Geometry IV

This volume of the Encyclopaedia contains two articles,which give a survey of modern research into non-regularRiemannian geometry, carried out mostly by Russianmathematicians.The first article written by Reshetnyak is devoted to thetheory of two-dimensional Riemannian manifolds of boundedcurvature. Concepts of Riemannian geometry, such as the areaandintegral curvature of a set, and the length and integralcurvature of a curve are also defined for these manifolds.Some fundamental results of Riemannian goemetry like theGauss-Bonnet formula are true in the more general caseconsidered in the book.The second article by Berestovskij and Nikolaev is devotedto the theory of metric spaces whose curvature lies betweentwo given constants. The main result is that these spacesare infact Riemannian. This result has importantapplications in global Riemanniangeometry.Both parts cover topics, which have not yet been treated inmonograph form. Hence the book will be immensely useful tograduate students and researchers in geometry, in particularRiemannian geometry.