Research On Riemannian Manifold With Parallel Ricci Curvature

In this paper, we discuss several problems of Riemannian manifold with parallel Ricci curvature. Firstly, we study the peculiarity of Riemannian manifold with parallel Ricci cuvature, classify the conformally flat manifold with parallel curvature then research the gap phenomenon of the length of curvature tensor of Riemannian manifold with parallel Ricci curvature, and obtain some pinching theorems about pointwise and glolal property. Secondly, we study some problems of Riemannian mianifold with paralled Ricci curvature as primary image space of map, and get a sufficient condition under which the harmonic map is totally geodesic map, then we estimate the length of Riemannian curvature of minimal submanifold in the space of coustant curvature and find a necessary condition under which the Riemannian manifold with parallel Ricci curvature minimally immerse the space of constant curvature , after that, we tersely prove a conclusion in Chinese Jounal of mathematics by computing the Laplace of the square of the length of Ricci curvature tensor. At last, we study some problems of the Riemanian manifold with parallel Ricci curvature as image space of map, including some researches of some pinching phenomenon about minimal sub-manifold and the estimation of the upper boundary of scalar curvature on conformal metric (g|~) of this manifold.