Complex Spaces in Finsler, Lagrange and Hamilton Geometries

From a historical point of view, the theory we submit to the present study has its origins in the famous dissertation of P. Finsler from 1918 ([Fi]). In a the classical notion also conventional classification, Finsler geometry has besides a number of generalizations, which use the same work technique and which can be considered self-geometries: Lagrange and Hamilton spaces. Finsler geometry had a period of incubation long enough, so that few math­ ematicians (E. Cartan, L. Berwald, S.S. Chem, H. Rund) had the patience to penetrate into a universe of tensors, which made them compare it to a jungle. To aU of us, who study nowadays Finsler geometry, it is obvious that the qualitative leap was made in the 1970's by the crystallization of the nonlinear connection notion (a notion which is almost as old as Finsler space, [SZ4]) and by work-skills into its adapted frame fields. The results obtained by M. Matsumoto (coUected later, in 1986, in a monograph, [Ma3]) aroused interest not only in Japan, but also in other countries such as Romania, Hungary, Canada and the USA, where schools of Finsler geometry are founded and are presently widely recognized.

Table of Contents:
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.- 5.2 The generalized complex Lagrange spaces.- 5.3 Lagrange geometry via complex Lagrange geometry.- 5.4 Holomorphic subspaces of a complex Lagrange space.- 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.- 7 Complex Finsler vector bundles.- 7.1 The geometry of total space of a holomorphic vector bundle.- 7.2 Finsler structures and partial connections.