Author has written several excellent Springer books.; This book is a sequel to Introduction to Topological Manifolds; Careful and illuminating explanations, excellent diagrams and exemplary motivation; Includes short preliminary sections before each section explaining what is ahead and why
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.
Table of contents:
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.
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.
Table of contents:
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.