The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. Metric spaces, convexity and nonpositive curvature athanase papadopoulos this book is about metric spaces of nonpositive curvature in the sense of busemann, that is, metric spaces whose distance function satisfies a convexity condition. Roughly speaking, this means that in these spaces, if two observers move at the same constant speed from a com. Groups acting on spaces of nonpositive curvature arxiv. Then the universal cover ing space x of y, with the induced interior metric, is a hadamard space. Probability measures on metric spaces of nonpositive curvature. A complete riemannian manifold with all its sectional curvatures bounded by. Manifolds of nonpositive curvature and their buildings. We recall the definition of such spaces following 28. The book is about metric spaces of nonpositive curvature in the sense of busemann, that is, metric spaces whose distance function is convex. If k0, then the perimeter of each triangle in rk is less than 2. A notion of nonpositive curvature for general metric spaces. In the broader setting of geodesic metric spaces, we have the notion of a locally cat0 metric.
Similarly to the definitions of busemann and cat0 spaces, it is based on comparing triangles in the metric space in question with triangles in the euclidean plane, but in contrast it does not require the space to be geodesic. Each interior point is a riemannian manifold m,g with dimm n, diamm. We study geodesic ows on compact rank 1 manifolds and prove that su ciently regular potential functions have unique equilibrium states if the singular set does not carry full pressure. Uniqueness theorems for harmonic maps into metric spaces chikako mese.
Domains rk and spaces of curvature bounded from above. We prove that not every metric space embeds coarsely into an alexandrov space of nonpositive curvature. This condition has a meaning in any metric space in which the geodesic connection is locally unique. Spaces with nonpositive curvature and their ideal boundaries christopher b. We introduce a new definition of nonpositive curvature or more general curvature bounds in metric spaces. Because the cat o condition captures the essence of nonpositive curvature so well. We define a complete and separable length metric d on the family of all isomorphism classes of normalized metric measure spaces. We present an introduction to metric spaces of nonpositive curva ture npc spaces and a discussion of barycenters. A polyhedral space is a usually finite simplicial complex in which every simplex has a flat metric.
Bruce kleinery february 6, 1998 abstract we construct a pair of. In particular, the open star stv of a vertex v of x is an admissible polyhedron, and so is clearly the closed star stv. Metric spaces of nonpositive curvature, chapters 1 and 2. The proof of the following result should be done by using the second variation formula of geodesics but i do not know how to start or what is the main idea of the proof. The idea of what it means for a geodesic metric space to have nonpositive curvature or, more generally, curvature bounded above by a real number goes back to the independent, but very similar. Bb00 werner ballmann and michael brin, rank rigidity of euclidean polyhedra, amer. Metric characterization of berwald spaces of nonpositive. Alexandrov, and the structure of groups which act on such spaces by isometries. Since the representation p 2pm of g is the isotropy representation of the grassmannian manifold of. Recall that a curve 0,t x is said to be absolutely continuous if there. So an analytical property is characterized by synthetic concepts of nonpositively curved metric spaces. Nonpositive curvature is not coarsely universal springerlink. This inequality encapsulates the concept of nonpositive curvature in riemannian geometry and allows one to reflect the same concept faithfully in a much wider setting that of geodesic metric spaces. An rk domain is a metric space with the following properties 1.
The new edition is an expanded and revised version. In the category of riemannian manifolds, one can consider the sectional curvature of the manifold and require that. A space of curvature bounded from above by kin the. Metric spaces, convexity and nonpositive curvature second. Because of the counterexamples we need to restrict the class of manifolds with boundarythatweconsider. Calculus, heat flow and curvaturedimension bounds in. Metric spaces of nonpositive curvature springerlink. The only spaces of nonconstant curvature that are known to be boundary rigid are two dimensional spaces of nonpositive curvature see section 5, subdomains of negatively curved symmetric spaces see section 6, and some products see section 9. The purpose of this book is to describe the global properties of complete simply connected spaces that are nonpositively curved in the sense of a. Thepadic metric on q is an ultrametric, and q p is an ultrametric space too. Request pdf metric spaces of nonpositive curvature this book describes the global properties of simplyconnected spaces that are nonpositively curved in. The starting point for our paper is the manifold view.
Bal95 werner ballmann, lectures on spaces of nonpositive curvature, dmv seminar, vol. The calculation of the mean curvature of a hypersurface requires only the following data. Aleksandrov domains of a space of nonpositive curvature has been provided. Metric spaces of nonpositive curvature request pdf. Gromovhausdorff space each point is a compact metric space. This is actually a part of geometric group theory and we recommend the survey 22 for a panoramic view on that larger subject. The existence of metrics of nonpositive curvature on the bradykrammer complexes for. The existence of metrics of nonpositive curvature on the. We show the equivalence of the nonpositivity of the flag curvature with the nonpositive curvature properties of busemann and pedersen for not necessarily reversible berwald manifolds. As an application, we can extend the mostow and prasad rigidity theorems to compact singular orbi spaces of nonpositive curvature which are homotopy equivalent to a quotient of a symmetric space or euclidean building by a cocompact group of isometries. Bridson and andr e hae iger, metric spaces of nonpositive.
The metric target space y,d y is required throughout except in sec tion 5 to be a compact geodesic space of nonpositive alexandrov curvature. This answers a question of gromov geometric group theory, cambridge university press, cambridge, 1993 and is in contrast to the fact that any metric space embeds coarsely into an alexandrov space of nonnegative curvature, as shown by andoni et al. Homotopy problems for harmonic maps 685 of other indication that u is given the induced riemannian metric. Geodesic convexity gconvexity is a classical concept in mathematics and is used extensively in the study of hadamard manifolds and metric spaces of nonpositive curvature 7, 24 i. The metric spaces of nonpositive curvature we have in mind are the socalled cat0 spaces. This answers a question of gromov 1993 and is in contrast to the fact that any metric space embeds coarsely into an alexandrov. Spaces with nonpositive curvature and their ideal boundaries. Moreover, if the curvature upper bound x satisfies x nonpositive or neg ative curvature for a geodesic metric space defined via the catxinequality.
A1 a2 a3 a4 a5 a6 a7 a8 a9 b2 b3 b4 b5 b6 b7 b8 b9 d4 d5 d6 d7 d8 d9 i2m h3 f4 h4 e6 e7 e8 woonjung choi. The study of spaces of nonpositive curvature originated with the discovery of hyperbolic spaces, the work of jacques hadamard at the beginning of. An ultrametric space is a metric space m,dx,y in which 12 dx,z. In mathematics, spaces of nonpositive curvature occur in many contexts and form a generalization of hyperbolic geometry. Uniqueness theorems for harmonic maps into metric spaces. Lectures on spaces of nonpositive curvature werner. Other spaces of interest are spherical and hyperbolic polyhedral spaces, where every simplex has a metric of constant positive or negative curvature. Recent developments extend much of the known theory of classical harmonic maps between smooth riemannian manifolds to the case when the target is a metric space of curvature bounded from above.
Lectures on spaces of nonpositive curvature werner ballmann. International audiencethis is the second edition of a book which appeared in 2005. Besides manifolds, the most important examples of npc spaces are trees, in. Introduction 3 if x is a p,q space, then the interior metric d, which turns every face of x into a regular euclidean polygon of side length 1, has nonpositive alexandrov curvature. Besides manifolds, the most important examples of npc spaces are trees, in particular, spiders. Introduction in this course, we are interested in the geometry of metric spaces which are negatively or, more generally, nonpositively curved in some sense. Metric spaces of nonpositive curvature martin r bridson, andre haefliger. Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. In mathematics, a metric space is a set together with a metric on the set. On the existence of flat planes in spaces of nonpositive curvature 225 triangle for a is a triangle a in the euclidean plane e2 with vertices p,q,r. The metric in the tangent space at the identity is given by the trace of the algebra. These notes largely follow martin bridson and andre haefligers. Singular metrics of nonpositive curvature on branched.
Alexandrov and to examine the structure of groups that act properly on such spaces by isometries. Aleksandrov domains were characterized by introducing a quasilinearization for an arbitrary metric space and formulating a cauchyschwarz inequality, which implies an upper curvature. Metric spaces of nonpositive curvature npc spaces provide another setting for which barycenters exist. Metric spaces, convexity and nonpositive curvature. The local structure induced by the metric is smooth and quadratic.
Introduction riemannian manifolds of nonpositive sectional curvature are a class of manifolds featuring a rich interplay between their geometry, their topology, and their dynamics. In combinatorial group theory, p,q spaces arise as seifertvan kampen diagrams or cayley complexes of small cancellation groups, see lysc, ghsh, bb2. Metric spaces, convexity, and nonpositive curvature. The crucial point is that our lower curvature bounds are stable under an appropriate notion of dconvergence of metric measure spaces. Thus the central objects of study are metric spaces in which every pair of points can be joined by an arc isometric to a compact interval of.
1430 1326 40 623 1491 315 388 904 865 1067 249 714 955 1084 898 1317 74 330 18 1474 1164 336 189 1316 1018 126 1533 488 961 64 1213 709 781 214 695 1110 1014