Laplacian of distance function

Question

Suppose $\Sigma$ is a hypersurface in the complete Riemannian manifold $M$ with positive ricci curvature. Denote the distance function to $\Sigma$ in $M$ by d. Now for small distance $c$, consider the level hypersurface $\Sigma_1:\{d=c\}$, how is $\bar{\Delta} d$ on $\Sigma_1$ related to the mean curvature of $\Sigma_1$ in $M$? Here $\bar{\Delta}$ means the Laplacian in $M$. My idea is to choose an local O.N. basis $\{e_i\}$ for $i=1,\cdots,n+1$ such that $e_{n+1}=N$ is just the normal vector of $\Sigma_1$ in $M$, then \begin{eqnarray*}\bar{\Delta}d&=&\sum e_ie_id-(\bar{\nabla}_{e_i}e_i)d\\&=& \Delta d-H_{\Sigma}N(d)+NN(d)-(\bar{\nabla}_NN)d \end{eqnarray*} I think $\Delta d=0$ since it is constant $c$ on $\Sigma_1$. How do I deal with other terms? I guess but not pretty sure that $N(d)=1$ due to Gauss lemma. My expectation is $$\bar{\Delta}d=-H_{\Sigma_1}.$$ Any help will be appreciated.