TITLE: Normal coordinates near the boundary
QUESTION [7 upvotes]: Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal coordinate $(x,t)$, where $x$ is the coordinate on $\partial M$ while $t$ is the coordinate for the geodesic $exp_p(te_n)$. Now, the question is whether we can assume in such coordinate, the metric has following form $g=dt^2+g^t_{\partial M}$, where $g^t_{\partial M}$ is just a familiy of metric on $\partial M$ depending on the normal coordinate $t$. It seems to be a folklore for people working on APS index theorem (non-product case). But I cannot find a proof for this result on any Riemannian geometry textbook. In fact, I suspect certain condition should be added for the existence of such coordinate. I think $(\nabla e_n)|_{\partial M}=0$ will work.
Edit: I make a rather simple mistake, the condition we need is $\langle \nabla_{\partial x_i} e_n, e_n \rangle =0$ for any local coordinate $x_i$ on $\partial M$, which is right clearly. I'm quite depressed that several months were wasted for such foolish mistake. For the people who are concerned, thank you all the same.

REPLY [4 votes]: Let $d(x)$ be the distance from $x$ to $\partial M$ and for each $t \ge 0$ let $H_t = d^{-1}(t)$. If $M$ is compact with boundary, then for any $t$ sufficiently close to $0$, $H_t$ is a smooth hypersurface with metric $g_t$ and outer unit normal $\nabla d$. It follows from this that $g = dt^2 + g_t$. You can write $g_t$ in terms of inner products of Jacobi fields obtained by varying geodesics normal to $\partial M$. This is a great exercise in understanding the relationship between Jacobi fields and the metric tensor along a geodesic.
This is used and perhaps explained in this classic and important paper: Heintze, Ernst; Karcher, Hermann A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451–470