On Differential Geometry Of Orthogonal Frame Bundle

In the paper,we calculate the curvature on the orthogonal frame bundle,and obtain that the boundedness of the sectional curvature on the orthogonal frame bundle is controlled by the boundedness of the sectional curvature of the Riemannian manifold itself and its curvature derivative.An example is given that the sectional curvature of a Riemannian manifold is bounded,but that of its orthogonal frame bundle is unbounded.The concrete process is as follows: according to the method of O.Kowalski and M.Sekizawa,the metric of linear frame bundle is given and its curvature formula is calculated.Then based on the embedding relationship between the orthogonal frame bundle and the linear frame bundle,the metric on the orthogonal frame bundle is induced and the curvature expression of orthogonal frame bundle is calculated by Gauss equation.Finally,an example of Riemannian manifolds with bounded sectional curvature and unbounded sectional curvature on its orthogonal frame bundle is constructed.