  Natural metric structures on the tangent bundle and tangent sphere bundles
$S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many
important unsolved problems. Admitting metric connections on $M$ with torsion,
we deduce the equations of induced metric connections on those bundles. Then
the equations of reducibility of $TM$ to the almost Hermitian category. Our
purpose is the study of the natural contact structure on $S_rM$ and the
$G_2$-twistor space of any oriented Riemannian 4-manifold.
