On the Geometry of the Tangent Bundle with the Cheeger-Gromoll Metric

Abstract

Let $(M,g)$ be a Riemannian manifold of constant sectional curvature $\kappa$ and $(TM, \tilde{g})$ be the tangent bundle of $M$ equipped with the Cheeger-Gromoll metric induced by $g$. We give necessary and sufficient conditions for $TM$ having positive scalar curvature. This gives counterexamples to a stated theorem of Sekizawa.