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.
