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Mathematics - Geometry & Topology | Introduction to Smooth Manifolds

Introduction to Smooth Manifolds

Series: Graduate Texts in Mathematics, Vol. 218

Lee, John M.

2003, XVII, 631 p.

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Manifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for under­ standing "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, com­ puter graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics-theoretical physics. No longer a specialized subject that is studied only by differential geometers, manifold theory is now one of the basic skills that all mathematics students should acquire as early as possible. Over the past few centuries, mathematicians have developed a wondrous collection of conceptual machines designed to enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional zero set of a polynomial in four complex variables, or the lO-dimensional manifold of 5 x 5 orthogonal ma­ trices, as easily as we think about the familiar 2-dimensional sphere in ]R3.

Content Level » Graduate

Keywords » Algebra - Cohomology - De Rham cohomology - Fundamental group - foliation - homology - vector bundle

Related subjects » Geometry & Topology

Table of contents 

Preface * Smooth Manifolds * Smooth Maps * Tangent Vectors * Vector Fields * Vector Bundles * The Cotangent Bundle * Submersions, Immersions, and Embeddings * Submanifolds * Lie Groups Actions * Embedding and Approximation Theorems * Tensors * Differential Forms * Orientations * Integration on Manifolds * De Rham Cohomology * The de Rham Theorem * Integral Curves and Flows * Lie Derivatives * Integral Manifolds and Foliations * Lie Groups and Their Lie Algebras * Appendix: Review of Prerequisites * References * Index

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