New features of this revised and expanded second edition include: Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.Įlementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included.ĭifferential geometry, as its name implies, is the study of geometry using differential calculus. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. A knowledge of de Rham cohomology is required for the last third of the text. After the first chapter, it becomes necessary to understand and manipulate differential forms. Initially, the prerequisites for the reader include a passing familiarity with manifolds. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory.
The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. This text presents a graduate-level introduction to differential geometry for mathematics and physics students.
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