Geometry of tangent bundle with Cheeger–Gromoll type metric

Abstract

In this paper we study the structure of tangent bundle TM of a Riemannian manifold (M,g) with a general metric Ga,b. We prove that (TM,Ga,b) is flat if and only if it is Kählerian, and (TM,Ga,b) is Kählerian if and only if it is almost Kählerian and M is flat. We also prove that (TM,Ga,b) Einstein if and only if both (TM,Ga,b) and (M,g) are flat. Finally, for M to be a space form with constant curvature, we obtain a necessary and sufficient condition for (TM,Ga,b) having the constant scalar curvature.