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Two-Dimensional Manifolds of Bounded Curvature

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Geometry IV

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 70))

Abstract

The theory of two-dimensional manifolds of bounded curvature is a generalization of two-dimensional Riemannian geometry. Formally a two-dimensional manifold of bounded curvature is a two-dimensional manifold in which there are defined the concepts of the length of a curve, the angle between curves starting from one point, the area of a set, and also the integral curvature of a curve and the integral curvature of a set. For the case when the given manifold is Riemannian, the integral curvature of a curve is equal to the integral of the geodesic curvature along the length of the curve, and the integral curvature of a set is equal to the integral of the Gaussian curvature of the manifold with respect to the area. The remaining concepts in this case have the meaning that is usual in Riemannian geometry. For an arbitrary two-dimensional manifold of bounded curvature the integral curvature is a completely additive set function, which may not admit representations in the form of an integral with respect to area.

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  1. Institute of Mathematics, Siberian Branch of the Russian, Academy of Sciences, Universitetskij pr. 4, 630090, Novosibirsk, Russia

    Yu. G. Reshetnyak

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Reshetnyak, Y.G. (1993). Two-Dimensional Manifolds of Bounded Curvature. In: Reshetnyak, Y.G. (eds) Geometry IV. Encyclopaedia of Mathematical Sciences, vol 70. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02897-1_1

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