Riemannian Geometry Of Tangent Bundles And Tangent Sphere Bundles

In this paper, we study the geometric properties of tangent bundles and tangent sphere bundles of Riemannian manifolds.Firstly, we study the geometric properties of tangent bundles endowed with the Cheeger-Gromoll type metrics Ga,b.In particular, the complex structures and the Kahler structures com-patible with Gaa,b are studied. We also prove that there exist some Cheeger-Gromoll type metrics such that the tangent bundles are of constant scalar curvature which give the necessary condition of curvature homogeneity of some Cheeger-Gromoll type metrics.Secondly, we study the extrinsic geometric properties of normal bundles of surfaces in Riemannian spaces induced by the Cheeger-Gromoll type metrics. In particular, the conditions of minimality and constant mean curvature are studied.Thirdly, we study the extrinsic geometric properties of tangent bundles of hypersurfaces in Riemannian spaces induced by the Sasaki metrics and which give a geometrical description of the isoparametric hypersurfaces Miyaoka[115] studied. The integrabilities of the almost complex structures and the Kahler forms in the tangent bundles of hypersurfaces compatible with the induced metrics are also studied.Finally, the slant geodesics in the unit tangent bundles of surfaces endowed with the Sasaki metrics are studied. We prove that such geodesics are determined by either the geodesics in the surfaces and the parallel vector field along them, or the curves in the surfaces with constant velocity and constant curvature and the directions or anti-directions of such curves.