  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.
