Loring W. Tu Bücher

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.

This text serves as a graduate-level introduction to differential geometry for mathematics and physics students, tracing the historical evolution of connection and curvature concepts to elucidate the Chern–Weil theory of characteristic classes on a principal bundle. Key milestones in differential geometry, such as Gauss' Theorema Egregium and the Gauss–Bonnet theorem, are explored. The book includes exercises that challenge the reader's understanding and highlight extensions of the theory. A basic familiarity with manifolds is required, with a deeper knowledge of differential forms necessary after the first chapter. Understanding de Rham cohomology is essential for the final third of the text. Prerequisite material is found in the author's earlier work, which can be mastered in one semester. Appendix A reviews fundamental manifold theory to aid readers. To enhance self-containment, sections on algebraic constructs like the tensor product and exterior power are also included. Differential geometry, rooted in the seventeenth century with Newton and Leibniz, gained prominence in the nineteenth century through Gauss and Riemann's contributions. Today, it is crucial for comprehending physical theories, including Einstein's general relativity, and has applications in various mathematical fields, such as topology and algebraic geometry. Its relevance extends to group theory and probability theory, making it an essential tool for math