Let $(M^n,g)$ be a complete Riemannian manifold with $n \geq 2$, let $p \in M$, and let
paragraph
admin
\begin{align*}
r_p: M &\to [0,\infty) \\
x &\mapsto d_g(p,x)
\end{align*}
latex_env
admin
be the distance function from $p$. Let $\Omega_p \subset M \setminus \{p\}$ denote the [open set](/page/Open%20Set) where $r_p$ is smooth, equivalently the complement of $p$ and the cut locus of $p$.
paragraph
admin
For $k \in \mathbb{R}$, define
paragraph
admin
\begin{align*}
s_k(t) :=
\begin{cases}
\frac{1}{\sqrt{k}}\sin(\sqrt{k}t), & k > 0,\\
t, & k = 0,\\
\frac{1}{\sqrt{-k}}\sinh(\sqrt{-k}t), & k < 0,
\end{cases}
\qquad
\operatorname{ct}_k(t) := \frac{s_k'(t)}{s_k(t)}
\end{align*}
latex_env
admin
on the interval
paragraph
admin
\begin{align*}
I_k :=
\begin{cases}
(0,\pi/\sqrt{k}), & k > 0,\\
(0,\infty), & k \leq 0.
\end{cases}
\end{align*}