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Since the times of Gauss, Riemann, and Poincare, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. This title is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and...

Addresses the interplay between topology, functions, geometry, and algebra. This title offers serious mathematics in a reader-friendly style. It includes exercises and many figures illustrating the main concepts. It is suitable for advanced high-school stu

Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fibre bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side.