Let $S$ be a smooth closed submanifold of $M$. Let $U$ be a tubular neighborhood such that for any $x\in U\setminus S$ there is a unique minimizing geodesics. We now consider the distance squared to $S$, $d_S^2(x):=dist^2(x,S)$, $x\in U$. Can we write down the taylor expansion of $d_S^2(\exp_p(t\nu))$, for any $t$ as long as $\exp_p(t\nu)\in U$? Here $\nu=\nabla d_S(p)$ or in general $\nu$ is any vector in $T_pM$). When we consider the distance function from a point $p$, $d(p,.)$, and write the Teylor expansion of $d_p(\exp_p(tv),\exp_p(tw))$, we see the Ric(v,w) in the expansion.
When instead of a point, we have a distance function from a submanifold $S$, I think that the Hessian of $\nabla^2 d_S$ on $S$ should be equal to the second fundamental form of $S$ and its eigenvalues give the principal curvatures. I wondering if we can choose some direction and calculate the Taylor expansion and see other curvature terms in the expansion. What would other curvature terms that might appear?