Ebook: 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.

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This book presents the most recent advances in complex Finsler geometry and related geometries: the geometry of complex Lagrange, Hamilton and Cartan Spaces. The last three spaces were initially introduced to and have been investigated by the author of the present volume over the past several years. This book will acquaint the reader with:

• a survey of some basic results from complex manifolds and the complex vector bundles theory,
• the geometry of holomorphic tangent bundles,
• an analysis of the main results in complex Finsler geometry,
• a study of the geometry of complex Lagrange and generalized Lagrange Spaces. Of special interest are their holomorphic subspaces,
• the construction of the complex Hamilton geometry,
• the complex Finsler vector bundles.

Audience: Geometers, complex analysts, and physicists in quantum field theory and in theoretical mechanics will find this book of interest. The volume can be also used as a supplementary graduate text.