Perturbing a Riemannian metric near the boundary to a product-type metric

Question

Let $M$ be a Riemannian manifold with metric $g$ and boundary $\partial M$ and metric $g$. I understand that it is always possible to perturb $g$ on a small tubular neighbourhood of $\partial M$ to obtain a new metric $g'$ that is a product near the boundary.

Question 1: Is there a basic reference for how one can do this?

The second question is about whether this can be done while minimally disturbing volume.

Question 2: Fix a tubular neighborhood $U\cong\partial M\times[0,\delta)$. Then for any $\varepsilon>0$, does there exist such a perturbation $g'_\varepsilon$ of $g$ on $U$ so that $$\big|\left|g(x)\right|-\left|g'_\varepsilon(x)\right|\big|\leq\epsilon$$ for all $x\in U$, where $|g(x)|$ and $|g'_\epsilon(x)|$ denote $\det g(x)$ and $\det g'_\epsilon(x)$ respectively?

Here $g'_\epsilon|_U$ takes the form $g'_\epsilon|_{\partial M}\oplus dr^2$, where $r\in(-\delta,\delta)$.

Answer 1

Let $p \in \partial M$. Then you can write $T_pM = T_p\partial M \overset{\perp}{\oplus} \mathbb{R}\nu(p)$ where $\nu(p)$ is a unit vector normal to $T_p\partial M$. We also require that $\nu(p)$ points inside $M$, that is the geodesic $\gamma(t) = \exp_p(t\nu(p))$ is defined for small values of $t \geqslant 0$, say $t\in [0,\varepsilon(p))$. Suppose a sort of orientability condition, that is we have a smooth function $p \mapsto \nu(p)$. Then we can defined $E : \bigcup_{p\in\partial M} \{p\}\times [0,\varepsilon(p)) \to M$ by $E(p,t) = \exp_p(t\nu(p))$.

If $\partial M$ is compact, then there exists a uniform $\varepsilon>0$ such that $E$ is defined on all $\partial M \times [0,\varepsilon)$. Moreover, as the differential $\mathrm{d}\exp_p(0)$ of the exponential map is the identity map of $T_pM$, its differential is invertible on a neighbourhood. By the inverse map theorem, $E$ is then a local diffeomorphism. Moreover, as $\partial M$ is supposed compact, there exists $0 < \delta < \varepsilon$ such that $E$ is a diffeomorphism from $\partial M \times [0,\delta)$ onto its image. Thus we have a diffeomorphism \begin{align} E : \partial M \times [0,\delta) &\to U \\ (p,t) &\mapsto E(p,t) \end{align} $U$ is a tubular neighbourhood of $\partial M$ in $M$. Then the pullback metric $E^*g$ on $\partial M \times [0,\delta)$ is written \begin{align} E^*g = g_t \oplus \mathrm{d}t^2 \end{align} This is because $t \mapsto E(p,t)$ are geodesics (so the "$t$ coordinate" are geodesics) and $g_t$ is the pullback on $\partial M \times \{t\}$ of the metrics $\left.g\right|_{E(\partial M \times \{t\})}$

To answer your questions in order :

• I don't have a particular reference for that, but I guess it can be found in Gallot, Hullin, Lafontaine, Riemannian Geometry, or in Petersen, Riemannian Geometry
• the metric here is exactly as you wanted, and not a perturbation, so the volume is preserved

In the non compact case, or if you cannot choose a smooth normal $\nu$ on $\partial M$ pointing inside $M$, I do not know how to answer your question.