On R^n endowed with a riemannian metric of bounded nonpositive curvature, the
weakly convex closed subsets are topologically trivial. The stability of such
subsets under intersection characterizes the euclidean spaces

Grognet, Stephane

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arXiv

On R^n endowed with a riemannian metric of bounded nonpositive curvature, the weakly convex closed subsets are topologically trivial. The stability of such subsets under intersection characterizes the euclidean spaces

HAL-Univ-Nantes

Weakly convex closed subsets of spaces with boundednonpositive curvatureStephane GrognetTo cite this version:Stephane Grognet. Weakly convex closed subsets of spaces with bounded nonpositive curvature.2005. <hal-00005878>HAL Id: hal-00005878https://hal.archives-ouvertes.fr/hal-00005878Submitted on 8 Jul 2005HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00005878, version 1 - 8 Jul 2005Weakly convex closed subsets of spaces withbounded nonpositive curvatureSte´phane GrognetJuly 8, 2005Universite´ de Nantes, Laboratoire Jean Leray, U. M. R. 6629, 2, rue de laHoussinie`re, BP 92208, F-44322 Nantes cedex 03.Stephane.Grognet@univ-nantes.frRE´SUME´. - Sur Rn muni d’une me´trique riemannienne a` courbure ne´gativeborne´e, les ferme´s faiblement convexes sont topologiquement triviaux. Leurstabilite´ par intersection caracte´rise les espaces euclidiens.ABSTRACT. - On Rn endowed with a riemannian metric of bounded non-positive curvature, the weakly convex closed subsets are topologically trivial.The stability of such subsets under intersection characterizes the euclideanspaces.AMS-MSC : 53C20, 52A55keywords : weak convexity, geodesic flow, negative curvature, horospheres1 IntroductionThe convexity of closed subsets in the euclidean spaces and in the riemannianmanifolds has been subject of many studies among which we will only cite [1,4, 5], the third presenting an important bibliography. In presence of negativecurvature, there appears a notion of weak convexity different from the geodesicconvexity [2, 3]. Horospheres are instances of closed subsets which are notgeodesically convex but are weakly convex ; others examples in dimension twocan be found in [3].In the whole paper we consider a complete, connected, simply connectedmanifold M of C∞-class, endowed with a riemannian metric with bounded non-positive sectional curvature : −k2 ≤ K ≤ 0. According to the Cartan-Hadamardtheorem, the manifold M is diffeomorphic to Rn.1We present here two expected properties of the weakly convex closed sub-sets : their topology is trivial, and their bad behaviour with intersection ischaracteristic.Given a closed subset of M , it is said that the projection is unique when thedistance to G of each point of the complementary GC is realized by a uniquepoint of G. The projection from GC to ∂G, which to a point x associates theunique point y such that d(x,G) = d(x, y) is written down pi.Definition 1.1 [2] A closed subset G of M is said weakly convex if the pro-jection is unique, and if for any point x of the complementary GC , the openhorobowl associated to the geodesic ray [pi(x), x) does not intersect the closedsubset G : the half geodesic [pi(x), x) projects itself on the point pi(x).Given a closed subset of M , the Motzkin theorem in bounded nonpositive sec-tional curvature [2] ensures the equivalence between the three properties of weakconvexity, uniqueness of projection, and differentiability of the distance to G onthe complementary GC .The condition of geodesic convexity is stronger than this weak convexity [2] ;weak and strong convexity coincide in euclidean spaces.We prove that nonempty weakly convex closed subsets are deformation re-tracts of M (theorem 2.1). In particular they are connected, simply connectedas expected (corollary 2.1). We deduce from it that the property of stabilityof the weakly convex closed subsets under finite intersection characterizes theeuclidean spaces (theorem 3.1).2 Topological viewpointLemma 2.1 Let M be a C∞-manifold diffeomorphic to Rn equipped with ariemannian metric of bounded nonpositive sectional curvature. For x, y ∈ Mand t ∈ [0, 1], let us define (1 − t)x + ty as a barycentre on the geodesic seg-ment [x, y]. Let G be a nonempty weakly convex closed subset of M . ThemappingsH : M × [0, 1] → G(x, t) 7→ x if x ∈ G(x, t) 7→ (1− t)x + tpi(x) if x ∈ GCandP : M → Gx ∈ G 7→ xx ∈ GC 7→ pi(x)are continuous.Proof : The continuity of P implies that of H . It is clear that P is continuousat every point of the complementary GC and the interior◦G. Let x be a pointof ∂G. We have P (x) = x. Let ε > 0. Let y be a point of B(x, ε) ∩ G. We2have P (y) = y ∈ B(x, ε). Let y be a point of B(x, ε)∩GC . We have P (y) = pi(y)withd (y, pi(y)) ≤ d(y, x) < εthusd (pi(y), x) = d (P (y), P (x)) ≤ d (pi(y), y) + d(y, x) < 2ε.Therefore P is continuous at x. ⋄Theorem 2.1 Let M be a C∞-manifold diffeomorphic to Rn equipped with ariemannian metric of bounded nonpositive sectional curvature. Every nonemptyweakly convex closed subset G of M is a deformation retract of M .Proof : This results from the previous lemma, since H(., 0) = IdM . ⋄As an immediate consequence, we get :Corollary 2.1 Let M be a C∞-manifold diffeomorphic to Rn equipped with ariemannian metric of bounded nonpositive sectional curvature. Every nonemptyweakly convex closed subset G of M is connected, simply connected.3 Geometrical viewpointThe behaviour with respect to intersection of the weakly convex closed subsetsis different from that of the geodesically convex ones.Definition 3.1 The property of stability under finite intersection for the weaklyconvex closed subsets is the following : for all weakly convex closed subsets G1, G2,the intersection G1 ∩G2 is weakly convex.In general, this property is not true.Theorem 3.1 Let M be a C∞-manifold diffeomorphic to Rn equipped with ariemannian metric of bounded nonpositive sectional curvature. The property ofstability under finite intersection for the weakly convex closed subsets is true ifand only if the metric is euclidean.Proof : For euclidean spaces, the condition of weak convexity coincides withthe geodesic convexity, so the property is true. In order to prove the otherimplication, we suppose the property to be true. If there exists a point x ∈M and a 2-plane Π ⊂ TxM such that K(Π) < 0, let v be a unit tangentvector of T 1xM , belonging to Π. Let (Hi)i=1,...,n−2 be a collection of horospherestangent to Π, such that their intersection admits Π as tangent space at x. Thehorospheres are weakly convex. Let G1 be the complementary of the openstable horobowl associated with v. The geodesic flow on T 1M is written ϕt.For some ε > 0, let G2 be the complementary of the open unstable horobowl3associated with ϕεv. The closed subsets G1 and G2 are weakly convex [2].Because of the property, the intersectionI = G1 ∩G2 ∩(n−2∩i=1Hi)is weakly convex. But because of the condition K(Π) < 0, for ε small enough,the second fondamental form of the horospheres ∂G1, ∂G2 are nonzero in theorthogonal of v in Π. Thus, for ε small enough, the intersection I is nonemptyand not connected, which contradicts the corollary 2.1. Therefore the curvatureis always zero. ⋄Bibliography[1] L. Calabi, W.E. Hartnett. A Motzkin-type theorem for closed nonconvexsets. Proc. Am. Math. Soc. 19 (1968), 1495–1498.[2] S. Grognet. The´ore`me de Motzkin en courbure ne´gative. Geometriae De-dicata 79 (2000), 219–227.[3] S. Grognet. Courbes et polygones faiblement convexes en courbure stricte-ment ngative pince. Beitr. Algebra Geom. 42 (2001), 595–599.[4] R. Walter. Some analytical properties of geodesically convex sets. Abh.Math. Sem. Univ. Hamburg 4 (1976), 263–282.[5] R. Walter. Konvexita¨t in riemannschen Mannigfaltigkeiten. Jber. d. Dt.Math.-Verein. 83 (1981), 1–31.4

Weakly convex closed subsets of spaces with bounded nonpositive curvature

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Weakly convex closed subsets of spaces with bounded nonpositive curvature

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