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Graduate Studies in Mathematics
2009; 671 pp; hardcover
List Price: US$89
Member Price: US$71.20
Order Code: GSM/107
Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle.
This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations.
The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry.
Graduate students and research mathematicians interested in differential geometry.
"This book is certainly a welcome addition to the literature. As noted, the author has an on-line supplement, so the interested reader can follow up on the development of further topics and corrections. One cannot begin to imagine the Herculean amount of work that went into producing a volume of this size and scope, over 660 pages! Future generations will be in the author's debt."
-- Mathematical Reviews
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