In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature extends to the category of geodesic metric spaces, where one can use comparison triangles to quantify the curvature of a space; in this context, non-positively curved spaces are known as (locally) CAT(0) spaces.
If is a closed, orientable Riemann surface then it follows from the Uniformization theorem that may be endowed with a complete Riemannian metric with constant Gaussian curvature of either , or . As a result of the Gauss–Bonnet theorem one can determine that the surfaces which have a Riemannian metric of constant curvature i.e. Riemann surfaces with a complete, Riemannian metric of non-positive constant curvature, are exactly those whose genus is at least . The Uniformization theorem and the Gauss–Bonnet theorem can both be applied to orientable Riemann surfaces with boundary to show that those surfaces which have a non-positive Euler characteristic are exactly those which admit a Riemannian metric of non-positive curvature. There is therefore an infinite family of homeomorphism types of such surfaces whereas the Riemann sphere is the only closed, orientable Riemann surface of constant Gaussian curvature .
The definition of curvature above depends upon the existence of a Riemannian metric and therefore lies in the field of geometry. However the Gauss–Bonnet theorem ensures that the topology of a surface places constraints on the complete Riemannian metrics which may be imposed on a surface so the study of metric spaces of non-positive curvature is of vital interest in both the mathematical fields of geometry and topology. Classical examples of surfaces of non-positive curvature are the Euclidean plane and flat torus (for curvature ) and the hyperbolic plane and pseudosphere (for curvature ). For this reason these metrics as well as the Riemann surfaces which on which they lie as complete metrics are referred to as Euclidean and hyperbolic respectively.
Generalizationsedit
The characteristic features of the geometry of non-positively curved Riemann surfaces are used to generalize the notion of non-positive beyond the study of Riemann surfaces. In the study of manifolds or orbifolds of higher dimension, the notion of sectional curvature is used wherein one restricts one's attention to two-dimensional subspaces of the tangent space at a given point. In dimensions greater than the Mostow–Prasad rigidity theorem ensures that a hyperbolic manifold of finite area has a unique complete hyperbolic metric so the study of hyperbolic geometry in this setting is integral to the study of topology.
In an arbitrary geodesic metric space the notions of being Gromov hyperbolic or of being a CAT(0) space generalise the notion that on a Riemann surface of non-positive curvature, triangles whose sides are geodesics appear thin whereas in settings of positive curvature they appear fat. This notion of non-positive curvature allows the notion of non-positive curvature is most commonly applied to graphs and is therefore of great use in the fields of combinatorics and geometric group theory.
Ballmann, Werner (1995). Lectures on spaces of nonpositive curvature. DMV Seminar 25. Basel: Birkhäuser Verlag. pp. viii+112. ISBN3-7643-5242-6. MR1377265
Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 319. Berlin: Springer-Verlag. pp. xxii+643. ISBN3-540-64324-9. MR1744486
Papadopoulos, Athanase (2014) [2004]. Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics Vol. 6. Zürich: European Mathematical Society. p. 298. ISBN978-3-03719-010-4. MR2132506
January 01, 1970
positive, curvature, mathematics, spaces, positive, curvature, occur, many, contexts, form, generalization, hyperbolic, geometry, category, riemannian, manifolds, consider, sectional, curvature, manifold, require, that, this, curvature, everywhere, less, than,. In mathematics spaces of non positive curvature occur in many contexts and form a generalization of hyperbolic geometry In the category of Riemannian manifolds one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero The notion of curvature extends to the category of geodesic metric spaces where one can use comparison triangles to quantify the curvature of a space in this context non positively curved spaces are known as locally CAT 0 spaces Contents 1 Riemann Surfaces 2 Generalizations 3 See also 4 ReferencesRiemann Surfaces editIf S displaystyle S nbsp is a closed orientable Riemann surface then it follows from the Uniformization theorem that S displaystyle S nbsp may be endowed with a complete Riemannian metric with constant Gaussian curvature of either 0 displaystyle 0 nbsp 1 displaystyle 1 nbsp or 1 displaystyle 1 nbsp As a result of the Gauss Bonnet theorem one can determine that the surfaces which have a Riemannian metric of constant curvature 0 displaystyle 0 nbsp 1 displaystyle 1 nbsp i e Riemann surfaces with a complete Riemannian metric of non positive constant curvature are exactly those whose genus is at least 1 displaystyle 1 nbsp The Uniformization theorem and the Gauss Bonnet theorem can both be applied to orientable Riemann surfaces with boundary to show that those surfaces which have a non positive Euler characteristic are exactly those which admit a Riemannian metric of non positive curvature There is therefore an infinite family of homeomorphism types of such surfaces whereas the Riemann sphere is the only closed orientable Riemann surface of constant Gaussian curvature 1 displaystyle 1 nbsp The definition of curvature above depends upon the existence of a Riemannian metric and therefore lies in the field of geometry However the Gauss Bonnet theorem ensures that the topology of a surface places constraints on the complete Riemannian metrics which may be imposed on a surface so the study of metric spaces of non positive curvature is of vital interest in both the mathematical fields of geometry and topology Classical examples of surfaces of non positive curvature are the Euclidean plane and flat torus for curvature 0 displaystyle 0 nbsp and the hyperbolic plane and pseudosphere for curvature 1 displaystyle 1 nbsp For this reason these metrics as well as the Riemann surfaces which on which they lie as complete metrics are referred to as Euclidean and hyperbolic respectively Generalizations editThe characteristic features of the geometry of non positively curved Riemann surfaces are used to generalize the notion of non positive beyond the study of Riemann surfaces In the study of manifolds or orbifolds of higher dimension the notion of sectional curvature is used wherein one restricts one s attention to two dimensional subspaces of the tangent space at a given point In dimensions greater than 2 displaystyle 2 nbsp the Mostow Prasad rigidity theorem ensures that a hyperbolic manifold of finite area has a unique complete hyperbolic metric so the study of hyperbolic geometry in this setting is integral to the study of topology In an arbitrary geodesic metric space the notions of being Gromov hyperbolic or of being a CAT 0 space generalise the notion that on a Riemann surface of non positive curvature triangles whose sides are geodesics appear thin whereas in settings of positive curvature they appear fat This notion of non positive curvature allows the notion of non positive curvature is most commonly applied to graphs and is therefore of great use in the fields of combinatorics and geometric group theory See also editMargulis lemmaReferences editBallmann Werner 1995 Lectures on spaces of nonpositive curvature DMV Seminar 25 Basel Birkhauser Verlag pp viii 112 ISBN 3 7643 5242 6 MR1377265 Bridson Martin R Haefliger Andre 1999 Metric spaces of non positive curvature Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences Vol 319 Berlin Springer Verlag pp xxii 643 ISBN 3 540 64324 9 MR1744486 Papadopoulos Athanase 2014 2004 Metric Spaces Convexity and Nonpositive Curvature IRMA Lectures in Mathematics and Theoretical Physics Vol 6 Zurich European Mathematical Society p 298 ISBN 978 3 03719 010 4 MR2132506 Retrieved from https en wikipedia org w index php title Non positive curvature amp oldid 1200515559, wikipedia, wiki, book, books, library,
