- 416 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android
eBook - ePub
Curvature in Mathematics and Physics
About this book
This original text for courses in differential geometry is geared toward advanced undergraduate and graduate majors in math and physics. Based on an advanced class taught by a world-renowned mathematician for more than fifty years, the treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool.
Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds. A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civita's theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer's, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms.
Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds. A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civita's theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer's, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms.
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Please note we cannot support devices running on iOS 13 and Android 7 or earlier. Learn more about using the app.
Yes, you can access Curvature in Mathematics and Physics by Shlomo Sternberg in PDF and/or ePUB format, as well as other popular books in Mathematics & Differential Geometry. We have over one million books available in our catalogue for you to explore.
Information
Table of contents
- Cover Page
- Title Page
- Copyright Page
- 0.1 Introduction
- Contents
- 1. Gauss’s theorema egregium.
- 2. Rules of calculus.
- 3. Connections on the tangent bundle.
- 4. Levi-Civita’s theorem.
- 5. Bi-invariant metrics on a Lie group.
- 6. Cartan calculations
- 7. Gauss’s lemma.
- 8. Variational formulas.
- 9. The Hopf-Rinow theorem.
- 10. Curvature, distance, and volume.
- 11. Review of special relativity.
- 12. The star operator and electromagnetism.
- 13. Preliminaries to the Einstein equations.
- 14. Die Grundlagen der Physik.
- 15. The Frobenius theorem.
- 16. Connections on principal bundles.
- 17. Reduction of principal bundles.
- 18. Superconnections.
- 19. Semi-Riemannian submersions.
- Bibliography
- Index
- Back Cover