Geometry of tangent bundle with Cheeger–Gromoll type metric

In this paper we study the structure of tangent bundle TM of a Riemannian manifold ( M , g ) with a general metric G a , b . We prove that ( T M , G a , b ) is flat if and only if it is Kنhlerian, and ( T M , G a , b ) is Kنhlerian if and only if it is almost Kنhlerian and M is flat. We also prove that ( T M , G a , b ) Einstein if and only if both ( T M , G a , 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 ( T M , G a , b ) having the constant scalar curvature.