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Lectures on Finsler geometry / Zhongmin Shen.

By: Shen, Zhongmin, 1963-Material type: TextTextPublication details: Singapore ; River Edge, NJ : World Scientific, ©2001. Description: 1 online resource (xiv, 307 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9789812811622; 9812811621; 1281960659; 9781281960658; 9789810245313; 9810245319Subject(s): Finsler spaces | Geometry, Differential | Finsler, Espaces de | Géométrie différentielle | MATHEMATICS -- Geometry -- Analytic | Finsler spaces | Geometry, Differential | Finsler, Espaces de | Géométrie différentielleGenre/Form: Electronic book. | Electronic books. Additional physical formats: Print version:: Lectures on Finsler geometry.DDC classification: 516.3/73 LOC classification: QA689 | .S54 2001ebOnline resources: Click here to access online
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.
Summary: 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 curvature to regular metric spaces and introduced an important non-Riemannian curvature using his connection for regular metrics. Since then, Finsler geometry has developed steadily. In his Paris address in 1900, D. Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler's category. Finsler geometry has broader applications in many areas of science and will continue to develop through the efforts of many geometers around the world. Usually, the methods employed in Finsler geometry involve very complicated tensor computations. Sometimes this discourages beginners. Viewing Finsler spaces as regular metric spaces, the author discusses the problems from the modern geometry point of view. The book begins with the basics on Finsler spaces, including the notions of geodesics and curvatures, then deals with basic comparison theorems on metrics and measures and their applications to the Levy concentration theory of regular metric measure spaces and Gromov's Hausdorff convergence theory.
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Includes bibliographical references (pages 299-304) and index.

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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 curvature to regular metric spaces and introduced an important non-Riemannian curvature using his connection for regular metrics. Since then, Finsler geometry has developed steadily. In his Paris address in 1900, D. Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler's category. Finsler geometry has broader applications in many areas of science and will continue to develop through the efforts of many geometers around the world. Usually, the methods employed in Finsler geometry involve very complicated tensor computations. Sometimes this discourages beginners. Viewing Finsler spaces as regular metric spaces, the author discusses the problems from the modern geometry point of view. The book begins with the basics on Finsler spaces, including the notions of geodesics and curvatures, then deals with basic comparison theorems on metrics and measures and their applications to the Levy concentration theory of regular metric measure spaces and Gromov's Hausdorff convergence theory.

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.

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