Research On Several Problems Of Non-Space Form And A Class Of Projectively Flat Polynomial (α, β)-Metrics

This paper consists of two parts, three chapters totally. In the first part, which contains the first and the second chapter, we mainly study two special non-space forms (locally symmetrics spaces and locally conformally flat spaces), and obtain series of results. In the first chapter, by introducing a self-adjoint second order differedtial operator, we investigate some hyper-surfaces in locally symmetrics spaces, obtain some rigidity theorems and a classification of the hypersurface, generalize some previous results. In the second chapter, we study a Schouten tensor on the locally conformally flat manifold, get some sufficient conditions for M to be space forms, which im-prove known results. In the second part (the third chapter), inspired by article [24] and [25], we investigate polynomial (α,β)-metrics, obtain a class of projectively flat Finsler metrics, and then study the flag curvature of this metric.