp o (tv ) to γ(t) = exp x (tv) and because N = G.o and γ N we deduce γ N. In particular, maximal, totally geodesic flats in X passing through the origin are parametrized by maximal abelian subspaces a p. Definition 2.1. We call Cartan subspace any maximal abelian subspace of p. Fact 2.1. Any two Cartan subspaces are conjugate by an element of K. Otherwise stated, for any fixed Cartan subspace a, we have p = Ad(k)a. k K Fact 2.2. If a is a Cartan subspace, then the family {ad(x), X a} End(g) acts codiagonally on g. Proof. This is due to the fact that any ad X, X p, is a symmetric linear map of g with respect to the Euclidean scalar product (X, Y ) B(θX, Y ) on g. Thus, there is a finite number of linear forms a such that for all α, g α := {X g H a, [H, X] = α(h)x} 0 and we have the (vector space) decomposition, called restricted root-space decomposition: g = a m α g α, where m = z k (a) = {X k : H a, [H, X] = 0}. Remark that z(a) = a m, and that this direct sum is the decomposition of z(a) (which is θ-stable) with respect to g = k p. Remark moreover that θ sends g α to g α. Definition 2.2. The elements α are called restricted roots (with respect to a), and the subspaces g α are called restricted root-spaces. Remark 2.1. If a = Ad(k)a is another Cartan subspace, then the roots of a are deduced from those of a by precomposition by Ad(k). More pompously, t Ad(k) : (a ) a is an isometry sending restricted roots to restricted roots. Modulo such identifications, the restricted roots are defined independently of a. The structure of the set of restricted roots is central in the geometric structure of the visual boundary X, so let s look a bit further to them. Fact 2.3. The space a comes with a natural Euclidean structure, provided by the Killing form B of g. Indeed, since X is of non-compact type, B p p is positive definite. So, it induces a natural identification a a. Notably, if H α a represents the linear form α, then for all α, β a, < α, β >= α(h β ) = β(h α ) is the Euclidean scalar product we are talking about. Fact 2.4. The restricted-roots satisfy the following properties: (1) 0 / and spans linearly a ; 7

8 (2) For all α, the orthogonal symmetry s α with respect to α preserves ; (3) For all α, β, the quantity 2 <α,β> α 2 is in Z. Note that s α is given by s α (v) = v 2 <α,v> α. So, for all α, β, s α 2 α (β) is another root γ of the form γ = β + nα, with n Z. Using Cauchy-Schwarz s inequality, we can see that if α is not colinear to β, this n belongs to {0, ±1, ±2, ±3}. Definition 2.3. Given a Euclidean vector space V and a finite set V, we say that is an abstract root system of rank dim V if it satisfies these three properties. So, the last fact says that the restricted root-system of a semi-simple Lie algebra without compact factor is an abstract root-system (logic..). Remark 2.2. In the light of a previous remark, we say that two root systems (V, ) and (V, ) are isomorphic when there is an isometry f : V V sending to. Definition 2.4. A root system V is irreducible if there does not exist non-trivial orthogonal decomposition V = V 1 V 2 and disjoint union = 1 2 with 1 V 1 and 2 V 2. Fact 2.5. Irreducible root systems are classified. Namely, they are classified into five infinite families and five additional exceptions: Regular infinite families Exceptional root systems A n, B n, C n, D n, (BC) n, n 1 E 6, E 7, E 8, F 4, G 2 The index indicates the rank of the root-system, for instance E 7 is living in a 7- dimensional Euclidean space. Remark 2.3. There are repetitions in this list, for instance all root-systems of rank 1 are isomorphic - except (BC) 1, B 2 C 2 and D 3 A 3. In fact I lied because, D 2 A 1 A 1 is not irreducible, but it s the only mistake in this list! Going back to semi-simple Lie algebras without compact factor, an important remark is that restricted root-system is not a classifying notion: several Lie algebra may have the same restricted root-system. For instance, we will see that A n is the restricted rootsystem of sl(n + 1, R), sl(n + 1, C) (seen as a real Lie algebra) and sl(n + 1, H), which are of course non-isomorphic. Let s now give explicit instances of the regular root-systems: (1) A n. Let V = {x x n+1 = 0} R n+1 with standard inner product and (e 1,..., e n+1 ) the standard basis. Then, a root-system of type A n is a root-system isomorphic to = {e i e j, i j} V. 8 A 1 A 2 A 3

9 9 (2) B n. A root-system of type B n is a root-system isomorphic to = {±e i ± e j, i j} {e k } V = R n. B 2 (3) C n. A root-system of type C n is a root-system isomorphic to = {±e i ± e j, i j} {2e k } V = R n. (4) D n. A root-system of type D n is a root-system isomorphic to = {±e i ± e j, i j} V = R n. (5) (BC) n. A root-system of type (BC) n is a root-system isomorphic to = {±e i ± e j, i j} {±e i, ±2e i } V = R n. If Rk X 2, the presence of restricted roots indicates some anisotropy inside any maximal flat: the directions in a contained in the kernels of roots will be singular. This idea of singularity/regularity of tangent vectors is in fact intrinsic as the following shows. Proposition 2.2. Let X p. Then, the following assertions are equivalent. (1) z(x) p is abelian. (2) There exists a Cartan subspace a such that X a and for all restricted root α (relative to a), α(x) 0. (3) For all Cartan subspace a containing X and for all restricted root α (relative to a), α(x) 0. (4) The geodesic γ(t) = exp o (tx) X is contained in a unique maximal flat. Definition 2.5. An element satisfying these conditions is said to be regular. Proof. In any event, any Cartan subspace containing X must be included in z(x) p. If the latter is abelian, then by maximality there is exactly one Cartan subspace containing X, namely z(x) p. If (1) is true, let α be a restricted root relatively to a = z(x) p. Then, (g α g α ) a = 0 by definition. For all Y α g α, we have Y α θy α p, implying [X, Y α θy α ] = α(x)y α + α(x)θy α 0. Necessarily, α(x) 0 and we have proved (1) (2) and (1) (3). If (2) is true, let Y z(x) p. We decompose it with respect to the restricted root-spaces of a: Y = Y a + Y m + α Y α. By hypothesis, 0 = [X, Y ] = α α(x)y α. Because all α(x) are non-zero, we obtain Y a m. And since Y p, we get Y a. Thus, z(x) p = a is abelian. This proves (2) (1), and the three first assertions are equivalent. The fourth is a geometric translation of these properties.

10 Definition 2.6. If a is a Cartan subspace, then a Weyl chamber of a is a connected component of the set of regular vectors contained in a. Equivalently, it is a connected component of a \ α Ker α. Definition 2.7. If (V, ) is a root-system, we define its Weyl group W = W ( ) as being the subgroup of O(V ) generated by the reflections s α, α. It is immediate to see that W must be finite: by axioms, W permutes the roots, which form a finite set, and no non-trivial element of W can act trivially on, since the latter generates V linearly. Fact 2.6. W permutes the Weyl chambers, and its action on the set of Weyl chambers is simply transitive. 10

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