Some Questions In Finsler Geometry And Dipolarization Of 2-step Nilpotent Lie Algebra

The thesis includes two parts:(1) Finsler geometry (2) dipolarizations of Lie groups.Finsler manifold is the generalization of Riemannian manifold. Finsler geometry is just Riemannian geometry without the quadratic restriction. (α,β)-metrics form a rich class of computable Finsler metrics. They play an important role in Finsler geometry. Flag curvature is the most important curvature in Finsler geometry because it is the extension of section curvature in Riemannian geometry. The S-curvature and the flag curvature are subtly related with each other. Therefore S-curvature is one of the most important non-Riemannian quantities in Finsler geometry.The study of homogeneous Finsler manifold was initiated by Professor Shaoqing Deng. In this thesis, we give a formula of the S-curvature of homogeneous (α.β)-metrics first. Then we use this formula to deduce a formula of the mean Berwald cur-vature Eij of the simplest (α,β)-metrics—Randers metrics. Then by investigating the third order nonlinear differential equation of (α.β)-metrics of isotropic S-curvature. we show that the existence of (α,β)-metrics with arbitrary constant S-curvature which is not Randers type.A dipolarization in a Lie algebra g consist of two polarizations g±in g at a com-mon linear form f on g satisfying g=g++g-. A 2-step nilpotent Lie algebra g is a Lie algebra which satisfies [g,[g,g]]=0 and [g,g]≠0. In this paper, we study dipo-larizations of Heisenberg algebras, which are 2-step nilpotent, and construct a series of dipolarizations of Quaternionic Heisenberg algebras.The paper is organized as follows:In Chapter 1, we introduce some basic definitions in Finsler geometry briefly and recallsome results.In Chapter 2, we study the invariant (α,β)-metrics on homogeneous manifolds, and give the formula of S-curvature and the formula of mean Berwald curvature on homogeneous manifolds. In Chapter 3, we study the existence of Finsler metrics with constant S-curvature ,which is not Randers type, on general Finsler manifolds, not on homogeneous mani-folds.In Chapter 4, we study dipolarizations of Heisenberg algebras, and construct a series of dipolarizations of Quaternionic Heisenberg algebras.