Curvatures of the Tangent Bundle with a Special Almost Product Metric

We study a class of Riemannian almost product metrics on the tangent bundle of a smooth manifold. This class includes the Sasaki and Cheeger–Gromoll metrics as special cases. For this class of metrics, we find the dependence of the scalar curvature of the tangent bundle on objects of the base manifold. For the case in which the base manifold is a space of constant sectional curvature, we obtain conditions on the metric and the dimension of the base under which the scalar curvature of the tangent bundle is constant. For special cases of metrics of the class considered, we find the intervals on which the scalar curvature of the tangent bundle treated as a function of the sectional curvature of the base has constant sign.