for every $v \in TM$, where $k \in \mathbb{R}$. Fix a point $p \in M$ and a unit vector $\theta \in S_pM$. Let $c(\theta) \in (0,\infty]$ denote the cut time of the geodesic
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\begin{align*}
\gamma: [0,c(\theta)) &\to M,\\
r &\mapsto \exp_p(r\theta).
\end{align*}
Let $J_p(r,\theta)$ be the Riemannian polar volume density in the direction $\theta$, equivalently the Jacobian determinant of the radial exponential map on the transverse space $\theta^\perp \subset T_pM$ at radius $r$. Then, on the interval
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\begin{align*}
0<r<c(\theta)
\end{align*}
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with the additional restriction $r<\pi/\sqrt{k}$ when $k>0$, the function
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\begin{align*}
r \longmapsto \frac{J_p(r,\theta)}{\operatorname{sn}_k(r)^{n-1}}
\end{align*}