Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$. Fix $p \in M$, and let $\Omega_p \subset M \setminus \{p\}$ be an open normal domain on which the distance function
paragraph
admin
\begin{align*}
r: \Omega_p &\to (0,\infty) \\
x &\mapsto d(p,x)
\end{align*}
latex_env
admin
is smooth and satisfies $|\nabla r|_g = 1$. For $x \in \Omega_p$, set $S(p,r(x)) := \{y \in M : d(p,y)=r(x)\}$, and let