Riemannian geometry in an orthogonal frame : from lectures delivered by Elie Cartan at the Sorbonne in 1926-1927 / translated from Russian by Vladislav V. Goldberg ; foreword by S. S. Chern.

Format:

Book

Author/Creator:

Cartan, Elie, 1869-1951.

Contributor:

Finikov, S. P. (Sergeĭ Pavlovich), 1883-1964.

Standardized Title:

Rimanova geometrii͡a v ortogonalʹnom repere. English

Language:

English

Subjects (All):

Geometry, Riemannian.

Geometry.

Physical Description:

1 online resource (280 p.)

Edition:

1st ed.

Place of Publication:

River Edge, NJ : World Scientific, c2001.

Language Note:

English

Summary:

Foreword by <i>S S</i> <i>Chern</i> 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

Contents:

Contents ; Foreword ; Translator's Introduction ; Preface to the Russian Edition ; PRELIMINARIES ; Chapter 1 Method of Moving Frames ; 1. Components of an infinitesimal displacement ; 2. Relations among 1-forms of an orthonormal frame

3. Finding the components of a given family of trihedrons 4. Moving frames ; 5. Line element of the space ; 6. Contravariant and covariant components ; 7. Infinitesimal affine transformations of a frame ; Chapter 2 The Theory of Pfaffian Forms ; 8. Differentiation in a given direction

9. Bilinear covariant of Frobenius 10. Skew-symmetric bilinear forms ; 11. Exterior quadratic forms ; 12. Converse theorems. Cartan's Lemma ; 13. Exterior differential ; Chapter 3 Integration of Systems of Pfaffian Differential Equations ; 14. Integral manifold of a system

15. Necessary condition of complete integrability 16. Necessary and sufficient condition of complete integrability of a system of Pfaffian equations ; 17. Path independence of the solution

18. Reduction of the problem of integration of a completely integrable system to the integration of a Cauchy system 19. First integrals of a completely integrable system ; 20. Relation between exterior differentials and the Stokes formula ; 21. Orientation ; Chapter 4 Generalization

22. Exterior differential forms of arbitrary order

Notes:

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

Includes bibliographical references and index.

ISBN:

9786611948030

9781281948038

1281948039

9789812799715

9812799710

OCLC:

879025492