Riemannian geometry in an orthogonal frame. From lectures delivered by Élie Cartan at the Sorbonne 1926--27. With a preface to the Russian edition by S. P. Finikov. Translated from the 1960 Russian edition by Vladislav V. Goldberg and with a foreword by S. S. Chern (Q2781751)

S. P. Finikov took notes of É. Cartan's lectures on the geometry of Riemannian manifolds he delivered at the Sorbonne in 1926-1927. In this course Cartan introduced exterior differential forms as the outset and adapted orthogonal coordinate systems. Finikov translated his notes in Moscow in 1960 and published them as the ``Rimanova geometriya v ortogonal'nom repere'' (1960; Zbl 0099.37201). By the suitable choice of examples the reader could know that Cartan's moving frame (repère mobile) can be applied to affine frames, or indeed to the frames subjected to general Lie transformation groups. In 2001 V. V. Goldberg translated this book into English.NEWLINENEWLINENEWLINEThe book begins with preliminaries consisting of four chapters. The reader immediately sees Cartan's \(\omega\)'s, the components of infinitesimal displacement of vectors. For a moving frame formed by the position vector \({\mathbf M}\) (\(=\vec{OM}\)) and three vectors \(e_i\) (\(i=1,2,3\)), in the three-dimensional Euclidean space its displacement yields the structure equations in the space expressed in terms of \(\omega^i\) and \(\omega^i_j\). With the inner product \(g_{ij}=(e_i,e_j)\), the length of the line element takes the form \(ds^2=g_{ij} \omega^i\omega^j\). When the \(e\)'s are orthonormal, \(g_{ij}= \delta_{ij}\) and \(\omega^i_j+\omega_i^j=0\). For solving the structure equations and also for the use of all future discussions Pfaffian forms, exterior products and exterior differentiation are introduced in the remaining three chapters.NEWLINENEWLINENEWLINEThe book consists of five main sections A, B, C, D and E. A states the geometry of Euclidean space. The application of exterior differentiation to the system of structure equations gives rise to the condition of complete integrability of the system. Knowing that the condition assures the unique solutions \({\mathbf M}\) and \(e\)'s, Weyl's rigidity theorem on the \(\omega^i\) follows as fundamental. The last half of A is concerned with the vector and tensor analysis in an \(n\)-dimensional Euclidean space.NEWLINENEWLINENEWLINESection B gives the definition of manifolds characterized by the existence of a coordinate neighborhood \((u^i)\) at each point. A manifold metrized by \(ds^2=g_{ij} du^idu^j\) is a Riemannian manifold. It is regular, if the form is positive definite, and is locally Euclidean, if in each region the structure equations satisfy the complete integrability condition. Then, for a regular, locally compact and locally Euclidean manifold a small region can be developed in the Euclidean space, and if it is done along a cycle, the resulting displacement \(S^i\) (\(i=1,2,\dots\)) of the frame forms Cartan's holonomy group of the manifold. The osculating Euclidean space of a manifold yields \(\omega^i_j=\Gamma^i_{jk}du^k\), which leads to defining Euclidean connection and then to the absolute differentiation.NEWLINENEWLINENEWLINEThe curvature and torsion that result from the connection are discussed in Section C.NEWLINENEWLINENEWLINESection D deals with the variation of the length of geodesics. The first and second variation are formulated in terms of the orthogonal system \(e^1,e^2,\dots,e^n\) and in the latter, of the sectional curvature along geodesics. The discussion extends to geodesic and totally geodesic subspaces. NEWLINENEWLINENEWLINEThe last section E reveals the geometry of submanifolds. Beginning with the theory of curves in a Riemannian manifold that is based on the associated Frenet apparatus, the surface theory in three-dimensional Riemannian manifolds is developed. Therein, the second fundamental form is introduced, giving the second structure equation of the surfaces. These are generalized to the case of \(p\)-dimensional submanifolds in an \(n\)-dimensional Riemannian manifold. As examples and particular cases, hypersurfaces and subspaces in a four-dimensional manifold are treated.NEWLINENEWLINENEWLINEHerewith, on behalf of students of advanced graduate knowledge and of overall geometers now and in the future the reviewer expresses his admiration to Professors S. P. Finikov and V. V. Goldberg for their utmost endeavor of translating É. Cartan's ever valuable lectures second to those that appear in ``Lȩcons sur la géométrie des espaces de Riemann'', Paris, Gauthier-Villars (1928; JFM 54.0755.01).