Introduction to smooth manifolds
This book is an introductory graduate-level textbook on the theory of smooth manifolds, for students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. It is a natural sequel to the author's last book, Introduction to Topological Manifolds(2000). While the subject is often called "differential geometry," in this book the author has decided to avoid use of this term because it applies more specifically to the study of smooth manifolds endowed with some extra structure, such as a Riemannian metric, a symplectic structure, a Lie group structure, or a foliation, and of the properties that are invariant under maps that preserve the structure. Although this text addresses these subjects, they are treated more as interesting examples to which to apply the general theory than as objects of study in their own right. A student who finishes this book should be well prepared to go on to study any of these specialized subjects in much greater depth
1 online resource (xvii, 628 pages) : illustrations
9780387217529, 9780387954950, 9781280189784, 9780387954486, 0387217525, 0387954953, 1280189789, 0387954481
666929817
Preface
Smooth Manifolds
Smooth Maps
Tangent Vectors
Vector Fields
Vector Bundles
The Cotangent Bundle
Submersions, Immersions, and Embeddings
Submanifolds
Embedding and Approximation Theorems
Lie Group Actions
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