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Product Code:  GSM/107.S 
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Electronic ISBN:  9781470411701 
Product Code:  GSM/107.E 
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Book DetailsGraduate Studies in MathematicsVolume: 107; 2009; 671 ppMSC: Primary 58; 53; 22; 55;
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 graduatelevel 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 indepth chapter on semiRiemannian 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 onesemester course on manifolds. There is more than enough material for a yearlong course on manifolds and geometry.ReadershipGraduate students and research mathematicians interested in differential geometry.

Table of Contents

Chapters

Chapter 1. Differentiable manifolds

Chapter 2. The tangent structure

Chapter 3. Immersion and submersion

Chapter 4. Curves and hypersurfaces in Euclidean space

Chapter 5. Lie groups

Chapter 6. Fiber bundles

Chapter 7. Tensors

Chapter 8. Differential forms

Chapter 9. Integration and Stokes’ theorem

Chapter 10. De Rham cohomology

Chapter 11. Distributions and Frobenius’ theorem

Chapter 12. Connections and covariant derivatives

Chapter 13. Riemannian and semiRiemannian geometry

Appendix A. The language of category theory

Appendix B. Topology

Appendix C. Some calculus theorems

Appendix D. Modules and multilinearity


Additional Material

Reviews

This book is certainly a welcome addition to the literature. As noted, the author has an online 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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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 graduatelevel 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 indepth chapter on semiRiemannian 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 onesemester course on manifolds. There is more than enough material for a yearlong course on manifolds and geometry.
Graduate students and research mathematicians interested in differential geometry.

Chapters

Chapter 1. Differentiable manifolds

Chapter 2. The tangent structure

Chapter 3. Immersion and submersion

Chapter 4. Curves and hypersurfaces in Euclidean space

Chapter 5. Lie groups

Chapter 6. Fiber bundles

Chapter 7. Tensors

Chapter 8. Differential forms

Chapter 9. Integration and Stokes’ theorem

Chapter 10. De Rham cohomology

Chapter 11. Distributions and Frobenius’ theorem

Chapter 12. Connections and covariant derivatives

Chapter 13. Riemannian and semiRiemannian geometry

Appendix A. The language of category theory

Appendix B. Topology

Appendix C. Some calculus theorems

Appendix D. Modules and multilinearity

This book is certainly a welcome addition to the literature. As noted, the author has an online 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