Manifolds and Differential Geometry
About this Title
Jeffrey M. Lee, Texas Tech University, Lubbock, TX
Publication: Graduate Studies in Mathematics
Publication Year 2009: Volume 107
ISBNs: 978-0-8218-4815-9 (print); 978-1-4704-1170-1 (online)
MathSciNet review: MR2572292
MSC: Primary 53-01; Secondary 58-01
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.
Table of Contents
- 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