Let $ (M,g)$ be a compact Riemannian manifold with boundary and let $ f:M\rightarrow \mathbb{R}$ be a smooth function. Let $ r:M\rightarrow \mathbb{R}$ denote the distance to the boundary $ d(\cdot, \partial M)$ .Let $ N\varepsilon = \{ x\in M : r(x)<\varepsilon\}$ . It is well known that for small enough $ \varepsilon$ we can form boundary normal coordinates.

I was interested in Taylor expanding an integral operator of $ f$ near the boundary in terms of $ r$ . We can write $ f$ in boundary geodesic coordinates $ (x^1,…x^n)$ on a neighborhood $ U$ . Where $ x^1$ is equal to the distance from the boundary and $ (x^2,…,x^n)$ are normal coordinates in a neighborhood $ U_\partial$ of $ \partial M$ . We then have: $ $ \int_{U} f(x^1,…,x^n) \sqrt{|g|}dx^1,…,dx^n = \int_{U} f(0,x^2,…,x^n) + \frac{\partial f}{\partial x^1}\Big|_{(0,x^2,…x^n)}x^1 +\frac{\partial^2 f}{\partial (x^1 )^2}\Big|_{(0,\zeta)}(x^1)^2\sqrt{|g|}dx^1,…,dx^n. $ $

We then can separate out the terms involving $ x^1$ to obtain: $ $ \int_{U} f(x^1,…,x^n) \sqrt{|g|}dx^1,…,dx^n = \int_{0}^\varepsilon dx^1 \int_{U_\partial} f(0,x^2,…,x^n) \sqrt{|g|_\partial}dx^2…dx^n + \int_{0}^\varepsilon x^1 dx^1 \int_{U_\partial}\frac{\partial f}{\partial x^1}\Big|_{(0,x^2,…x^n)}\sqrt{|g|_\partial}dx^2,…,dx^n + \mathcal{O}(\varepsilon^3) $ $

I am trying to “patch together” this local argument into a global result. The first term (up to constant) is really just the integral of $ f$ over the boundary. The second term seems like it is $ df(\partial_1)$ where $ \partial_1$ is the gradient of $ r$ .

I suspect that I am mistaken. If $ \partial M$ is not orientable, then this suggests that grad $ r$ is an outward facing globally-defined vector field near the boundary.

I was wondering if there was anyone who could clarify the obstruction here. Is grad $ r$ simply not smooth on the boundary? Is that the only problem?

If there is also a good reference on doing analysis on tubular neighborhoods, that would be much appreciated!