Let $(M,g)$ be a Riemannian manifold, and let $S\subset M$ be a closed embedded totally geodesic submanifold. On any tubular neighbourhood where the nearest-point projection $\pi:U\to S$ is well-defined and smooth, the function

\begin{align*} r^2:U&\to \mathbb R, & r^2(x)&=d(x,S)^2 \end{align*}

satisfies the following second-variation formula, with a fixed parametrisation convention. For $s\in S$ and a unit vector $n\in \nu_s(S)$, let $\operatorname{inj}^{\perp}_S(s,n)$ denote the supremum of all $a>0$ such that the normal geodesic $t\mapsto \exp_s(tn)$ is minimizing from $S$ and has no focal point to $S$ on $[0,a)$. If $x=\exp_s(\ell n)$ with $s\in S$, $n\in \nu_s(S)$, $|n|=1$, and $0<\ell<\operatorname{inj}^{\perp}_S(s,n)$, let

\begin{align*} \alpha(u)=\exp_s(u\ell n),\qquad 0\le u\le 1. \end{align*}

For $X\in T_xM$, let $J$ be the Jacobi field along $\alpha$ with $J(1)=X$ and initial boundary condition

\begin{align*} J(0)\in T_sS,\qquad (\nabla_{\dot\alpha}J(0))^\top=0, \end{align*}

where the second condition is the transversality condition for variations whose initial point is allowed to move in $S$. Then

\begin{align*} \frac{1}{2}\operatorname{Hess}_x(r^2)(X,X) = I_\alpha(J,J), \end{align*}

where, for this $[0,1]$ parametrisation,

\begin{align*} I_\alpha(J,J)=\int_0^1\left(|\nabla_{\dot\alpha}J|^2-\langle R(J,\dot\alpha)\dot\alpha,J\rangle\right)\,du. \end{align*}

Equivalently, if the same geodesic is parametrised by arclength $t\in[0,\ell]$, the right-hand side is $\ell$ times the corresponding arclength index form. Consequently, for every relatively compact tubular subneighbourhood $U'\Subset U$, there is a constant $C_{U'}>0$ such that

\begin{align*} \operatorname{Hess}_x(r^2)(X,X)\ge 2|X^\perp|^2-C_{U'} r(x)^2|X|^2 \end{align*}

for every $x\in U'$ and $X\in T_xM$, where $X^\perp$ denotes the component tangent to the normal fibre of $\pi$. This estimate gives a positive leading term in normal-fibre directions; it does not assert convexity in all directions, since tangent directions may be affected by the $O(r(x)^2)$ error. In the exact normal product model $S\times \mathbb R^k$, this inequality becomes the equality $\operatorname{Hess}(r^2)(X,X)=2|X^\perp|^2$.