## Handbook of Finsler geometry. 1 (2003)There are several mathematical approaches to Finsler Geometry, all of which are contained and expounded in this comprehensive Handbook. The principal bundles pathway to state-of-the-art Finsler Theory is here provided by M. Matsumoto. His is a cornerstone for this set of essays, as are the articles of R. Miron (Lagrange Geometry) and J. Szilasi (Spray and Finsler Geometry). After studying either one of these, the reader will be able to understand the included survey articles on complex manifolds, holonomy, sprays and KCC-theory, symplectic structures, Legendre duality, Hodge theory and Gauss-Bonnet formulas. Finslerian diffusion theory is presented by its founders, P. Antonelli and T. Zastawniak. To help with calculations and conceptualizations, a CD-ROM containing the software package FINSLER, based on MAPLE, is included with the book. |

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### Contents

Finsler Metrics 1347 | 7 |

Kahler Fibrations | 9 |

Complex Finsler Bundles | 23 |

Kobayashi Metrics | 59 |

The Geometry of Lagrange Spaces 969 | 89 |

The Geometry of the Tangent Bundle | 91 |

Nonlinear Connections | 97 |

Finsler Connections on the Tangent Bundle | 109 |

Appendix A Diffusion and Laplacian on the Base Space | 335 |

Appendix B TwoDimensional Constant Berwald Spaces | 343 |

The Geometry of TM and TM | 363 |

Symplectic Transformations of the Differential Geometry | 385 |

The Duality Between Lagrange and Hamilton Spaces | 413 |

Symbolic Finsler Geometry 1125 | 449 |

Holonomy of Positively Homogeneous Connections | 453 |

Holonomies of Finsler V Connections | 463 |

Connection | 112 |

Parallelism | 116 |

Second Order Differential Equations | 123 |

Homogeneous Systems of Second Order Differential | 135 |

The Classical Projective Geometry of Paths | 151 |

Normal Spray Connection | 161 |

Lagrange Spaces 1013 | 173 |

Finsler Spaces | 187 |

Introduction to Stochastic Calculus on Manifolds | 213 |

Stochastic Development on Finsler Spaces | 227 |

VolterraHamilton Systems of Finsler Type | 249 |

Finslerian Diffusion and Curvature | 295 |

Diffusion on the Tangent and Indicatrix Bundles | 319 |

Holonomies of the Finsler Vector Bundle | 469 |

Holonomies of Special Finsler Manifolds | 477 |

Topological Preliminary | 497 |

The Correction Term | 503 |

Differential Operators 1191 | 515 |

Modules and Exact Sequences 1403 | 517 |

Elliptic Complexes | 521 |

The Weitzenbock Formula | 533 |

Finsler Metrics | 565 |

Connections in Finsler Spaces | 601 |

Important Finsler Spaces | 677 |

### Common terms and phrases

Berwald connection Berwald space Brownian motion called Cartan connection chart components connection coefficients connection-pair consider constant convex Finsler coordinate system covariant derivative curvature tensor defined denote differential equations diffusion dual equivalent fibre Finsler connection Finsler geometry Finsler manifold Finsler metric Finsler space Finsler tensor field Finsler vector field Finslerian follows formula frame bundle geodesic given Hence holomorphic holonomy group homogeneous of degree horizontal lift implies induced initial conditions integral invariant Kahler Lemma linear connection locally Minkowski Math metric function metric tensor Minkowski space nonlinear connection HTM obtain orthonormal parallel paths positive definite positively homogeneous projective Proof Proposition respect Riemannian manifold Riemannian metric satisfies scalar curvature semispray smooth curve solution spray structure subspace symmetric tangent bundle tangent space tangent vector tensor field Theorem theory torsion tensor transformation vanishes vector field Volterra-Hamilton system X(TM