Let $M$ be a manifold with boundary $\partial M$. Form the manifold $M'$ by attaching a half-infinite cylinder $\partial M\times[0,\infty)$ to $M$ along its boundary. In other words, $$M'=M\cup_{\partial M}\partial M\times[0,\infty),$$ where we identify $\partial M\sim\partial M\times\{0\}$.
Let $g$ be a Riemannian metric on $M$.
Question: Does there always exist a Riemannian metric $g'$ on $M'$ such that the restriction of $g'$ to $M$ is equal to $g$?
Comment added later: Now that I think about it, perhaps the required property is built into the definition of smoothness of $g$ at the boundary, namely that it's extendible slightly beyond the boundary of the chart into some open neighborhood; one then uses a partition of unity to get a metric on all of $M'$.
Take a collar neighborhood of $\partial M$. This is given by a flow $\phi:\partial M \times[0,\infty)\rightarrow M$, generated by vector field $X$ such that $\phi_{*}(\partial_{t})_{(p,t)}=X_{\phi(p,t)}$. Define $g'$ on $M'$ as follows:
$$g' = \begin{cases} g & p\in M\\ \phi_{t}^{*}g_{\phi_{t}(p)} & (p,t)\in\partial M \times[0,\infty) \end{cases}$$
Since $\phi(p,0) = id$, at all points in $\partial M$, $g'$ restricts to g as desired.