1. Differential Geometry Dominic Leung 梁树培 Lectures 19-21

2. A Brief Introduction to Symmetric Spaces

3. References [C] S.S. Chern, W.H. Chen and K.S. Lam, Lectures on differential Geometry, (World Scientific, 2000) [H] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press 1978) [J] Jurgen Jost, Riemannian Geometry and Geometric Analysis (5th Edition, Springer, 2008) [K] Kobayashi, Transformation Groups in Differential Geometry (Springer, 1972) [L1] D.S.P. Leung, The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geometry 8, no, 1 (1973), 163-160. [L2] D.S.P. Leung, On the classification of reflective submanifolds of Riemannian symmetric spaces, Indiana University Mathematics Journal, Vol.24, No. 4 (1974), 327-339. [L3] D.S.P. Leung, Reflective Submanifolds IV. Classification of Real forms of Hermitian Symmetric spaces, J. Differential Geometry 14, no, 1 (1979), 179-185. [L4] D.S.P. Leung, A Reflection Principle for Harmonic Mappings with Applications for Holomorphic Functions on and Holomorphic mappings, MSC Tsinghua Preprints (October, 2012) [W] J.A. Wolf, Spaces of constant curvature (6 th Edition), AMS Chelsea (2011)

4. 1 Addition notes on Lie groups, Isometries and geodesic submanifolds Relationship between Lie groups and Lie algebra. The Lie algebra consists of right invariant vector fields on a Lie group G. Let e  G be the identity element of the group G and X . Let  : (- ,  )x U  G be the local one-parameter group generated by X defined in a neighborhood U of e in G. Let exp(tX) =  (t,e) for |t|   and =  (t,e) n for |t|/ n  . Hence {exp(tX)} -   t   is the one parameter subgroup of G generated by X. The case t = 1, for all X , gives us the exponential map Exp: T e G  G. For the classical Lie groups, consisting of only nonsingular matrices, we have in fact Exp(X) = e X = Id + X + (1/2)X 2 +(1/3!)X 3 +...

5. Definition1.1. A diffeomorphism h: M  N between Riemannian manifolds is an isometry if it preserves the Riemannian metrics. Thus for each p  M, v,w  T p M and if G M and G N denotes the scalar products of T p M and T h(p) N respectively then G M (v,w) = G N (h * (v), h * (w)). Theorem 1.1 The set of all isometries of a Riemannian manifolds M is Lie group G. Theorem 1.1 is classical theorem of Myer and Steenrod. In the case when M is a symmetric space a proof can be found in [H]. Definition1.2. A vector field X on a Riemannian manifold with metric tensor G is called a Killing field if L X (G) = 0. Lemma 1.1. A vector field X on a Riemannian manifold M is a Killing field if and only if the local 1-parameter group generated X consists of local isometries. Theorem 1.2 The Killing fields of a Riemannian manifold M constitute a Lie algebra which is the Lie algebra of the Lie group G consisting of the isometries of M.

9. 3 Geometry of Symmetric Spaces Definition 3.1. A Riemannian manifold is called symmetric if for every point p  M there exists an isometry  p : M  M with  p (p) = p  p* | p = -id (as a self-map of T p M). Such an isometry is also called an involution. Examples of symmetric spaces. R d, equipped with the Euclidean metric, i.e, the d-dim Euclidean space E d. The involution at p E d is the map  p (x) = 2p – x. The sphere S d : Since its isometry group operates transitively on S d, it suffices to display an involution at the north pole (1,0,…,0); such an involution is given by  (x 1, …,x d+1 ) = (x 1,x 2,…,-x d+1 )

10. The hyperbolic spaces H n. Equip R n+1 with the quadratic form  x,x  := -(x 0 ) 2 + (x 1 ) 2 +  + (x n ) 2 where (x = (x 0,..., x n )). We define H n := {x  R n+1 :  x,x  := -1, x 0  0 } Thus, H n is a hyperbola of revolution; the condition x 0  0 ensures that H n is connected. The symmetric bilinear form I := -(dx 0 ) 2 + (dx 1 ) 2 +... + (dx n ) 2 induces a positive definite symmetric bilinear form on H n. Namely, if p  H n T p H n is orthogonal to p w.r.t.  ·,· . Therefore the restriction of I to T p H n is positive definite. We thus obtain a Riemannian metric  ·,·  on H n of constant curvature -1. The resulting Riemannian manifold is called real hyperbolic space. Again, the isometry group operates transitively and it suffices to consider the points (1,0,..., 0), the sometry here is  (x 1, …,x n+1 ) = (x 1, -x 2, , -x n+1 )

12. Corollary 3.1. A symmetric space is geodesically complete, i.e., each geodesic can be indefinitely extended in both directions, i.e. defined for all R. Proof: This follows readily from Lemma 3.2. q.e.d. The Hopf-Rinow Theorem implies the following corollary. Corollary 3.2. In a symmetric space, any two points can be connected by a geodesic. By Lemma 3.1, the operation of  p on geodesic through p is given by a reversal of the direction. Since by corollary 3.2 any point can be connected to p by a geodesic, we conclude that the following corollary is true. Corollary 3.3.  p is uniquely determined.

13. Definition 3.2 Let M be a symmetric space, c : R  M a geodesic. The translation along c by the amount t  R is τ t =  c(t/2)   c(0). By Lemma 3.2 thus τ t maps c(s) onto c(s+t), and τ t* is parallel transport along c from c(s) to c(s+t). Remark. τ t is an isometry defined on all of M. τ = τ t maps the geodesic onto itself. The operation of t on geodesic other than c in general is quite different, and in fact τ need not map any other geodesic onto itself. One may see this for M = S n Convention: For the rest of this paragraph, M will be a symmetric space. G denotes the isometry group of M. G 0 is the following subset of G : G 0 = {g t for t  R, where s ı  g s is a group homomorphism from R to G } i.e. the union of all one-parameter subgroups of G. (It may be shown that G 0 is a subgroup of G.) Examples of such one-parameter subgroups are given by the families of translations (τ t ) t  R along geodesic lines. Theorem 3.1. G 0 operates transitively on M Proof: By Corollary 3.2, any two points p, q ϵ M can be connected by a geodesic c. Let p = c(0) and q = c(s). If (τ t ) t ϵR is the family of translation along c, then q = τ s (p). We have thus found an isometry from G 0 that maps p to q. q.e.d.

15. Definition 3.5. Let be the Lie algebra of Killing fields on the symmetric space M, and let p  M. We put := {X  : X(p) = 0 }, := { X  : DX(p) = 0 } Theorem 3.3.  =  = {0} Theorem 3.4. As a vector space, is isomorphic to T p M. The one-parameter subgroup of isometries generated by Y  is the group of translations along the geodesics exp p tY(p).

16. We now define a group homomorphism s p : G  G by s p (g) = s p  g  s p where s p : M  is the involution at p. Since s p 2 = id, we have s p (g) = s p  g  s p -1. We obtain a map  p :  by  p (X) = (d/dt)(s p (e tX ))| t=0. Theorem 3.5.  p | = id  p | = -id

17. Theorem 3.6. [, ] , [, ] . Corollary 3.4 is a ie subalgebra of. Corollary 3.5 With the identification T p M  from Theorem 3.4, the curvature tensor of M satisfies R(X,Y)Z(p) = -[[X,Y],Z](p) Proof: See [J].

18. Corollary 3.6. The sectional curvature of the plane in T p M spanned by the orthogonal vectors Y 1 (p), Y 2 (p) (Y 1, Y 2  ) satisfies K(Y 1 (p)  Y 2 (p)) = -  [[Y 1,Y 2 ],Y 2 ],Y 1  (p). For a Lie group G with Lie algebra = T e G, let ad be the adjoint representation of the Lie algebra as defined in Theorem 2.4 of §6-2 in [C]. Recall that for X  ad(X) is a linear transformation of the vector space. Definition 3.7. The Killing form of the Lie algebra is the bilinear form B : x  R where B(X,Y) = tr(ad X  ad Y). For the Lie algebra (and likewise G) is called semisimple if the Killing form of is nondegenerate. A semisimple Lie algebra (and likewise G) is called simple if it has no ideals except {0} and.

19. Definition 3.8. Let =  be the usual decomposition of the space of Killing fields of the symmetric space M. M is called of Euclidean type if [, ] = 0, (i.e. if the restriction of the Killing form vanishes identically). M is called semisimple, if is semisimple. M is called of compact (resp. noncompact) type, if it is semisimple and of nonnegative (resp, nonpositive) sectional curvature. Corollary 3.7. A semisimple symmetric space is of (non-) compact type if andonly if is Killing form B is negative (positive) definite on. Let K be the Lie subgroup of G that consisting elements of G that leaves p fixed. K is called the isotropy subgroup of G at p. Then the coset space G/K (homogeneous space) can be naturally given a manifold structure (for example through the restriction of the exponential map exp to ) such that G/K is diffeomorphic to M.

20. Totally Geodesic submanifolds of Riemannian symmetric Spaces. Let be a real Lie algebra over R and be a subspace of ; is called a Lie triple system if X, Y, Z ϵ implies [X, [Y, Z]] ϵ. Theorem 3.7. Let M be a globally symmetric space as defined above with its group of isometries G with Lie algebra. Identifying T p M with the subspace of, let be a Lie triple system contained in. Let S = Exp. Then S has a natural differentiable structure such that it is a totally geodesic submanifold of M with T p S =. A proof of this Theorem can be found in [H]. A linear subspace of is said to be reflective if and its orthogonal complement ┴ are Lie triple systems such that [[, ], ┴ ]  ┴, [[ ┴, ┴ ], ] , [[, ┴ ], ]  ┴, [[, ┴ ], ┴ ] . Theorem 3.8. There is a one-to-one correspondence between the set of reflective subspaces of of and the set of complete globally reflective submanifolds B through the point p of M, the correspondeence being given by = T p B  T p M.

21. A proof of Theorem 3.8 can be found in [L1]. All the reflective submanifolds of an irreducible Riemannian symmetirc space have been completely classified in [L2]. [L2] in fact provides the longest list of totally geodesic submanifolds of all irreducible Riemannian symmetric spaces in addition to its uses in the reflection principle of minimal submanifolds. §4 Extension of a reflective submanifolds B in a non-compact Riemannian symmetric spaces H to a minimal submanifolds Z with dim Z = dim B +1. In addition to fact that there are numerous reflective submanifolds among the Riemannian symmetric spaces, we also have the following theorem about the existence of minimal submanifolds in a Riemannian symmetric space. Theorem 4.1. (a) Let M be a locally symmetric space, S  M a reflective submanifolds of codimension greater than one, and p  S. For a non-zero vector X  T p S ┴, let {  t } denote the local one parameter group of isometries generated by X. Then there exists  > 0 and a neighborhood U of p in S such that the map f : (- ,  ) x U  M, f(t, x) =  t (x) is a minimal immersion. (b) If, in addition to the above assumptions, M is a simply connected noncompact globally symmetric space, and S is complete, then the map f extends to a complete minimal embedding f : R x S  M. A proof of theorem is given in [G], under some weaker assumptions on S

30. (8.71 )

35. Definition 4.6. Let =  be the usual decomposition of the space of Killing fields of the symmetric space M. M is called of Euclidean type if [, ] = 0, (i.e. if the restriction of the Killing form vanishes identically). M is called semisimple, if is semisimple. M is called of compact (resp. noncompact) type, if it is semisimple and of nonnegative (resp, nonpositive) sectional curvature. Corollary 4.7. A semisimple symmetric space is of (non-) compact type if and only if is Killing form B is negative (positive) definite on. Let K be the Lie subgroup of G that consisting elements of G that leaves p fixed. K is called the isotropy subgroup of G at p. Then the coset space G/K (homogeneous space) can be naturally given a manifold structure (for example through the restriction of the exponential map exp to ) such that G/K is diffeomorphic to M. For the Riemannian symmetric space M, the pair of Lie algebras,, and the involutive map  p constructed is called an orthogonal symmetric algebra (,,  p ). The most difficult task in the classification of Riemannian symmetric spaces is the classification of all orthogonal symmetric algebra. Riemannian symmetric spaces were introduced and studied by E. Cartan who also completed the task of its classification

36. A summary of the classification of Riemannian Symmetric spaces. (See [H, W] for details) A simply connected Riemannian symmetric space is isometric (up to homothetic equivalence) to a product E d x  M i x  M * j x  N i x  N * j where M i are type I symmetric spaces consisting of simple, compact, connected Lie groups U; the metric on U is two-sided invariant and is uniquely determined up to a positive factor. M * j are type IV symmetric spaces (see [H] for detailed descriptions). N i are simply connected compact symmetric spaces, G/K, of type III, with G a simple compact Lie group and K a compact subgroup of G. N * j are simply connected non-compact symmetric spaces, G/K, of type I, with G a simple non-compact Lie group and K a compact subgroup of G. [H] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press 1978) [W] J.A. Wolf, Spaces of constant curvature (6 th Edition), AMS Chelsea (2011)

37. For each symmetric space M of type I has exactly one dual partner M * a symmetric space of type IV. Similarly, for each symmetric space M of type III has exactly one dual partner M * a symmetric space of type I. Compact symmetric spaces of type III and their corresponding dual partners, compact symmetric spaces of type III are described in Table 1. From the rank of an irreducible symmetric space is the dimension of its maximum totally geodesic submanifolds. The groups SU(m), SU(p,q), SO(m), SO(p,q), Sp(m), Sp(p,q), SL(m,R) and Sp(m,R) are all matrix groups called the classical groups. Their detailed descriptions can be found in [H]. The groups E6, E7, E8, F4 and G2 are the compact exceptional Lie groups their detailed descriptions can be found in [H] and [Y].

38. Specializing to the Riemannian symmetric spaces of type I and their type III dual partners, Cartan found that there are the following seven infinite series and twelve exceptional Riemannian symmetric spaces G/K. They are here given in terms of G and K. The labelling of these spaces is the one given by Cartan. The following gives only the compact symmetric spaces and their geometric interpretations.

40. In the following table taken from [H], a list of irreducible symmetric spaces of Type I and their corresponding Type III partners are given, using more recent, or the Helgason, notations. The Hermitian symmetric spaces are also identified.

43. Typedim C Ambient spaceReal forms A IIIpq SU(p. q)/S(U p x U q ) p = q = 1 or p < q p, q not both even p, q both even p = q > 1 p odd p even SO(p, q)/SO(p) x SO(q) SO(p, q)/SO(p) x SO(q); Sp((p + q)/2)/Sp(p/2) x Sp(q/2) SO(2p)/SO(p) x SO(p); SL(p, C) x R SO(2p)/SO(p) x SO(p); SL(p, C) x R; Sp(p)/Sp(p/2) x Sp(p/2) D III n(n-1)/2 SO*(2n)/U(n) n odd n even SO(n, C) SO(n, C); [SU*(n)/Sp(n/2)] x R BD I p SO(2, p)/SO(2) x SO(p) [SO(1, k)/SO(1) x SO(k)] x[SO(1, p – k)/SO(1) x SO(p – k)], 0 ≤ k ≤ [p/2] C I n(n+1)/2 Sp(n, R)/U(n) n odd n even [SL(n, R)/SO(n)] x R [SL(n, R)/SO(n)] x R; Sp(n/2, C) E III16 ; Sp(6)/Sp(2) x Sp(4) E VII27SU*(8)/Sp(4); Real forms of noncompact Hermitian Symmetric Spaces

44. The reflection principle for minimal submanifolds has been recently generalized to harmonic mappings [L4]. Using this new generalization the following reflection principle for holomorphic functions defined on a subclass of Hermition symmetric spaces have been proved.

45. Theorem 5.2. Let M n be a noncompacr Hermitian symmetric space of complex dimension n of one of the following types with the its associated real forms: Type Noncompact Hermitian Symmetric Spaces Real Forms 1 EuclideanCnCn RnRn 2 Hermitian Hyperbolic SU(n, 1)/S(U n x U 1 ) SO(n, 1)/SO(n) 3 BD ISO(2, n)/SO(2) x SO(n)

46. Real forms of Classical bounded symmetric domains. Typedim C Ambient spaceReal forms A IIIpq SO(p, q)/SO(p) x SO(q) SO(p, q)/SO(p) x SO(q); Sp(p/2, q/2)/Sp(p/2) x Sp(q/2) SO(p, p)/SO(p) x SO(p); SL(p, C) x R SO(p, p)/SO(p) x SO(p); SL(p, C) x R; Sp(p/2, p/2)/Sp(p/2) x Sp(p/2) D III n(n-1)/2 SO(n, C) SO(n, C); [SU*(n)/Sp(n/2)] x R BD I (q=2) p {X ϵ M 2,p (R): X t X < I 2 } = SO(2, q)/SO(p) x SO(2) P ≠ 2 [SO(1, k)/SO(1) x SO(k)] x [SO(1, p – k + 1)/SO(1) x SO(p-k+1)], 0 ≤ k ≤ [p/2] C I n(n+1)/2 [SL(n, R)/SO(n)] x R [SL(n, R)/SO(n)] x R; Sp(n/2, C)