The Researches On Some Important Projective Properties And Conformal Properties Of (α,β)-spaces

The study on Projective geometry and conformal geometry have a long his-tory, which are applied in the different fields of physical theory widely from thebeginning. The projective geometry and conformal geometry of Finsler metrics al-ways deserve extra attention. Rund has shown that the projective and conformalproperties of a Finsler space determine the structures of the metric uniquely [54].（α,β）-metrics form a rich and computable class of Finsler metrics, which play avery important role in Finsler geometry and have many important applicationin general relativity and biology（ecology）. In recent years, the researches on theproperties of （α,β）-metrics have a great development which in turn promotes theprogress of Finsler geometry. In this paper, we mainly study some importantprojective properties and conformal properties of （α,β）-spaces.The paper is divided into four parts, corresponding to the four chaptersrespectively. In chapter1, we introduce the fundamental concepts of Finsler ge-ometry and related curvatures. In chapter2, we study some projective propertiesof （α,β）-spaces. Firstly, we study （α,β）-metrics in the form=（α+β）^s/α^s-1which are projectively related to a Randers metric. This class of （α,β）-metricshas extremely strong application background and includes many important met-rics, including Riemannian metrics, Randers metrics, Matsumoto metrics. Wecharacterize the local structure of this class of （α,β）-metrics projectively relatedto a Randers metric. Secondly, we study projectively flat weak Einstein （α,β）-metrics and completely classify this class of polynomial （α,β）-metrics. Moreover,we obtain a rigidity result about projectively flat weak Einstein （α,β）-metrics.In chapter3, we study the conformally flat （α,β）-metrics. Firstly, we studypolynomial （α,β）-metrics and prove that the conformally flat weak Einstein poly-nomial （α,β）-metric must be either a locally Minkowski metric or a Riemannian metric. Secondly, we characterize conformally flat （α,β）-metrics with isotropic-curvature and prove that such metrics are also either locally Minkowski metricsor Riemannian metrics.In chapter4, we study the conformal transformations between two （α,β）-metrics and prove that the conformal transformations between two non-Randerstype （α,β）-metrics which are both Douglas metrics must be a homothety. Atthe same time, we also prove that the conformal transformations between two（α,β）-metrics which are both of isotropic-curvatures must be a homothety.