Shigeyuki Morita

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 book is a comprehensive introduction to differential forms. It begins with a quick presentation of the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results about them, such as the de Rham and Frobenius theorems.The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated in the book is a detailed description of the Chern-Weil theory. With minimal prerequisites, the book can serve as a textbook for an advanced undergraduate or a graduate course in differential geometry.

Geometry of Differential Forms

Toshikazu Sunada

Kenji Ueno

Kōji Shiga

Three volumes originating from a series of lectures in mathematics given by professors of Kyoto University in Japan for high school students.

The Interplay Between Topology, Functions, Geometry, and Algebra

A Mathematical Gift, III

This is the first of three volumes originated from a series of lectures in mathematics given by professors of Kyoto University in Japan for high school students. The main purpose of the lectures was to show the listeners the beauty and liveliness of mathematics using the material that is accessible to people with little preliminary knowledge. The first chapter of the book talks about the geometry and topology of surfaces. Among other topics the authors discuss the Poincare-Hopf theorem about critical points of vector fields on surfaces and the Gauss-Bonnet theorem about the relation between the curvature and topology (Euler characteristics). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincare approach to dimension. It also discusses the structure of three-dimensional manifolds, proving, in particular, that the three-dimensional sphere is the union of two doughnuts.

Shigeyuki Morita

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