Orthogonal Connections On Riemannian Manifold And Some Results

In this paper,we give an n-dimensional manifold M equipped with some Riemanni-an metric g and orthogonal connections ?.We invoke Cartan's results about orthogonal connection to do some work.In the first chapter,we introduce Cartan's work,that is the orthogornal connection can be split into three components,then induce our new work.In the second chapter,we introduce some basic knowledge.In the third chapter,we calculate the fundamental equations under orthogonal connection,and then obtain the expression under orthonormal frame field.In the last chapter,we begin our discussion in 3-dimensional manifold equipped with orthogonal connection.And we also assume the torsion tensor A is totally anti-symmetric.we obtain the results about sectional curvatures and totally umbilical sub-manifold of constant sectional curvature,which are similar with that under Levi-civita connection.