Flag Curvature And Cartan Torsion Of A Class Of Finsler Manifolds

In this dissertation, we mainly discuss two important quantities in Finsler geome-try. One is Riemannian quantity: flag curvature, the other is non-Riemannian quantity:Cartan torsion. We use the equation of constant flag curvature to study a class of Finslermetrics with constant flag curvature and give a local classification; then making use ofBerwald frame we discuss the bound of such metrics'Cartan torsion.This dissertation is mainly made up of following three chapters. In the first chap-ter, we introduce some basic conceptions and theorems in Finsler geometry which arerelative to the next several chapters so that it can be self-contained.In the second chapter, we compute Riemannian curvature and Ricci curvature of(α,β) metrics in local coordinates. Then we apply these formulae to discuss a specialclass (α,β) metrics F =α(1 +β/α)~p (|p|≥1) which have constant flag curvature.We obtain the sufficient and necessary conditions that F = ((α+β)~2)/αhave constant flagcurvature and prove that such metrics must be locally projectively flat. Thus we com-plete their local classification and answer the question of B.Li and Z.Shen [21]. Whenp = -1, we find a necessary condition that flag curvature of F =α~2/(α+β) is constant andprove that there are no non-trivial Matsumoto metrics. Furthermore, we give a negativeanswer whether there are non-trivial metrics F =α(1 +β/α)~p (|p|≥1) of constantflag curvature except for p = 1, 2 whenβis closed.In the last chapter, we compute Cartan torsion and mean Cartan torsion of (α,β)metrics and give the relationship between them. Then we continue to discuss the Cartantorsion of Finsler metrics F =α(1 +β/α)~p (|p|≥1). We prove that when p belongsto an interval [1, 2), Cartan torsion of such metrics is bounded. Using this theorem, weobtain two natural corollaries about curvature.