CODIMENSION REDUCTION IN SYMMETRIC SPACES ANTONIO J. DI SCALA AND FRANCISCO VITTONE 3 1 0 2 Abstract. In this paper we give a short geometric proof of a gen- eralization of a well-known result about reduction of codimension for n submanifolds of Riemannian symmetric spaces. a J 0 1. Introduction 1 The goal of this paper is to give a short geometric proof of the following ] generalization of the reduction of codimension theorem for submanifolds of G space forms [Er71, page 339]: D . Theorem 1. Let M be a submanifold of a symmetric space S and let ν(M) h t be its normal bundle. Assume that there exists a ∇⊥-parallel subbundle a m V ⊂ ν(M) containing the first normal space, i.e. N1 ⊂ V where N1 = α(TM ×TM). If TM ⊕V is invariant by the curvature tensor of S then [ there exists a totally geodesic submanifold of S of dimension rank(TM ⊕V) 2 containing M. v 6 As a corollary of this result one can obtain several well-known special 2 cases [Ce74], [CHL78], [Ka93], [Ka94], [KP08], [KP99], [Ok82] . 6 1 The hypothesis about the curvature invariance of TM ⊕V is redundant 2. if S is a space form. We will give an example showing that such condition 1 can not be omitted in general, see Section 4. 2 Our proof of the above theorem was mainly inspired by the proof, due 1 to C. Olmos, of the existence theorem of a totally geodesic submanifold : v with prescribed tangent space usually attributed to E. Cartan, see [BCO03, i X Theorem8.3.1.,page231]. Olmos’proofisbasedinLemma8.3.2in[BCO03, r page 232]. We include the slight modification we need of this lemma as an a appendix. Theorem 1 does not hold for submanifolds of locally symmetric spaces. The problem is that under the same hypothesis, the “totally geodesic sub- manifold” containing M may intersect itself, see example in Section 4. One Date: January 11, 2013. 1991 Mathematics Subject Classification. Primary 53C29, 53C40. Key words and phrases. Codimensionreduction,ErbacherTheorem,Parallelfirstnor- mal space, Totally geodesic submanifolds. This work was partially supported by ERASMUS MUNDUS ACTION 2 programme, through the EUROTANGO II Research Fellowship. The second author would like to thanks Politecnico di Torino for the hospitality during his research stay. 1 2 A.J.DISCALAANDF.VITTONE can prove a slightly different version of this theorem for locally symmetric spaces by either assuming that the submanifold M is embedded or allowing totallygeodesicimmersions(notnecessarily1−1)insteadoftotallygeodesic submanifolds. Finally, we want to point out that Theorem 1 can be obtained, besides our proof, by following two other different approaches. The first one makes use of the Grassmann bundle theory and the integration theory of differen- tiable distributions, see [JR06, Prop. 3, page 90]. The second one is based on a generalization of the classical theorem of existence and uniqueness of isometric immersions into space forms, see [ET93]. 2. Basic definitions We will say that a Riemannian manifold M is a submanifold of a Rie- mannian manifold S if there is a 1 − 1 isometric immersion f : M → S. In order to simplify the notation, we shall assume that M is a subset of S, eventually endowed with a different topology, and f is the inclusion map. If in addition M has the induced topology from S we say that M is an embedded submanifold. We identify the tangent space to M at a point p with a subspace of T S p and consider the orthogonal splitting T S = T M ⊕ν M. Here ν M is the p p p p normal space and νM will denote the normal bundle of M. We denote by ∇ the Levi-Civita connection of S and by ∇ and ∇⊥ the Levi-Civita and the normal connections of M respectively. Let α and A be the second fundamental form and shape operator of M respectively. They are defined taking tangent and normal components by the Gauss and Co- dazzi formulas (1) ∇ Y = ∇ Y +α(X,Y), ∇ ξ = −A X +∇⊥ξ X X X ξ X and related by (cid:104)α(X,Y),ξ(cid:105) = (cid:104)A X,Y(cid:105), for any tangent vector fields X and ξ Y to M and any normal vector field ξ. 3. Proof of Theorem 1 First notice that it suffices to prove the theorem locally around each point. Namely, to show the existence of a totally geodesic submanifold N of S containing a neighbourhood U of p in M whose tangent space is p T N = T M ⊕V for all q ∈ U. Indeed, the global result follows since a q p q q complete totally geodesic submanifold of a symmetric space S with a pre- scribed tangent space is unique as a global object [KN63, Lemma 2, page 235]. SowemayassumethatM issmallenoughsothatthenormalexponential map exp⊥ : V → S is an immersion from a small neighbourhood V of the 0 0 zero section of V. Set N = exp⊥(V ). Since M is the image of the zero section of V we get 0 thatM isasubmanifoldofN. NowwearegoingtoprovethatN isatotally geodesicsubmanifoldbyasimilarargumentasintheproofofTheorem8.3.1 CODIMENSION REDUCTION IN SYMMETRIC SPACES 3 in [BCO03, page 231]. It will suffice to prove that the parallel transport in S along any curve in N preserves the tangent bundle TN. To do this we will fix a point p ∈ M and we will show the following two properties: i) for any point q in N, there is a curve γ joining p with q such that the parallel transport along γ in S of T N is T N; p γ(t) ii) the tangent space T N is preserved by parallel transport in S along p any loop in N based at p. Inordertoprovei), wewillstartshowingthatTN isparallelwithrespect to the connection ∇ of S in directions tangent to M. Since TN = TM ⊕V, a section X of TN splits as X = X +X , |M |M 1 2 with X ∈ TM and X ∈ V. So if v ∈ T M, then 1 2 p ∇ X = ∇ X +α(v,X )+∇⊥X −A v. v v 1 1 v 2 X2 This shows that ∇ X belongs to TN since α(v,X ) ∈ N1 ⊂ V and V is v 1 parallel with respect to the normal connection of M. The second step is to prove that TN moves parallel along any normal geodesic γ(t) = exp (tξ ) for p ∈ M and ξ ∈ V . Observe that the tangent p p p 0 spaces to N along γ are generated by the Jacobi fields J(t) along γ(t) with initial conditions J(0) ∈ T M and J(cid:48)(0) ∈ V. γ(0) Denote by W the parallel transport of T N along γ from γ(0) to γ(t). t p Let J(t) be any Jacobi vector field along γ with J(0) ∈ T M and γ(0) J(cid:48)(0) ∈ V . Since TM ⊕ V is invariant under the curvature tensor of γ(0) the symmetric space S one gets that J(t) ∈ W for every t. Indeed, W is t t curvature invariant and so the Jacobi equation can be solved in W . This t shows that T N ⊂ W , hence W = T N, since both are linear spaces γ(t) t t γ(t) of the same dimension. Now, if q is any point in N, there exists a point q in M and a normal 0 vector ξ ∈ V such that q = exp⊥(ξ ). From the above discussion, any q0 0 q0 curve in M connecting p to q followed by a normal geodesic from q in the 0 0 direction of ξ gives one curve joining p with q satisfying i). 0 Now we prove ii) by using Lemma 8.3.2 in [BCO03, page 232]. Let c(s) be any loop in N based at p ∈ M. There exists a loop cˆ(s) in M based at p and a normal vector field ξ(s) ∈ V along cˆsuch that c(s) = exp⊥(ξ(s)). cˆ(s) For each s ∈ I, define the transformation τ(s) ∈ SO(T S) obtained by p ∇-parallel transport along the curve c from p = c(0) to c(s), then along the normal geodesic γ (t) = exp⊥(tξ(s)) backwards from γ (1) to γ (0) and s s s finally backwards along cˆ, from cˆ(s) to cˆ(0) = p. Observe that τ(0) is the identity transformation of T S and τ(1) is the p ∇-parallel transport along the loop c followed by the ∇-parallel transport along the loop cˆ−1. Consider now the function f : I ×I → N defined by (cid:26) cˆ(2st) if 0 ≤ t ≤ 1, s ∈ I f(s,t) = 2 exp⊥((2t−1)ξ(s)) if 1 ≤ t ≤ 1, s ∈ I 2 4 A.J.DISCALAANDF.VITTONE Observe that f(s,0) = p for all s ∈ I and the transformation τ(s) defined above is the ∇-parallel transport along the curve t (cid:55)→ f(0,t) from t = 0 to t = 1, then along the curve s (cid:55)→ f(s,1) from 0 to s, and finally along the curve t (cid:55)→ f(s,t), backwards from t = 1 to t = 0. For each s ∈ I set A(s) = τ(cid:48)(s)◦τ(s) ∈ so(T S). We can apply Lemma p 8.3.2in[BCO03](withoutalmostanymodification, seeAppendix)toobtain that for each u, v ∈ T S, p (cid:90) 1(cid:28) (cid:18)∂f ∂f (cid:19) (cid:29) (2) (cid:104)A(s)u,v(cid:105) = R (s,t), (s,t) U (t),W (t) dt s s ∂s ∂t 0 whereU (t)andW (t)are∇-parallelvectorfieldsalongthecurvet (cid:55)→ f(s,t) s s with U (0) = u and W (0) = w. s s Observe that for each fixed s, the curve t (cid:55)→ f(s,t) is the concatenation of a curve in M and a normal geodesic. We have seen that for each s, the tangent space T N is invariant under ∇-parallel transport along these f(s,t) curves and, since S is symmetric, equation (2) implies that (cid:104)A(s)u,v(cid:105) = 0 for each u ∈ T N, w ∈ ν N. That is, A(s)(T N) ⊂ T N for all s ∈ I. Since p p p p τ(s) is defined by the system of differential equations τ(cid:48)(s) = A(s)τ(s) and τ(0) is the identity transformation, one gets that τ(s) preserves T N. In p particular, τ(1) preserves T N, but by construction τ(1) is the ∇-parallel p transport along the concatenation of the loops c and cˆ−1. Since by i) the ∇-parallel transport along cˆpreserves T N, we obtain ii). (cid:3) p Remark 3.1. The same proof shows that Theorem 1 is still true if the ambi- ent space S is locally symmetric, as long as the submanifold M is embedded. However if S is locally symmetric and M is not embedded, the theorem is not true as we will show in the example below. 4. Further Remarks As we noticed in the Introduction, Theorem 1 does not hold under the weaker assumption of the ambient space S being locally symmetric even if S is compact. For example let S = S1 × Σ be the product of a circle S1 with a compact Riemann surfaces Σ of genus 2 endowed with the metric of constantnegativecurvature. Itiswell-knownthatthereisaself-intersecting geodesic γ in Σ. Then the product N := I×γ is a subset of S but it is not a submanifold. However N can be regarded as the image of a totally geodesic (non injective!!) immersion. So we can regard N as in Figure 1. Consider the 1−1 immersed curve M starting at O going through the point A coming back to B and finally approaching the point A along the self-intersection of N. The vector field V is clearly continuous hence it gen- erates a subbundle V of ν(M). Since N is the image of a totally geodesic immersion it follows that the subbundle V is ∇⊥-parallel and contains the first normal space of M as in Theorem 1. Now it is clear that there is not CODIMENSION REDUCTION IN SYMMETRIC SPACES 5 M O V B A V N Figure 1 a totally geodesic submanifold containing M as in Theorem 1. Indeed, such totally geodesic submanifold should be contained in N and self-intersect near the point A. This is so because near A, coming from the point O, it should contain an open subset of the surface N tangent to V. But when M approaches A coming from B such totally geodesic submanifold should be contained in the leaf of N tangent to V which is transversal to the first one. This shows that such ‘totally geodesic submanifold’ intersects itself which is a contradiction since by submanifolds we intend 1-1 immersions. WewantalsotoremarktheimportanceofthehypothesisofTM⊕Vbeing curvature invariant. By using the existence theorem for curves by means of its Frenet-Serret curvatures, see for example [Gu11, Page 2158, Lemma 4], let γ be a regular curve in CP2 (the complex space form of dimension 2) with κ = κ = 1 and κ ≡ 0. Then the mean curvature vector field 1 2 3 H of γ and its normal derivative ∇⊥ H are linearly independent. Since γ(cid:48)(t) κ ≡ 0 the rank 2 vector subbundle V = span{H,∇⊥ H}, which contains 3 γ(cid:48)(t) the first normal space N1 = span{H}, is ∇⊥-parallel. Since CP2 has no 3-dimensional totally geodesic submanifolds we conclude that hypothesis of Tγ ⊕V being curvature invariant can not be removed from Theorem 1. Appendix We present here a slight variation of Lemma 8.3.2 in [BCO03, page 232] with a sketch of its proof. 6 A.J.DISCALAANDF.VITTONE LetM beadifferentiablemanifoldandf : [0,1]×[0,1] → M acontinuous map. We say that f is piecewise-smooth if there exist points 0 = t < t < 0 1 ··· < t = 1 such that f is smooth for i = 0,··· ,n−1. n |[0,1]×(ti,ti+1) Lemma 4.1. Let M be a Riemannian manifold and p ∈ M. Let f : [0,1]× [0,1] → M be a piecewise-smooth map with f(s,0) = p for all s ∈ [0,1]. For each s ∈ [0,1], we define f : [0,1] → M, t (cid:55)→ f(s,t) and for each t ∈ [0,1] s we define ft : [0,1] → M, s (cid:55)→ f(s,t). For each s ∈ [0,1], denote by τ(s) ∈ SO(T M)the orthogonal transformation of T M obtained by parallel p p translation along f from p = f (0) to f (1) = f1(0), then along f1 from 0 0 0 f1(0) to f1(s) = f (1) and finally along f from f (1) to f (0) = p. Let s s s s A(s) ∈ so(T M) be the skew-symmetric transformation of T M defined by p p A(s) = τ(cid:48)(s)◦τ(s)−1 for all s ∈ [0,1]. Then for each u,w ∈ T M, p (cid:90) 1(cid:28) (cid:18)∂f ∂f (cid:19) (cid:29) (3) (cid:104)A(s)u,w(cid:105) = R (s,t), (s,t) U (t),W (t) dt, s s ∂s ∂t 0 where U (t) and W (t) are the parallel vector fields along f with U (0) = u s s s s and W (0) = w respectively. s Proof. Using the same argument as in the proof of Lemma 8.3.2 in [BCO03] one can see that it suffices to prove formula (3) only for s = 0. Let U(s,t) bethevectorfieldalongf(s,t)obtainedbyparalleltranslationofualongf 0 from p = f (0) to f (1) = f1(0), then along f1 from f1(0) to f1(s) = f (1) 0 0 s and finally along f from f (1) to f (t). Then s s s U(s,0) = τ(s)u, A(0)u = τ(cid:48)(0)u = Z(0), whereZ isthevectorfieldalongf definedbyZ(t) = (cid:0)DU(cid:1)(0,t). Sincethe 0 ∂s vector field t (cid:55)→ U(s,t) is parallel along f , DU(s,t) = 0 for t ∈ (t ,t ) i = s ∂t i i+1 0,··· ,n−1 and so (cid:18) (cid:19) (cid:18) (cid:19) D D ∂f ∂f Z(cid:48)(t) = U (0,t) = R (0,t), (0,t) U (t), t ∈ I −{t }n−1. ∂t∂s ∂t ∂s 0 i i=1 Consider the piecewise smooth function g(t) = (cid:104)Z(t),W (t)(cid:105). 0 For each t ∈ I −{t }n−1, i i=1 (cid:28) (cid:18) (cid:19) (cid:29) g(cid:48)(t) = (cid:10)Z(cid:48)(t),W (t)(cid:11) = R ∂f(0,t), ∂f(0,t) U (t),W (t) . 0 0 0 ∂t ∂s CODIMENSION REDUCTION IN SYMMETRIC SPACES 7 Since g(t) is continuous on [t ,t ], i = 0,··· ,n − 1, we can repeatedly i i+1 apply Barrow’s law and get (cid:104)A(0)u,w(cid:105) = g(0) (cid:90) t1(cid:28) (cid:18)∂f ∂f (cid:19) (cid:29) = g(t )− R (0,t), (0,t) U (t),W (t) dt 1 0 0 ∂t ∂s 0 (cid:88) (cid:90) ti+1(cid:28) (cid:18)∂f ∂f (cid:19) (cid:29) = g(t )− R (0,t), (0,t) U (t),W (t) dt 2 0 0 ∂t ∂s i=0,1 ti . . . n(cid:88)−1(cid:90) ti+1(cid:28) (cid:18)∂f ∂f (cid:19) (cid:29) = g(1)− R (0,t), (0,t) U (t),W (t) dt 0 0 ∂t ∂s i=0 ti (cid:90) 1(cid:28) (cid:18)∂f ∂f (cid:19) (cid:29) = R (0,t), (0,t) U (t),W (t) dt 0 0 ∂s ∂t 0 since Z(1) = 0 by construction and so g(1) = (cid:104)Z(1),W (1)(cid:105) = 0. 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[Ok82] Okumura,M.Reducingthecodimensionofasubmanifoldofacomplexprojective spaceGeom.Dedicata13(1982)277-289.http://link.springer.com/article/ 10.1007%2FBF00148233 Authors’ Addresses: A. J. Di Scala, Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy [email protected] http://calvino.polito.it/~adiscala/ F. Vittone, Depto. de Matem´atica, ECEN, FCEIA, Universidad Nacional de Rosario, Rosario, Argentina CONICET [email protected]