An Introduction to Finsler Geometry (Peking University Series in Mathematics)

this Introductory Book Uses The Moving Frame As A Tool And Develops Finsler Geometry On The Basis Of The Chern Connection And The Projective Sphere Bundle. It Systematically Introduces Three Classes Of Geometrical Invariants On Finsler Manifolds And Their Intrinsic Relations, Analyzes Local And Global Results From Classic And Modern Finsler Geometry, And Gives Non-trivial Examples Of Finsler Manifolds Satisfying Different Curvature Conditions. Contents......Page 8 Preface......Page 6 1.1 Historical remarks......Page 10 1.2 Finsler manifolds......Page 11 1.3 Basic examples......Page 13 1.4 Fundamental invariants......Page 15 1.5 Reversible Finsler structures......Page 18 2.1 The Cartan tensor......Page 20 2.2 The Cartan form and Deicke's Theorem......Page 21 2.3 Distortion......Page 23 2.4 Finsler submanifolds......Page 24 2.5 Imbedding problem of submanifolds......Page 26 3.1 The adapted frame on a Finsler bundle......Page 32 3.2 Construction of Chern connection......Page 35 3.3 Properties of Chern connection......Page 41 3.4 Horizontal and vertical subbundles of SM......Page 46 4.1 Horizontal and vertical covariant derivatives......Page 48 4.2 The covariant derivative along geodesic......Page 49 4.3 Landsberg curvature......Page 51 4.4 S-curvature......Page 53 5.1 Curvatures of Chern connection......Page 62 5.2 Flag curvature......Page 67 5.3 The first variation of arc length......Page 68 5.4 The second variation of arc length......Page 73 6.1 Riemannian connection and curvature of projective sphere bundle......Page 78 6.2 Integrable condition of Finsler bundle......Page 84 6.3 Minimal condition of Finsler bundle......Page 86 7.1 The relation between Cartan tensor and flag curvature......Page 90 7.2 Ricci identities......Page 92 7.4 Finsler manifolds with constant S-curvature......Page 94 8.1 Finsler manifolds with isotropic S-curvature......Page 98 8.2 Fundamental equation on Finsler manifolds with scalar curvature......Page 100 8.3 Finsler metrics with relatively isotropic mean Landsberg curvature......Page 104 9.1 Some definitions and lemmas......Page 108 9.2 The first variation......Page 111 9.3 Composition properties......Page 117 9.4 The stress-energy tensor......Page 121 9.5 Harmonicity of the identity map......Page 123 Bibliography......Page 126 Index......Page 128 1. Finsler manifolds. 1.1. Historical remarks. 1.2. Finsler manifolds. 1.3. Basic examples. 1.4. Fundamental invariants. 1.5. Reversible Finsler structures -- 2. Geometric quantities on a Minkowski space. 2.1. The Cartan tensor. 2.2. The Cartan form and Deicke's theorem. 2.3. Distortion. 2.4. Finsler submanifolds. 2.5. Imbedding problem of submanifolds -- 3. Chern connection. 3.1. The adapted frame on a Finsler bundle. 3.2. Construction of Chern connection. 3.3. Properties of Chern connection. 3.4. Horizontal and vertical subbundles of SM -- 4. Covariant differentiation and second class of geometric invariants. 4.1. Horizontal and vertical covariant derivatives. 4.2. The covariant derivative along geodesic. 4.3. Landsberg curvature. 4.4. S-curvature -- 5. Riemann invariants and variations of arc length. 5.1. Curvatures of Chern connection. 5.2. Flag curvature. 5.3. The first variation of arc length. 5.4. The second variation of arc length -- 6. Geometry of projective sphere bundle. 6.1. Riemannian connection and curvature of projective sphere bundle. 6.2. Integrable condition of Finsler bundle. 6.3. Minimal condition of Finsler bundle -- 7. Relation among three classes of invariants. 7.1. The relation between Cartan tensor and flag curvature. 7.2. Ricci identities. 7.3. The relation between S-curvature and flag curvature. 7.4. Finsler manifolds with constant S-curvature -- 8. Finsler manifolds with scalar curvature. 8.1. Finsler manifolds with isotropic S-curvature. 8.2. Fundamental equation on Finsler manifolds with scalar curvature. 8.3. Finsler metrics with relatively isotropic mean Landsberg curvature -- 9. Harmonic maps from Finsler manifolds. 9.1. Some definitions and lemmas. 9.2. The first variation. 9.3. Composition properties. 9.4. The stress-energy tensor. 9.5. Harmonicity of the identity map Uses the moving frame as a tool and develops Finsler geometry on the basis of the Chern connection and the projective sphere bundle. This book introduces three classes of geometrical invariants on Finsler manifolds and their intrinsic relations, analyzes local and global results from classic and modern Finsler geometry, and more.