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Manifolds and Differential Geometry

Jeffrey M. Lee Texas Tech University, Lubbock, TX
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Softcover ISBN: 978-1-4704-6982-5
Product Code: GSM/107.S
List Price: $95.00 MAA Member Price:$85.50
AMS Member Price: $76.00 Electronic ISBN: 978-1-4704-1170-1 Product Code: GSM/107.E List Price:$89.00
MAA Member Price: $80.10 AMS Member Price:$71.20
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List Price: $142.50 MAA Member Price:$128.25
AMS Member Price: $114.00 Click above image for expanded view Manifolds and Differential Geometry Jeffrey M. Lee Texas Tech University, Lubbock, TX Available Formats:  Softcover ISBN: 978-1-4704-6982-5 Product Code: GSM/107.S  List Price:$95.00 MAA Member Price: $85.50 AMS Member Price:$76.00
 Electronic ISBN: 978-1-4704-1170-1 Product Code: GSM/107.E
 List Price: $89.00 MAA Member Price:$80.10 AMS Member Price: $71.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$142.50 MAA Member Price: $128.25 AMS Member Price:$114.00
• Book Details

Volume: 1072009; 671 pp
MSC: 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 graduate-level 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 in-depth chapter on semi-Riemannian 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 one-semester course on manifolds. There is more than enough material for a year-long 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 semi-Riemannian geometry
• Appendix A. The language of category theory
• Appendix B. Topology
• Appendix C. Some calculus theorems
• Appendix D. Modules and multilinearity

• Reviews

• This book is certainly a welcome addition to the literature. As noted, the author has an on-line 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1072009; 671 pp
MSC: 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 graduate-level 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 in-depth chapter on semi-Riemannian 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 one-semester course on manifolds. There is more than enough material for a year-long 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 semi-Riemannian 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 on-line 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.