By Yu.G. Reshetnyak, Yu.G. Reshetnyak, E. Primrose, V.N. Berestovskij, I.G. Nikolaev

This quantity of the Encyclopaedia includes articles, which provide a survey of contemporary examine into non-regular Riemannian geometry, conducted typically by means of Russian mathematicians. the 1st article written through Reshetnyak is dedicated to the speculation of two-dimensional Riemannian manifolds of bounded curvature. thoughts of Riemannian geometry, akin to the realm and quintessential curvature of a suite, and the size and quintessential curvature of a curve also are outlined for those manifolds. a few basic result of Riemannian goemetry just like the Gauss-Bonnet formulation are actual within the extra basic case thought of within the e-book. the second one article through Berestovskij and Nikolaev is dedicated to the idea of metric areas whose curvature lies among given constants. the most result's that those areas are actually Riemannian. This consequence has very important purposes in international Riemannian geometry. either components disguise themes, that have now not but been handled in monograph shape. for that reason the ebook can be immensely precious to graduate scholars and researchers in geometry, specifically Riemannian geometry.

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I. Two-Dimensional Manifolds of Bounded Curvature 65 Let K, and K, be rectifiable simple closed curves in the metric spaces M, and M,, and cpa map of K, onto K,. We shall say that cpmaps K, onto K, with the lengths of arcs preserved if cpmaps K i onto K, one-to-one and takes any arc of K, into an arc of K, of the same length. Suppose we are given metric spaces (M, , pi) and (M,, p2). 1. (Reshetnyak (1961a)). Then there is a convex cone Q and a map cp: Q -+ R such that the following conditions are satisfied: 1) the curvature of Q doesnot exceed w’(R); 2) cpis a contracting map; 3) the boundary of the cone Q is transformed by the map cpinto the boundary of R with the lengths of arcs preserved.

For the case when X is a boundary point of M the constructions are similar. Manifolds of Bounded Curvature 51 The proof of the theorem reduces to a consideration of neighbourhoods of points of M that arise as a result of pasting together boundary points of the manifolds M,. Suppose, say, that a point X is obtained by pasting together points Xi E M,i, i = 1, 2, . . , m. In M,, the point Xi has a neighbourhood isometric to some circular sector A(&, h). By pasting together these neighbourhoods we obtain a neighbourhood of X.

The given theorem establishes a necessary condition that is satisfied by any two-dimensional manifold of bounded curvature. This condition is also suflicient. In the part that touches on the sufficiency, the conditions imposed on the sequence of polyhedral metrics pn can be weakened. Namely, the following theorem, which we shall call the second theorem on approximation, is true. 2.

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Geometry IV by Yu.G. Reshetnyak, Yu.G. Reshetnyak, E. Primrose, V.N. Berestovskij, I.G. Nikolaev

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