Smooth manifolds-the higher-dimensional analogues of smooth curves and surfaces-are fundamental objects in modern mathematics. Drawing on algebra, topology, and analysis, they also play key roles in classical mechanics, general relativity, quantum field theory, and data analysis. This streamlined introduction develops the theory of manifolds with the goal of helping readers achieve a rapid mastery of the essential topics. By the end of the book, readers will be able to compute, for simple spaces, one of the most basic ...
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Smooth manifolds-the higher-dimensional analogues of smooth curves and surfaces-are fundamental objects in modern mathematics. Drawing on algebra, topology, and analysis, they also play key roles in classical mechanics, general relativity, quantum field theory, and data analysis. This streamlined introduction develops the theory of manifolds with the goal of helping readers achieve a rapid mastery of the essential topics. By the end of the book, readers will be able to compute, for simple spaces, one of the most basic topological invariants of a manifold: its de Rham cohomology. Along the way, they will gain the knowledge and skills needed for further study in geometry and topology. The third edition emphasizes clarity and simplification. While preserving the overall structure of the second edition, every section has been rewritten, with new or simplified proofs, clearer exposition, and additional exercises, hints, and solutions. This book is suitable for a one-semester graduate or advanced undergraduate course, as well as for independent study. The necessary point-set topology appears in a twenty-page appendix; other appendices review material from real analysis and linear algebra. Hints and solutions accompany many exercises and problems. Requiring only minimal undergraduate prerequisites, An Introduction to Manifolds also provides an excellent foundation for the author's companion volumes: Differential Geometry: Connections, Curvature and Characteristic Classes; Differential Forms in Algebraic Topology (with Raoul Bott); Introductory Lectures on Equivariant Cohomology.
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