Lectures on Finsler geometry /
In 1854, B. Riemann introduced the notion of curvature for spaces with a family of inner products. There was no significant progress in the general case until 1918, when P. Finsler studied the variation problem in regular metric spaces. Around 1926, L. Berwald extended Riemann's notion of curva...
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| Format: | Electronic eBook |
| Language: | English |
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Singapore ; River Edge, NJ :
World Scientific,
2001
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| Local Note: | ProQuest Ebook Central |
Table of Contents:
- Ch. 1. Finsler spaces. 1.1. Metric spaces. 1.2. Minkowski spaces. 1.3. Finsler spaces
- ch. 2. Finsler m spaces. 2.1. Measure spaces. 2.2. Volume on a Finsler space. 2.3. Hyperplanes in a Minkowski m space. 2.4. Hypersurfaces in a Finsler m space
- ch. 3. Co-area formula. 3.1. Legendre transformations. 3.2. Gradients of functions. 3.3. Co-area formula
- ch. 4. Isoperimetric inequalities. 4.1. Isoperirnetric profiles. 4.2. Sobolev constants and first eigenvalue. 4.3. Concentration of Finsler m spaces. 4.4. Observable diameter
- ch. 5. Geodesies and connection. 5.1. Geodesies. 5.2. Chern connection. 5.3. Covariant derivatives. 5.4. Geodesic flow
- ch. 6. Riemann curvature. 6.1. Birth of the Riemann curvature. 6.2. Geodesic fields. 6.3. Projectively related Finsler metrics
- ch. 7. Non-Riemannian curvatures. 7.1. Cartan torsion. 7.2. Chern curvature. 7.3. S-curvature
- ch. 8. Structure equations. 8.1. Structure equations of Finsler spaces. 8.2. Structure equations of Riemannian metrics. 8.3. Riemann curvature of randers metrics
- ch. 9. Finsler spaces of constant curvature. 9.1. Finsler metrics of constant curvature. 9.2. Examples. 9.3. Randers metrics of constant curvature
- ch. 10. Second variation formula. 10.1. T-curvature. 10.2. Second variation of length. 10.3. Synge theorem
- ch. 11. Geodesies and exponential map. 11.1. Exponential map. 11.2. Jacobi fields. 11.3. Minimality of geodesies. 11.4. Completeness of Finsler spaces
- ch. 12. Conjugate radius and injectivity radius. 12.1. Conjugate radius. 12.2. Injectivity radius. 12.3. Geodesic loops and closed geodesies
- ch. 13. Basic comparison theorems. 13.1. Flag curvature bounded above. 13.2. Positive flag curvature. 13.3. Ricci curvature bounded below. 13.4. Green-Dazord theorem
- ch. 14. Geometry of hypersurfaces. 14.1. Hessian and Laplacian. 14.2. Normal curvature. 14.3. Mean curvature. 14.4. Shape operator
- ch. 15. Geometry of metric spheres. 15.1. Estimates on the normal curvature. 15.2. Convexity of metric balls. 15.3. Estimates on the mean curvature. 15.4. Metric spheres in a convex domain
- ch. 16. Volume comparison theorems. 16.1. Volume of metric balls. 16.2. Volume of tubular neighborhoods. 16.3. Gromov simplicial norms. 16.4. Estimates on the expansion distance
- ch. 17. Morse theory of loop spaces. 17.1. A review on the morse theory. 17.2. Indexes of geodesic loops. 17.3. Energy functional on a loop space. 17.4. Approximation of loop spaces
- ch. 18. Vanishing theorems for homotopy groups. 18.1. Intermediate curvatures. 18.2. Vanishing theorem for homotopy groups. 18.3. Finsler spaces of positive constant curvature
- ch. 19. Spaces of Finsler spaces. 19.1. Gromov-Hausdorff distance. 19.2. Precompactness theorem.
