Clifford Henry Taubes. Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here.
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Every chapter is preceded by a paragraph or two or even more leading the reader gently into the particular part of the forest that lies ahead: a very nice touch. Here are three samples. These introductions tend to go far to give the reader a sense, before the fact, of what the purpose and stresses are of what he is about to encounter.
This is clearly a very good pedagogical ploy. Speaking of pedagogy, I should like to note, in closing, three more pluses and one pseudo-minus. This approach considerably mitigates the single quibble I have, i. On the other hand, via the many, many fine references in the book this reader can easily find other sources for problems and exercises. Smooth manifolds 2. Matrices and Lie groups 3. Introduction to vector bundles 4. Algebra of vector bundles 5. Maps and vector bundles 6.
Vector bundles with fiber Cn 7. Metrics on vector bundles 8. Geodesics 9. Properties of geodesics Principal bundles Covariant derivatives and connections Covariant derivatives, connections and curvature Flat connections and holonomy Curvature polynomials and characteristic classes Covariant derivatives and metrics The Riemann curvature tensor Complex manifolds Holomorphic submanifolds, holomorphic sections and curvature The Hodge star Indexed list of propositions by subject Index.
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Clifford Henry Taubes born February 21,  is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry , and low-dimensional topology. His brother, Gary Taubes , is a science writer. Taubes received his Ph. Soon, he began applying his gauge-theoretic expertise to pure mathematics.
Differential Geometry: Bundles, Connections, Metrics and Curvature
Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. The book uses many of The book uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Keywords: bundles , metrics , curvature , differential geometry , theoretical physics , differential topology , differentiable manifolds , smooth maps , differential forms , vector fields. Forgot password? Don't have an account? All Rights Reserved.