Analysis and algebra on differentiable manifolds: a workbook for students and teachers
This volume intends to provide material for the practical side of standard courses on analysis and algebra on differentiable manifolds at the middle level, corresponding to advanced undergraduate and graduate years. The exercises focus mainly on Lie groups, fibre bundles, and Riemannian geometry. This is the second edition of this problem book for students, now containing 412 completely solved exercises on differentiable manifolds, Lie theory, fibre bundles and Riemannian manifolds. The exercises range from elementary computations to rather sophisticated tools. Many of the definitions and theorems used throughout are explained in the first section of each chapter where they appear. A 56-page collection of formulae is included which can be useful as an aide-mémoire, even for teachers and researchers on those topics. The abstract results and properties are illustrated with 68 figures. With respect to the first edition, the main novelties in this volume are the following: (i) 76 new problems; (ii) a section devoted to a generalization of Gauss' lemma; (iii) a short novel section dealing with some properties of the energy of Hopf vector fields; (iv) an expanded collection of formulae and tables; (v) an extended bibliography. The content of the volume is divided into six chapters: 1. Differentiable Manifolds; 2. Tensor Fields and Differential Forms; 3. Integration on Manifolds; 4. Lie Groups; 5. Fibre Bundles; 6. Riemannian Geometry. An additional final chapter contains some important formulae and tables. This reviewer appreciates that the present enlarged edition will be useful to advanced undergraduate and graduate students of mathematics, theoretical physics and some branches of engineering with a rudimentary knowledge of linear and multilinear algebra.
scientific article; zbMATH DE number 6111884
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