Several Results On Riemannian Manifolds Equipped With Orthogonal Connections

We consider an n-dimensional manifold M equipped with some Riemannian metric g and orthogonal connections.We use the method of Cartan to split the torsion tensor into three components.We calculate the norms of curvatures under the vectorial torsion and the totally anti-symmetric torsion,and investigate the fundamental equations with orthogonal connections.Especially,we consider the case of that torsion is vectorial type.We also consider the relations of the mean curvature vectors,and generalize the Takahashi theorem.This paper is divided into four chapters:Chapter I gives background and outlines our research result for the orthogonal connections;Chapter II gives the preliminary knowledge of the orthogonal connections;Chapter III calculates the norms of curvatures;Chapter IV introduce some results in submanifold.