Riemannian Manifolds With Harmonic Weyl Curvature And The Curvature Operator Of Second Kind

A fundamental topic in Riemannian geometry is to understand the relationship between curvature and topology of Riemannian manifolds.The Riemannian curvature tensor of a Riemannian manifold can induce two classes of curvature operators.This paper classifies Riemannian manifolds by studying one of the curvature operators,the specific structure of which is as follows:In the first chapter,the research background and research status of this paper are given,the main research content and some preliminary knowledge and symbols.In the second chapter,a new Bochner formula is obtained first,and secondly,based on this Bochner formula to prove that an Einstein manifold with k-non-negative curvature operators of the second kind is a constant curvature space.In the third chapter,a classification theorem for Riemannian manifolds with nonnegative curvature operators of second kind and harmonic Weyl curvature is given.