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

Isometries and curvatures of tangent sphere bundles