Let $(M,g)$ be a $n$ dimensional Riemannian manifold. The corresponding Sasaki metric on $TM$ is denoted by $g_s$. This enable us to measure the volume $\mathcal{A}(S_x)$ of each unit sphere $S_x\subset T_x M,\;x\in M$.(Each $S_x$ inherit the Sasaki metric so it has a natural volum form). The sphere of radius $r$ is denoted by $S^r_x$.

Under which sufficient and necessary conditions on $(M,g)$ the quantity $\mathcal{A}(S_x)$ is independent of $x\in M$

Under which necessary and sufficient conditions $\mathcal{A}(S^r_x) $ is a constant multiplier of $r^{n-1}$, for a constant independent of the base point $x\in M$?