Non-positive curvature

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.

See also

• Margulis lemma

References

• Ballmann, Werner (1995). Lectures on spaces of nonpositive curvature. DMV Seminar 25. Basel: Birkhäuser Verlag. pp. viii+112. ISBN 3-7643-5242-6.  MR 1377265
• Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319. Berlin: Springer-Verlag. pp. xxii+643. ISBN 3-540-64324-9.  MR 1744486
• Papadopoulos, Athanase (2004, Second edition 2014). Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics Vol. 6. Zürich: European Mathematical Society. p. 298. ISBN 978-3-03719-010-4.  Check date values in: |date= (help) MR 2132506