an:03262924
Zbl 0164.22001
Choquet-Bruhat, Y.
G??om??trie diff??rentielle et syst??mes ext??rieurs
FR
Monographies universitaires de math??matiques. 28. Paris: Dunod. xvii, 328 p. (1968).
1968
b
53-01
differential geometry; exterior systems
This latest book on differential geometry by Y. Choquet-Bruhat is based on her lectures given at Sorbonne from 1960 to 1967. It is a textbook which gives a relatively elementary and self-contained introduction to modern differential geometry. The following table of contents should give some idea of this book.
Chapitre I. Vari??t??s diff??rentiables. Fibr??s vectoriels. (A) Vari??t??s diff??rentiables; (B) Champs de vecteurs; (C) Tenseurs; (D) Compl??ments.
Chapitre II. Formes diff??rentielles ext??rieures. Int??gration. (A) Alg??bre ext??rieure. Diff??rentielle ext??rieure; (B) Int??gration des formes diff??rentielles; (C) Formule de Stokes; (D) Compl??ments.
Chapitre III. Vari??t??s riemaniennes. (A) D??finitions; (B) G??od??siques; (C) Compl??ments. Spineurs.
Chapitre IV. Groupes de transformations diff??rentiables. (A) Non-titr??; (B) Invariant int??graux; (C) Compl??ments. Op??rateurs diff??rentiels invariants sur un groupe de Lie.
Chapitre V. Syst??me diff??rentiels ext??rieurs. (A) G??n??ralit??s; (B) Syst??mes de Pfaff; (C) Syst??mes diff??rentiels ext??rieurs; (D) Equations aux d??riv??es partielles du premier ordre.
Chapitre VI. Connexions. (A) Connexions lin??aires. Premi??re d??finition; (B) Connexions sur un espace fibr?? principal; (C) Connexions lin??aires. Deuxi??me d??finition.
Chapitre VII. Applications aux sciences physiques. (A) M??canique analytique classique; (B) Relativit?? restreinte; (C) Electromagn??tisme; (D) Relativit?? g??n??rale et m??canique des fluides relativistes.
Here are some of the characteristics of this book.
(1) A systematic treatment of the hyperbolic geometry (i.e., the geometry of indefinite Riemannian metric with signature + -);
(2) An introduction to the theory of spinors on a hyperbolic Riemannian manifold;
(3) Applications to physics, particularly, to classical mechanics, relativity and electromagnetism;
(4) A fairly detailed treatment of differential systems;
(5) A large number of problems at the end of each chapter except Chapitre VII.
Definitions and theorems are generally stated clearly. Most of local results are proved in full. On the other hand, proofs of global results are often referred to other books, but rarely to original papers. This is a commendable approach since the book is written for physicists as well as to students of mathematics. This book will be most useful as an introductory textbook for advanced undergraduates and for beginning graduate students rather than as a reference book for researchers.
S. Kobayashi