Serge Lang (auth.), Serge Lang (eds.)0387943382, 9780387943381, 3540943382
This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes’ theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.). |
Table of contents :
Front Matter….Pages i-xiii
Differential Calculus….Pages 1-19
Manifolds….Pages 20-39
Vector Bundles….Pages 40-63
Vector Fields and Differential Equations….Pages 64-113
Operations on Vector Fields and Differential Forms….Pages 114-152
The Theorem of Frobenius….Pages 153-168
Metrics….Pages 169-190
Covariant Derivatives and Geodesics….Pages 191-224
Curvature….Pages 225-260
Volume Forms….Pages 261-283
Integration of Differential Forms….Pages 284-306
Stokes’ Theorem….Pages 307-320
Applications of Stokes’ Theorem….Pages 321-342
Back Matter….Pages 355-364 |
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