On tangent sphere bundles with small or large constant radius (Q1586047)

For an \(n\)-dimensional Riemannian manifold \(M\), the authors determine some curvature properties of a tangent sphere bundle \(\text{Tr M}\) endowed with the induced Sasaki metric. For instance, let \((M,g)\), \(n>2,\) be a Riemannian manifold with bounded sectional curvature. Then, for each sufficiently small radius \(r\), the tangent sphere bundle is a space of positive scalar curvature. For a compact Riemannian manifold with positive Ricci curvature, the tangent sphere bundle is a space of positive Ricci curvature, in the case when the radius \(r\) is sufficiently small. Let \((M,g),\) \(n>1,\) be a \(d\)-pinched Riemannian manifold. Then, for each sufficiently large radius \(r\), the tangent sphere bundle is a space of negative scalar curvature.