Riemannian geometry in an orthogonal frame : from lectures delivered by Élie Cartan at the Sorbonne in 1926-27 / translated from Russian by Vladislav V. Goldberg ; foreword by S.S. Chern.

Translated from the 1960 Russian ed., which was translated and edited from original lecture notes by S.P. Finikov as, Rimanova geometriya v orthogonalʹnom repere.

Includes bibliographical references and index.

Print version record.

Elie Cartan's book "Geometry of Riemannian Manifolds" (1928) was one of the best introductions to his methods. It was based on lectures given by the author at the Sorbonne in the academic year 1925-26. A modernized and extensively augmented edition appeared in 1946 (2nd printing, 1951; 3rd printing, 1988). Cartan's lectures in 1926-27 were different - he introduced exterior forms at the very beginning and used orthogonal frames throughout to investigate the geometry of Riemannian manifolds. In this course, he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. The lectures were translated into Russian in the book "Riemannian Geometry in an Orthogonal Frame" (1960). This book has many innovations, such as the notion of intrinsic normal differentiation and the Gaussian torsion of a submanifold in a Euclidean multidimensional space or in a space of constant curvature, an affine connection defined in a normal fibre bundle of a submanifold, and so on. This book was available neither in English nor in French. It has now been translated into English by Vladislav V. Goldberg, currently Distinguished Professor of Mathematics at the New Jersey Institute of Technology, USA, who edited the Russian edition.

Machine generated contents note: Method of Moving Frames -- Theory of Pfaffian Forms -- Integration of Systems of Pfaffian Differential Equations -- Generalization -- Existence Theorem for a Family of Frames with Given Infinitesimal Components w[superscript i] and w[superscript i][subscript j] -- Fundamental Theorem of Metric Geometry -- Vector Analysis in an n-Dimensional Euclidean Space -- Fundamental Principles of Tensor Algebra -- Tensor Analysis -- Notion of a Manifold -- Locally Euclidean Riemannian Manifolds -- Euclidean Space Tangent at a Point -- Osculating Euclidean Space -- Euclidean Space of Conjugacy Along a Line -- Space with a Euclidean Connection -- Riemannian Curvature of a Manifold -- Spaces of Constant Curvature -- Geometric Construction of a Space of Constant Curvature -- Variational Problems for Geodesics -- Distribution of Geodesics Near a Given Geodesic -- Geodesic Surfaces -- Lines in a Riemannian Manifold -- Surfaces in a Three-Dimensional Riemannian Manifold -- Forms of Laguerre and Darboux.

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