Hopf’s theorems and Riemannian finsler manifold

At the first, Restricting the concept of flag curvatures to Riemannian geometry, one gets sectional curvatures. In other words, in this paper, we discuss some special Riemannian manifolds. One of the most basic properties of Riemannian metrics, which general Finsler metrics do not have, is the splitting property. This feature is illustrated by examples have, at the guise of warped products. Actually, many standard Riemannian metrics can be realized as warped products. Therefore, there are two applications of proposition and three technical ingedients that we need for proving Hopf’s classification theorem. One holds only for complete Riemannian manifolds of constant sectional curvature, while the other two are valid on arbitrary Riemannian manifolds. Then, we defined the laplacian on functions. It is manifestly in divergence form. Here, we extent that definition to differential K-forms. We show that the result can be manipulated into a divergence form plus corrections terms involving the curvature. Such is known as a weitzenbock fomula

Keywords:

Riemannian manifolds , Bianchi Identity , Ricci curvature , Riemannian space , Hopf’s theorem