05188nam 2200673Ia 450 991045435430332120200520144314.01-281-94803-99786611948030981-279-971-0(CKB)1000000000537964(EBL)1681703(OCoLC)879025492(SSID)ssj0000238239(PQKBManifestationID)11228439(PQKBTitleCode)TC0000238239(PQKBWorkID)10222268(PQKB)11743758(MiAaPQ)EBC1681703(WSP)00004808(Au-PeEL)EBL1681703(CaPaEBR)ebr10255809(CaONFJC)MIL194803(EXLCZ)99100000000053796420020311d2001 uy 0engur|n|---|||||txtccrRiemannian geometry in an orthogonal frame[electronic resource] from lectures delivered by Élie Cartan at the Sorbonne in 1926-1927 /translated from Russian by Vladislav V. Goldberg ; foreword by S. S. ChernRiver Edge, NJ World Scientificc20011 online resource (280 p.)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.981-02-4746-X Includes bibliographical references and index.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 frame3. 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 direction9. 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 system15. Necessary condition of complete integrability 16. Necessary and sufficient condition of complete integrability of a system of Pfaffian equations ; 17. Path independence of the solution18. 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 Generalization22. Exterior differential forms of arbitrary orderForeword 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 innovationsGeometry, RiemannianGeometryElectronic books.Geometry, Riemannian.Geometry.516.3/73516.373Cartan Elie1869-1951.42832Finikov S. P(Sergeĭ Pavlovich),1883-1964.980470MiAaPQMiAaPQMiAaPQBOOK9910454354303321Riemannian geometry in an orthogonal frame2237144UNINA