On Some Special Projectively Flat Finsler Metrics

Finsler geometry is an ancient and newly developing subject, which developquickly under the development of natural science. It is widespread concerned for it's alsoconsidered as the promotion of Riemann geometry. What's more, it is also applicable toother subjects. Projectively flat metric is concerned by many geometer as a researchfocus in Finsler geometry. In this dissertation, we study the projectively flat m-th rootFinsler metrics.This article is divided into five parts, corresponding to five chapters.The first chapter introduces the Finsler geometry and the development history andbackground of the projectively flat metrics and the general times radical measure. Then,the results in this direction at home and abroad are presented too. The second chapterdescribes some basic knowledge of Finsler geometry. Firstly, two important content ascurvature and geodesics are intrduced. After that, we lead Finsler geometry in commonseveral special Finsler measures, through the introduction of several important curvature.Then the definition of projective flat measure properties and research is introduced. Inthe third part, the radical to m-th Finsler measure projective flat is studied based on theknown work. And we get that if A is irreducible or reducible , then the m-th metric F islocally projectively flat if and only if it is locally Minkowskian. In the fourth chapter,we study we research a generalized m-th root metric, which is more common than m-throot metric. By analysis we get the some similar conclusions as above. Finally, weintroduce the projection of projective related. From the definition , theorem andconclusions, we will study projective related of m-th root in the future.