Geometry Of Tangent Bundle With The Cheeger-Gromoll Metric

Let (M, g) be a Riemannian manifold and TM be the tangent bundle of M. Let (p, v) be a point on the tangent bundle TM and X, Y∈T_PM, then (?) is defined as following:Hereα= 1 + g(v, v), X^h, Y^h and X^v, Y^v be the horizontal and vertical lift of X, Y.In this paper, We deduce the scalar curvature of TM using moving frame method, and give a necessary and sufficient condition for TM having positive scalar curvature and negative scalar curvature when M having constant sectional curvature.The main result of this paper as following:Theorem 3.2.3 Let (M,g) be an n-dimensional Riemannian manifold and (TM,(?)) be the tangent bundle of M equiped with the Cheeger-Gromoll metric induced by g. Then the scalar curvature (?) of (TM, (?)) is of the form:Where S be the scalar curvature of M.Remark: We simply denote R_cmijoπand R_cpijoπby the symbol R_cmij and R_cpij,where R_cmij and R_cpij be the Riemannian curvature tensor of (M,g). v^m,v^p be the local coordinate of (p,v)∈π^-1(U).Theorem 3.3.2 Let (M,g) be an n-dimensional Riemannian manifold of constant sectional curvatureκwith n>1. Then(I) (TM, (?)) has positive scalar curvature if and only if(Ⅱ) (TM, (?)) has negative scalar curvature if and only ifIn the end, we point out the incorrectness of [20] and correct it.