Consider $M$ a riemannian manifold and $A \subset M$. Let $SM$ the sphere bundle of $M$. There exist any relation among the volume of $A$ in $M$ and $\pi^{-1}(A)$ in $SM$?. I guess that as the fibers are $S^{n-1}$ $$ vol_M(A)vol_{\mathbb{R}^n}(S^{n-1}) = vol(\pi^{-1}(A)). $$
Remark: If $\omega$ is the volume form in $M$ and $\omega_0$ is the canonical volume form in the fiber $S^{n-1}$ then $$ \pi^* \omega \wedge \omega_0 $$ is a volume form in $SM$, the canonical volume form of the bundle.
