Complex Spaces in Finsler, Lagrange and Hamilton Geometries

1 Complex Manifolds.- 1.1 Rudiments of several complex variables.- 1.2 Complex and almost complex manifolds.- 1.3 Hermitian and Kählerian manifolds.- 2 Complex and holomorphic vector bundles.- 2.1 Complex vector bundles.- 2.2 Holomorphic vector bundles.- 2.3 Chern classes.- 2.4 Einstein-Hermitian vector bundles.- 3 The geometry of holomorphic tangent bundle.- 3.1 T?M manifold.- 3.2 N-complex linear connections on T?M.- 3.3 Metric structures on T?M.- 4 Complex Finsler spaces.- 4.1 Complex Finsler metrics.- 4.2 Chern-Finsler complex connection.- 4.3 Transformations of Finsler N - (c.l.c.).- 4.4 The Chern complex linear connection.- 4.5 Geodesic complex curves and holomorphic curvature.- 4.6 v-cohomology of complex Finsler manifolds.- 5 Complex Lagrange geometry.- 5.1 Complex Lagrange spaces.- 5.1.1 Projective changes of complex metrics.- 5.2 The generalized complex Lagrange spaces.- 5.3 Lagrange geometry via complex Lagrange geometry.- 5.4 Holomorphic subspaces of a complex Lagrange space.- 5.4.1 Holomorphic subspaces.- 5.4.2 Induced nonlinear connection.- 5.4.3 Coupling of connections along a holomorphic subspace.- 5.4.4 Induced tangent and normal connections.- 5.4.5 Other approach.- 6 Hamilton and Cartan complex spaces.- 6.1 The geometry of T? M bundle.- 6.2 N-complex linear connection on T? M.- 6.3 Metric Hermitian structure on T? M.- 6.4 Complex Hamilton space.- 6.5 Complex Cartan spaces.- 6.6 Complex Legendre transformation.- 6.7 ?-dual complex Lagrange-Hamilton spaces.- 6.8 ?-dual N - (c.l.c.).- 6.9 ?-dual complex Finsler-Cartan spaces.- 6.10 The ?-dual holomorphic sectional curvature.- 6.11 Recovering the real Hamilton geometry.- 6.12 Holomorphic subspaces of a complex Hamilton space.- 6.12.1 The geometry of holomorphic subspaces of (M, H).- 6.12.2 ?-dual holomorphic subspaces.- 7 Complex Finsler vector bundles.- 7.1 The geometry of total space of a holomorphic vector bundle.- 7.2 Finsler structures and partial connections.