Contents -- Preface -- 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 -- Bibliography -- Index.