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New Jersey : Higher Education Press : World Scientific, 2016

xii, 393 stran ; 24 cm

ISBN 978-981-4704-90-8 (vázáno)

Obsahuje bibliografii na stranách 379-387 a rejstřík

Book is an introduction to the basics of Finsler geometry with recent developments in its area. It includes local geometry as well as global geometry of Finsler manifolds. In Part I, the authors discuss differential manifolds, Finsler metrics, the Chern connection, Riemannian and non-Riemannian quantities. Part II is written for readers who would like to further their studies in Finsler geometry. It covers projective transformations, comparison theorems, fundamental group, minimal immersions, harmonic maps, Einstein metrics, conformal transformations, amongst other related topics.The authors made great efforts to ensure that the contents are accessible to senior undergraduate students, graduate students, mathematicians and scientists. (z úvodu).

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Preface v // Foundations 1 // 1. Differentiable Manifolds 3 // 1.1 Differentiable manifolds ...3 // 1.1.1 Differentiable manifolds ...3 // 1.1.2 Examples of differentiable manifolds ...3 // 1.2 Vector fields and tensor fields ...6 // 1.2.1 Vector bundles ...6 // 1.2.2 Tensor fields ...7 // 1.3 Exterior forms and exterior differentials ...9 // 1.3.1 Exterior differential operators ...9 // 1.3.2 de Rham theorem ...10 // 1.4 Vector bundles and connections ...10 // 1.4.1 Connection of the vector bundle ...10 // 1.4.2 Curvature of a connection ...12 // Exercises ...13 // 2. Finsler Metrics 15 // 2.1 Finsler metrics ...15 // 2.1.1 Finsler metrics ...15 // 2.1.2 Examples of Finsler metrics ...17 // 2.2 Cartan torsion ...22 // 2.2.1 Cartan torsion ...22 // 2.2.2 Deicke theorem ...24 // ѴІІ // viii Introduction to Modem Finsler Geometry // 2.3 Hilbert form and sprays ...28 // 2.3.1 Hilbert form ...28 // 2.3.2 Sprays ...30 // 2.4 Geodesics ...31 // 2.4.1 Geodesics ...31 // 2.4.2 Geodesic coefficients ...32 // 2.4.3 Geodesic completeness ...33 // Exercises ...36 // 3. Connections and Curvatures 39 // 3.1 Connections ...39 // 3.1.1 Chern connection ...39 // 3.1.2 Berwald metrics and Landsberg metrics ...42 // 3.2 Curvatures ...44 // 3.2.1 Curvature form of the Chern connection ...44 // 3.2.2 Flag curvature and Ricci curvature ...45 // 3.3 Bianchi identities ...47 // 3.3.1 Covariant differentiation ...47 // 3.3.2 Bianchi identities ...51 // 3.3.3 Other formulas ...53 // 3.4 Legendre transformation ...54 // 3.4.1 The dual norm in the dual space ...54 // 3.4.2 Legendre transformation ...55 // 3.4.3 Example ...56 // Exercises ...57 //

4. S-Curvature ...59 // 4.1 Volume measures ...59 // 4.1.1 Busemann-Hausdorff volume element ...59 // 4.1.2 The volume element induced from SM ...62 // 4.2 S-curvature ...65 // 4.2.1 Distortion ...65 // 4.2.2 S-curvature and E-curvature ...65 // 4.3 Isotropic S-curvature ...68 // 4.3.1 Isotropic S-curvature and isotropic E-curvature . 68 // 4.3.2 Randers metrics of isotropic S-curvature ...69 // 4.3.3 Geodesic flow ...72 // Exercises ...75 // 5. Riemann Curvature 77 // 5.1 The second variation of arc length ...77 // 5.1.1 The second variation of length ...77 // 5.1.2 Elements of curvature and topology ...80 // 5.2 Scalar flag curvature ...83 // 5.2.1 Schur theorem ...83 // 5.2.2 Constant flag curvature ...85 // 5.3 Global rigidity results ...89 // 5.3.1 Flag curvature with special conditions ...89 // 5.3.2 Manifolds with non-positive flag curvature ...91 // 5.4 Navigation ...94 // 5.4.1 Navigation problem ...94 // 5.4.2 Randers metrics and navigation ...99 // 5.4.3 Ricci curvature and Einstein metrics ...104 // Exercises ...112 // Further Studies 115 // 6. Projective Changes 117 // 6.1 The projective equivalence ...117 // 6.1.1 Projective equivalence ...117 // 6.1.2 Projective invariants ...119 // 6.2 Projectively flat metrics ...123 // 6.2.1 Projectively flat metrics ...123 // 6.2.2 Projectively flat metrics with constant flag curvature 126 // 6.3 Projectively flat metrics with almost isotropic S-curvature 136 // 6.3.1 Randers metrics with almost isotropic S-curvature 136 // 6.3.2 Projectively flat metrics with almost isotropic // S-curvature ...141 // 6.4 Some special projectively equivalent Finsler metrics ...144 // 6.4.1 Projectively equivalent Randers metrics ...144 // 6.4.2 The projective equivalence of (a, /3)-metrics . . . . 148 // 6.4.3 The projective equivalence of quadratic (a,0)- // metrics ...152 // Exercises ...155 //

7. Comparison Theorems ...157 // 7.1 Volume comparison theorems for Finsler manifolds ...157 // 7.1.1 The Jacobian of the exponential map ...157 // 7.1.2 Distance function and comparison theorems ...162 // 7.1.3 Volume comparison theorems ...165 // 7.2 Berger-Kazdan comparison theorems ...174 // 7.2.1 The Kazdan inequality ...174 // 7.2.2 The rigidity of reversible Finsler manifolds ...180 // 7.2.3 The Berger-Kazdan comparison theorem ...185 // Exercises ...190 // 8. Fundamental Groups of Finsler Manifolds 191 // 8.1 Fundamental groups of Finsler manifolds ...191 // 8.1.1 Fundamental groups and covering spaces ...191 // 8.1.2 Algebraic norms and geometric norms ...194 // 8.1.3 Growth of fundamental groups ...199 // 8.2 Entropy and finiteness of fundamental group ...205 // 8.2.1 Entropy of fundamental group ...205 // 8.2.2 The first Betti number ...209 // 8.2.3 Finiteness of fundamental group ...213 // 8.3 Gromov pre-compactness theorems ...218 // 8.3.1 General metric spaces ...218 // 8.3.2 d-Gromov-Hausdorff convergence ...224 // 8.3.3 Pre-compactness of Finsler manifolds ...231 // 8.3.4 On the Gauss-Bonnet-Chern theorem ...233 // Exercises ...235 // 9. Minimal Immersions and Harmonic Maps 237 // 9.1 Isometric immersions ...237 // 9.1.1 Finsler submanifolds ...237 // 9.1.2 The variation of the volume ...241 // 9.1.3 Non-existence of compact minimal submanifolds . 245 // 9.2 Rigidity of minimal submanifolds ...246 // 9.2.1 Minimal surfaces in Minkowski spaces ...246 // 9.2.2 Minimal surfaces in (a, /?)-ѕрасеѕ ...249 // 9.2.3 Minimal surfaces in special Minkowskian (a,ß)- // spaces ...253 // 9.3 Harmonic maps ...259 // 9.3.1 A divergence formula ...259 // 9.3.2 Harmonic maps ...262 // 9.3.3 Composition maps ...267 // 9.4 Second variation of harmonic maps ...270 // 9.4.1 The second variation ...270 // 9.4.2 Stress-energy tensor ...273 //

9.5 Harmonic maps between complex Finsler manifolds ...276 // 9.5.1 Complex Finsler manifolds ...276 // 9.5.2 Harmonic maps between complex Finsler manifolds 280 // 9.5.3 Holomorphic maps ...287 // Exercises ...289 // 10. Einstein Metrics 293 // 10.1 Projective rigidity and m-th root metrics ...293 // 10.1.1 Projective rigidity of Einstein metrics ...293 // 10.1.2 то-th root Einstein metrics ...296 // 10.2 The Ricci rigidity and Douglas-Einstein metrics ...298 // 10.2.1 The Ricci rigidity ...298 // 10.2.2 Douglas (a, /?)-metrics ...300 // 10.3 Einstein (a,/?)-metrics ...305 // 10.3.1 Polynomial (a, / -metrics ...305 // 10.3.2 Kropina metrics ...307 // Exercises ...317 // 11. Miscellaneous Topics 319 // 11.1 Conformal changes ...319 // 11.1.1 Conformal changes ...319 // 11.1.2 Conformally flat metrics ...322 // 11.1.3 Conformally flat (a, /?)-metrics ...328 // 11.2 Conformal vector fields ...330 // 11.2.1 Conformal vector fields ...330 // 11.2.2 Conformal vector fields on a Randers manifold . . 338 // 11.3 A class of critical Finsler metrics ...344 // 11.3.1 The Einstein-Hilbert functional ...344 // 11.3.2 Some special -critical metrics ...349 // 11.4 The first eigenvalue of Finsler Laplacian and the // generalized maximal principle ...351 // 11.4.1 Finsler Laplacian and weighted Ricci curvature . . 351 // xii Introduction to Modem Finsler Geometry // 11.4.2 Lichnerowicz-Obata estimates ...354 // 11.4.3 Li-Yau-Zhong-Yang type estimates ...360 // 11.4.4 Mckean type estimates ...364 // Exercises ...365 // Appendix A Maple Program 367 // A.l Spray coefficients of two-dimensional Finsler metrics . . . 367 // A.2 Gauss curvature ...372 // A.3 Spray coefficients of (a, /?)-metrics ...374 // Bibliography 379 // Index 389