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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 $x\in M$?

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    $\begingroup$ According to en.wikipedia.org/wiki/Sasaki_metric, the Sasaki metric restricted to each fiber is the standard Euclidean metric. $\endgroup$
    – Deane Yang
    Commented Jul 28, 2019 at 18:57
  • $\begingroup$ @DeaneYang I am sorry if my question is elementary. May you elaborate the second point in the linked wikipedia? Does this second point answer my question? $\endgroup$
    – Ali Taghavi
    Commented Jul 28, 2019 at 19:07
  • $\begingroup$ @DeaneYang I think that the second point can be modified to "each vertical space is isometric to the corresponding tangent space", yes? But the base points vary. $\endgroup$
    – Ali Taghavi
    Commented Jul 28, 2019 at 19:21
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    $\begingroup$ My comment, which is the second point in the wikipedia article, along with a straightforward use of basic definitions and facts from differential and Riemannian geometry answers your question. I think you're able to work out the details yourself. $\endgroup$
    – Deane Yang
    Commented Jul 28, 2019 at 22:51
  • $\begingroup$ @DeaneYang Thanks for your comment. I think about that. $\endgroup$
    – Ali Taghavi
    Commented Jul 29, 2019 at 9:16

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