Finsler Metrics Which Are Invariant Under The Action Of Lie Group

Finsler geometry is just Riemannian geometry without quadratic restrictions,which is a generalization of Riemannian geometry.Finsler geometry has been attracting more and more attentions and studies because of its widely applications in theoretical physics,mathematical biology and information sciences.In 2002,Shaoqiang Deng and Zixin Hou generalized the famous Myers-Steenrod theorem from Riemannian manifolds to Finsler manifolds and proved that the isometry group of a Finsler metric is a Lie group,thus opened a door of the application of Lie group to the study of homogeneous Finsler geometry.Therefore making it possible to transform some geometric problems to relatively easy problems of Lie group and Lie algebra.In this paper,we mainly study some problems of a class of Finsler metrics which are invariant under actions of Lie groups,especially some relationship of non-Riemannian geometric quantities on homogeneous Finsler manifolds,for instance,the equivalence characterization of S-curvature and E-curvature and the existence or rigidity problems of some special non-Hermitian strongly convex complex Finsler metrics which are invariant under the action of unitary group.Let M be an n(n≥3)-dimensional manifold,F=αφ(s),s=β/α,be an(α,β)-metric on M,here α is a Riemannian metric and β is a 1-form on M.In real Finsler geometry,it is an open problem that under what condition does S=0 is equivalent to E=0.It is well-known that the non-Riemann quantity S=0 implies E=0,but the converse is not necessary true.In[21],it is proved that for a Randers metric,S=0 if and only if E=0.In this thesis we shall give a satisfactory answer to this question for a Finsler manifold(M,F)with the Finsler metric F comes from an(α,β)-metric.Firstly,under the assumption that M is a homogeneous reducible manifold and F is an invariant(α,β)-metric on M,we obtain the necessary and sufficient condition for F to be weakly Berwald metric;we prove that F is a weakly Berwald metric if and only if F has vanishing S-curvature,we prove that F has weakly isotropic S-curvature if and only if F has vanishing S-curvature,and then prove that F has vanishing S-curvature if and only if F has vanishing E-curvature.As an application of these results,we prove that if F is a Douglas metric with weakly isotropic S-curvature,then F must be Berwald type metric.Secondly,we study the Landsberg curvature problem of homogeneous(α,β)-space.We give a classification of homogeneous(α,β)-metric with weakly Landsberg curvature and prove that a homogeneous(α,β)-metric F with weakly Landsberg curvature if and only if F is a Berwald metric.We study the relatively isotropic mean Landsberg curvature of homogenous(α,β)-space and prove that a homogenous(α,β)-metric F with relatively isotropic mean Landsberg curvature if and only if F is a Berwald metric.Under the assumption that M is a smooth manifold,F=αφ(s),s=β/α,be a general(α,β)-metric such that φ(s)is a real analytic but non even function,we prove obtain the necessary and sufficient condition for such metrics to be weakly Berwald metrics.We prove that F is a weakly Berwald metric if and only if F has vanishing S-curvature.Based on these,we further prove that F has weakly isotropic S-curvature if and only if F has vanishing S-curvature,and moreover prove that F has vanishing S-curvature if and only if F has vanishing E-curvature.Finally,we study a class of unitary invariant strongly convex complex Finsler metrics on domains in Cn.We obtain the explicit expression of the real fundamental tensor and its inverse of strongly convex unitary invariant complex Finsler metrics,and obtain the necessary and sufficient conditions for a unitary invariant metric to be a strongly convex complex Finsler metric.We prove that among all strongly convex unitary invariant complex Finsler metrics there exist no real Berwald metric other than unitary invariant Hermitian metrics.Further more we obtain an explicit formula for the holomorphic sectional curvature of unitary invariant strongly pseudoconvex complex Finsler metrics and study the real geodesics of the restriction a unitary invariant metric on the unit sphere in Cn.