The Geometry of Lagrange Spaces: Theory and Applications

I. Fibre Bundles. General Theory.- 1. Fibre Bundles.- 2. Principal Fibre Bundles.- 3. Vector Bundles.- 4. Morphisms of Vector Bundles.- 5. Vector Subbundles.- 6. Operations with Vector Bundles.- 7. Principal Bundle Associated with a Vector Bundle.- 8. Sections in Vector Bundles.- II. Connections in Fibre Bundles.- 1. Non-linear Connections in Vector Bundles.- 2. Local Representations of a Non-linear Connection.- 3. Other Characterisations of a Non-linear Connection.- 4. Vertical and Horizontal Lifts.- 5. Curvature of a Non-linear Connection.- 6. Affine Morphisms of Vector Bundles.- III. Geometry of the Total Space of a Vector Bundle.- 1. d-Connections on the Total Space of a Vector Bundle.- 2. Local Representation of d-Connections.- 3. Torsion and Curvature of d-Connections.- 4. Structure Equations of a d-Connection.- 5. Metric Structures on the Total Space of a Vector Bundle.- IV. Geometrical Theory of Embeddings of Vector Bundles.- 1. Embeddings of Vector Bundles.- 2. Moving Frame on E? in E.- 3. Induced Non-linear Connections. Relative Covariant Derivative.- 4. The Gauss-Weingarten Formulae.- 5. The Gauss-Codazzi Equations.- V. Einstein Equations.- 1. Einstein Equations.- 2. Einstein Equations in the Case m = 1.- 3. Another Form of the Einstein Equations.- 4. Einstein Equations for some particular metrics on E.- VI. Generalized Einstein-Yang Mills Equations.- 1. Gauge Transformations.- 2. Gauge Covariant Derivatives.- 3. Metrical Gauge d-Connections.- 4. Generalized Einstein-Yang Mills Equations.- VII. Geometry of the Total Space of a Tangent Bundle.- 1. Non-linear Connections in Tangent Bundle.- 2. Semisprays, Sprays and Non-linear Connections.- 3. Torsions and Curvature of a Non-linear Connections.- 4. Transformations of Non-linear Connections.- 5. Normald-Connections on TM.- 6. Metrical Structures on TM.- 7. Some Remarkable Metrics on TM.- VIII. Finsler Spaces.- 1. The Notion of Finsler Space.- 2. Non-linear Cartan Connection.- 3. Geodesics.- 4. Metrical Cartan Connection.- 5. Structure Equations. Bianchi Identities.- 6. Remarkable Finslerian Connections.- 7. Almost Kählerian Model of a Finsler Space.- 8. Subspaces in a Finsler Space.- IX. Lagrange Spaces.- 1. The Notion of Lagrange Space.- 2. Euler-Lagrange Equations. Canonical Non-linear Connection.- 3. Canonical Metrical d-Connection.- 4. Gravitational and Electromagnetic Fields.- 5. Lagrange Space of Electrodynamics.- 6. Almost Finslerian Lagrange Spaces.- 7. Almost Kählerian Model of a Lagrange Space.- X. Generalized Lagrange Space.- 1. Notion of Generalized Lagrange Space.- 2. Metrical d-Connections in a GLn Space.- 3. Structure Equations. Parallelism.- 4. On h-Covariant Constant d-Tensor Fields.- 5. Gravitational Field.- 6. Electromagnetic Field.- 7. Almost Hermitian Model of a GLn Space.- XI. Applications of the GLn Spaces with the Metric Tensor e2?(x,y)?ji(x,y).- 1. EPS conditions and the Metric e2?(x,y)?ij(x).- 2. Canonical Metrical d-Connection.- 3. Electromagnetic and Gravitational Fields.- 4. Two Particular Cases.- 5. GLn Spaces with the Metric e2?(x,y)?ij(y).- 6. Antonelli’s Metrics.- 7. General Case.- XII. Relativistic Geometrical Optics.- 1. Synge Metric in Dispersive Media.- 2. A Post-Newtonian Estimation.- 3. A Non-linear Connection.- 4. Canonical Metrical d-Connection.- 5. Electromagnetic Tensors.- 6. Einstein Equations.- 7. Locally Minkowski GLn Spaces.- 8. Almost Hermitian Model.- 9. A Finslerian Approach to the Relativistic Optics.- XIII. Geometry of Time Dependent Lagrangians.- 1. Non-linear Connections in ? = (R x TM,?,R x M).- 2.Time Dependent Lagrangians.- 3. Non-linear Connections and Semisprays.- 4. Normal d-Connections on R x TM.- 5. Metrical Normal d-Connections on RxTM.- 6. Rheonomic Finsler Spaces.- 7. Remarkable Time Dependent Lagrangians.- 8. Metrical Almost Contact Model of a Rheonomic Lagrange Space.- 9. Generalized Rheonomic Lagrange Spaces.