signs by taking our definition as $h(X,Z) = g( \nabla_X Z, -\partial_r)$, where $\nabla$ is the connection in $M$, and $X, Z$ are vector fields on $Y$ extended arbitrarily to vector fields on $M$. Note that this definition uses the *inward* pointing unit normal. Given this convention, we say $Y$ is *convex* (respectively *strictly convex*) with respect to the outward unit normal if the scalar second fundamental form is positive semidefinite (respectively positive definite).

In Fermi coordinates the second fundamental form of $r$-level sets may be written as $(h_r)_{\beta \gamma} = \frac{1}{2} \partial_r g_{\beta \gamma}$. We raise an index to obtain a family of shape operators $S(r)$, where $S(r)^{\beta}_{\gamma} = g^{\beta \nu} (h_r)_{\nu \gamma}$. A computation shows that $S$ satisfies a Riccati equation involving curvature, namely $$\begin{aligned}
\label{eqn:ricatti}
( \partial_r S(r) + S(r)^2 )_{\beta}^{\nu} = -R_{0 \beta \; 0}^{\; \; \; \; \nu}.\end{aligned}$$

We will make use of Jacobi fields suitable to our coordinates. Fix $p \in Y$ and consider the outward normal geodesic $\gamma_p$. Choose any curve $\sigma$ in $Y$ such that $\sigma(0) = p$ and define a variation through geodesics by $\Gamma(s,t) = {E}(\sigma(s), t )$. This gives rise to a Jacobi field $J(t) = \partial_s \Gamma(s,t) |_{s=0}$ 