Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, let $p \in M$, and let $\Omega_p \subset M$ be a normal domain of $p$ on which the distance function $r: \Omega_p \to (0,\infty)$ from $p$ is smooth away from $p$. Let
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\begin{align*}
\gamma: [0,a) \to M
\end{align*}
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be a unit-speed geodesic with $\gamma(0)=p$ and $\gamma(t) \in \Omega_p$ for every $0<t<a$.
be the shape operator of the geodesic sphere $S_t := \{x \in \Omega_p : r(x)=t\}$ at $\gamma(t)$ with respect to the outward unit normal $\nabla r$, using the sign convention
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\begin{align*}
A(t)X := \nabla_X \nabla r
\end{align*}