The Studies On Some Projective Invariances And Conformal Invariances In Finsler Geometry

Abstract The projective properties and conformal properties of Finsler metricsuniquely determine the structure of the metrics. Hence, it has been a research hot topic inFinsler geometry to study the projective properties and conformal properties of Finslermetrics.In this paper, we first study two kinds of important (α,β)-metrics——Kropinametrics and Matsumoto metrics. We mainly study the projective equivalence betweenKropina metrics and Matsumoto metrics and get the sufficient and necessary conditions forthese two kinds of (α,β)-metrics to be projectively equivalent. Secondly, We studyproperties of the dually flat and conformally flat Finsler metrics. We characterize the duallyflat and conformally flat (α,β)-metrics. In particular, we prove that the dually flat andconformally flat Randers must be Minkowskian. Finally, we study weak EinsteinMatsumoto metrics. We obtain the result that weak Einstein Matsumoto metrics must beRicci-flat under an extra condition. At the same time, we prove that conformally flat weakEinstein Matsumoto metrics must be locally Minkowski metrics.