by É Cartan (from lectures delivered by É Cartan at the Sorbonne in 1926–27), translated from Russian V V Goldberg (New Jersey Institute of Technology, USA)
& foreword by S S Chern

In 1926–27, Cartan gave a series of lectures in which he introduced exterior forms at the very beginning and used extensively 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. In 1960, Sergei P Finikov translated from French into Russian his notes of these Cartan's lectures and published them as a book entitled Riemannian Geometry in an Orthogonal Frame. 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 fiber bundle of a submanifold, etc. 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 also edited the Russian edition.

• Method of Moving Frames
• The Theory of Pfaffian Forms
• Integration of Systems of Pfaffian Differential Equations
• Generalization
• The Existence Theorem for a Family of Frames with Given Infinitesimal Components ωⁱ and ωⁱ_j
• The Fundamental Theorem of Metric Geometry
• Vector Analysis in an n-Dimensional Euclidean Space
• The Fundamental Principles of Tensor Algebra
• Tensor Analysis
• The 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
• p-Dimensional Submanifolds in a Riemannian Manifold of n Dimensions