Isometries and curvatures of tangent sphere bundles

Abstract: Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the characteristic classes of some of those structures. We solve the question of when two given tangent sphere bundles S_rM of a Riemannian manifold M,g are homothetic, assuming different variable radius functions r and weighted metrics induced only by the conformal class of g. We determine their Riemannian, Ricci, scalar and mean curvatures in some cases. We find a family of positive scalar curvature metrics on S_rM when M has positive scalar curvature or when it has bounded sectional curvature and index of nullity 0. Our objective is the study of contact structures and gwistor spaces, a recently found natural G_2-structure on S_1M.