Differential Geometry

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] JurgenJost, 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 (6th Edition), AMS Chelsea (2011)

4. 1 Addition notes on Lie groups, Isometriesand 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: TeG G. For the classical Lie groups, consisting of only nonsingular matrices, we have in fact Exp(X) = eX = Id + X + (1/2)X2 +(1/3!)X3 +...

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TpM and if GM and GN denotes the scalar products of TpM and Th(p)N respectively then GM (v,w) = GN (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 LX(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.

6. 2 Fixed Points of Isometries Theorem 2.1 Let M be a Riemannian, p  M, v TpM. Then there exist δ 0 and precisely one geodesic c : [0, δ]  M with c(0) = p and ċ(0) = v. In addition, c depends smoothly on p and v. Proof: (2.1) of §5-2 in [1] is a system of second order ODE, and the Picard-Lindelof Theorem yields the local existence and uniqueness of a solution with prescribed initial values and derivatives, and the solution depends smoothly on the data. QED We note if x(t) is a solution of (2.1) of §5-2 in [C]. So is x(λt) for any constant λ  R. Denoting the geodesic in Theorem 2.1 with c(0) = p and ċ(0) = v by cv, we obtain cv(t) = cλv(t/λ) for λ 0, t  [0, δ]. In particular, cλv is defined on [0, δ/λ]. Since it depends smoothly on v and { v TpM : = 1} is compact, there exists 0 0 with the property that for = 1 cv is defined at least on [0, 0]. Therefore, for any 0, cw is defined at least on [0,1]. Definition 2.2. Let M be a Riemannian manifold, p  M, Vp:= {v TpM: cv is defined in [0,1]}, let expp: Vp M be by defined expp (v) = cv (1). The map expp is called the exponential map of M at p.

7. A connection is complete if every maximal geodesic is complete (has the form (t), - < t <  ). Let p and q be two points of a Riemannian M, let ρ(p, q) = inf where denotes the arc length of any curve connecting p and q with measurable arc length. The function ρ gives M a metric space structure. Theorem. (Hopf-Rinow) . Let M be a connected Riemannian manifold. Then the following conditions are equivalent. (i) M is a complete metric space. (ii) The Levi-Civita connection on M is complete. (iii) For some x ϵ M, expx is defined on all of TxM (iv) Every closed metric ball in M is compact. A proof of this fundamental theorem in Riemannian Geometry can be found in many books in differential geometry, like Lectures on Differential Geometry By S.S. Chern and others. In this book, the theorem is actually proved for a Finsler manifold.

8. Definition 2.1. Let M be a Riemannian manifold and S a connected submanifolds of M, Let p  S. The submanifold S is called to be geodesic at if each M-geodesic which is tangent to S at p is an S-geodesic. The submanifold S is called totally geodesic if it is geodesic at each of its points. Examples: Each linear subspace of dimension d  n of En is a totally geodesic submanifold of En. In the n-sphere Sn := {x  En+1 : =1} considered as Riemannian manifolds of constant curvature, the intersection of Sn with any linear subspace of dimension d +1 n+1, through 0  En+1 , is a d dimensional totally geodesic submanifolds of Sn . For a generally Riemannian manifold there are many geodesics, but there may not be any totally geodesic submanifolds of dimension greater than or equal to two. The following theorem could provide some examples of totally geodesic submaniflds in many cases. Theorem 2.1. Let M be a Riemannian manifold and B be any set of isometries of M. Let F be the set of points of M which are left fixed by all elements of B, Then each connected component of F is a closed totally geodesic submanifolds of M. A proof of this fact can be found in [K] (Theorem5.1 of [K])

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 TpM). Such an isometry is also called an involution. Examples of symmetric spaces. Rd, equipped with the Euclidean metric, i.e, the d-dim Euclidean space Ed. The involution at p Ed is the map p (x) = 2p – x. The sphere Sd: Since its isometry group operates transitively on Sd, it suffices to display an involution at the north pole (1,0,…,0); such an involution is given by (x1, … ,xd+1) = (x1,x2,…,-xd+1)

10. The hyperbolic spaces Hn. Equip Rn+1 with the quadratic form x,x:= -(x0)2 + (x1)2 + + (xn)2 where (x = (x0, . . ., xn)). We define Hn := {x  Rn+1 : x,x:= -1, x0  0 } Thus, Hn is a hyperbola of revolution; the condition x0  0 ensures that Hn is connected. The symmetric bilinear form I := -(dx0)2 + (dx1) 2 + . . . + (dxn) 2 induces a positive definite symmetric bilinear form on Hn. Namely, if p HnTpHn is orthogonal to p w.r.t. ·,·. Therefore the restriction of I to TpHn is positive definite. We thus obtain a Riemannian metric ·,· on Hnof 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 (x1, … ,xn+1) = (x1, -x2, , -xn+1)

11. We will now summarize some important facts about Riemannian symmetric spaces. The proofs not geiven here can be found in §5.3 and §5.4 in [J]. Lemma 3.1. An involution p : M  M of a symmetric space reverses the geodesics through p. Thus , if c : (-,)  M is geodesic with c(0) = p ( as always parametrized proportional to arc length), then pc(t) = c(-t). Proof: As an isometry, p maps geodesics to geodesics. If c is a geodesic through p (with c(0) = p), then (0) = -(0). The claims follows since a geodesic is uniquely deterinied by its initial point and initial direction. q.e.d Lemma 3.2. Let c be a geodesic in the symmetric space M, c(0) = p, c(τ) = q. Then qp(c(t)) = c(t+2τ) (for all t, for which c(t) and c(t+2τ) are defined). For v Tc(t)M, q*p* (v) Tc(t+2τ)M is the vector at c(t+2 τ) obtained by parallel transport of v along c. Proof: Let ). then is a geodesic with (0)= q. It follows that qp(c(t)) = q (c(-t)) by Lemma 3,1 = q((-t-τ) = (t + τ) = c(t + τ) . Let v ϵTpM and let V be the parallel vector field along c with V(p) = v . Since p is an isometry, pV is likewise parallel. Moreover p*V(p) = - V(p). Hence p*V(c(t)) = - V(c(-t)) q*∘p*V(c(t)) = V(c(t + 2τ) as before. q.e.d.

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 τt = c(t/2)c(0) . By Lemma 3.2 thus τt maps c(s) onto c(s+t), and τt* is parallel transport along c from c(s) to c(s+t). Remark.τt is an isometry defined on all of M. τ = τt 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 = Sn Convention: For the rest of this paragraph, M will be a symmetric space. G denotes the isometry group of M. G0 is the following subset of G : G0 = {gt for t R, where s ıgsis a group homomorphism from R to G } i.e. the union of all one-parameter subgroups of G. (It may be shown that G0 is a subgroup of G.) Examples of such one-parameter subgroups are given by the families of translations (τt)tR along geodesic lines. Theorem 3.1. G0 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 (τt)t ϵR is the family of translation along c, then q = τs (p) . We have thus found an isometry from G0 that maps p to q . q.e.d.

14. Definition 3.3. A Riemannian manifold with a transitive group of isometries is called homogeneous. Therefore a symmetric space is homogeneous. Theorem 3.2. The curvature tensor R of M is parallel, DR  0. Proof: Let c be a geodesic, and let X, Y, Z, W be parallel vector fields along c, p = c(t0), q = c(t0 + t). Let q = τt (p) and by Lemma 3.2 <R(X(q), Y(q))Z(q), W(q)> = <R(τt*X(p), τt*Y(p)) τt*Z(q), τt*W(q)> = < R(X(p), Y(p))Z(q), W(q)> since τt is an isometry, Let now v:= (t0) . The preceding relation gives v<R(X, Y)Z, W> = 0 . Since X, Y, Z, W are parallel, <(DvR)(X, Y)Z, W> = 0 . Since DvR like R is a tensor, (DvR)(X, Y)Z depends only on the values of X, Y, Z at p. Since this holds for all c, X, Y, Z, W we get DvR ≡ 0 . q.e.d. Definition 3.4 A complete Riemannian manifold with DR 0 is called locally symmetric. Remark. One can show that for each locally symmetric space N there exists a simply connected symmetric space M and a group  operating on M discretely, without fixed points, and isometrically such that N = M/ Conversely, it is clear that such a space is locally symmetric.

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 TpM. The one-parameter subgroup of isometries generated by Y  is the group of translations along the geodesics expptY(p).

16. We now define a group homomorphism sp : G  G by sp(g) = spg sp where sp: M  is the involution at p. Since sp2 = id, we have sp(g) = spg  sp-1. We obtain a map p :  by p (X) = (d/dt)(sp(etX))|t=0. Theorem 3.5. p| = id p| = -id

17. Theorem 3.6. [ , ]  , [ , ]  , [ , ]  . Corollary 3.4is a iesubalgebra of . Corollary 3.5With the identification TpM 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 TpM spanned by the orthogonal vectors Y1(p), Y2(p) (Y1, Y2 ) satisfies K(Y1(p) Y2(p)) = -[[Y1,Y2],Y2],Y1(p). For a Lie group G with Lie algebra = TeG, 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 andonlyif 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 expto ) 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 TpM 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 TpS = . 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 = TpB TpM .

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 TpS┴ , 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 expto ) 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 Ed x Mi x M*j x Ni x N*j where Mi 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). Ni 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 (6th 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.

42. For a arbitrary Riemannian manifold M, the fixed point set B of an global involutiveisometryσ of M is called a reflective submanifold of M. For every irreducible Riemannian symmetric space M, the set of reflective submanifolds of M has been classified [L2]. The complete list is quite long we do not have time to discuss it. We will however look at some special interesting cases of reflective submanifolds in the case of Hermitian symmetric spaces. Let J be the complex structure of a Hermitian symmetric space M, a smooth mapping h of M into itself is said to anti-holomorphic if ∘ = - ∘ . A reflective submanifold B, with involutiveisometryσ, of a Hermitian symmetric space M is called a real form of M if σ is also anti-holomorphic. The real line R  C and RnCn are simple examples of real forms. In particular, the real forms of compact Hermitian symmetric spaces are of special interests, in particular the non-exceptional Hermitian symmetric spaces which are also known as the classical bounded domains. They have been all classified [L3]. They listed in the following tables .

43. Real forms of noncompactHermitian 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 Mn be a noncompacrHermitian symmetric space of complex dimension n of one of the following types with the its associated real forms: For each such Hermitian symmetric space and associated real form pair (Mn, B), where B is one of the real forms associated with M, let σ denote the involutiveisometry associated with the real form B. Let f be a holomorphic function defined on M that takes on real values on the associated real form M, then f satisfies the following reflection principle: f(σ(p)) = for all p ϵ Mn .

46. Real forms of Classical bounded symmetric domains.