Usefulness of compact boundary proving uniqueness in gluing smooth manifolds I have a question about the following fact which used to prove the uniqueness(upto diffeomorphism) in gluing smooth manifolds:

Let $M$ be a smooth manifold and let $f_0,f_1:[0,1]\times \partial
M\to M$ be two smooth embedding that are identity map on $\partial
 M\times\{0\}=\partial M$. If $\partial M$ is compact then there is
a diffeotopy $H:M\times [0,1]\to M$ relative to $\partial M$ from
$H_0=\text{Id}_M$ to a diffeomorphism $\Phi=H_1:M\to M$ with
$\Phi\circ f_0=f_1$.

Now, proof of this can be found in the book Differential Topology by C.T.C. Wall or Introduction to Differential Topology by Klaus Jänich and T. Bröcker.
Both books assume $\partial M$ to be compact to prove the above fact, is it necessary? I mean is there any sort of counterexample without the assumption $\partial M$ is compact?
Remark: I noticed the same assumption that $\partial M$ is compact in the setting of topological manifold, for example, in Collars and concordances of topological manifolds written by M.A. Armstrong.
 A: Suppose $(M,g)$ is a riemannian manifold and suppose $\partial M$ has a unit normal vector field, pointing inside $M$, say $\nu$. For $p\in \partial M$ fixed, there is a little interval $[0,\varepsilon(p))$ on which the geodesic $\gamma_p(t)=\exp_p(t\nu(p))$ is defined. By regularity assumptions on $(M,g)$, one can chose $p \mapsto \varepsilon(p)$ as regular as $(M,g)$. Thus one can define a map
\begin{align}
E :\bigcup_{p\in \partial M}\{p\} \times [0,\varepsilon(p)) & \to  M \\ (p,t) \mapsto \gamma_p(t)
\end{align}
Because the differential of the exponential map at $0$ is the identity map, by the inverse map theorem, $E$ is a local diffeomorphism around $(q,0)\in\bigcup_{p\in \partial M}\{p\} \times [0,\varepsilon(p)) $: there exists an open neighbourhood $U_q$ of $(q,0)$ on which $E$ is a diffeomorphism onto its image. Such a neighbourhood contains an open subset $V_q \times [0,\varepsilon'(q))$.
If moreover, $\partial M$ is compact, one can chose a finite number of $q_i \in \partial M$ such that $V_{q_i}\times [0,\varepsilon'(q_i))$ covers $\partial M$. If $\varepsilon = \min \varepsilon'(q_i)$, then $\partial M \times [0,\varepsilon)$ is a neighbourhood on which $E$ is a local diffeomorphism, and we can choose $\varepsilon$ small enough such that it is injective. Thus, we have found a uniform tubular neighbourhood of $\partial M$ in $M$ isometrically embedded $E\left(\partial M \times [0,\varepsilon)\right)$.
If $M$ is not compact, one could not be able to chose $\varepsilon$ and to isometrically embed such a tubular neighbourhood. Because topology and curvature are closely related, maybe there can exist no riemannian metric on $M$ for which this construction can work. Right now I cannot figure out an example of such a manifold $M$, but one can easily construct a particular riemannian metric on a manifold with non-compact boundary such that you cannot define a uniform tubular neighbourhood. For example, the subset of $\mathbb{R}^2$ defined by
\begin{align}
M = \left\{(x,y)\in\mathbb{R}^2 ~|~ 0 \leqslant y < e^{-x^2} \right\}
\end{align}
with the induced euclidean metric is a submanifold with boundary $\partial M =\mathbb{R}\times\{0\}$, but there exists no isometric tubular neighbourhood of the form $\mathbb{R}\times [0,\varepsilon)$. This is a bad example, because here the metric is flat and we can easily deform it... But still, it is an example.
I quickly read some parts of the chapter about tubular neighbourhoods of C.T.C Wall, and there is a systematic use of riemannian metrics for the constructions. I did not find exactly where the compactness assumption is used in the lemmas before the theorem you stated, but I am almost sure it is used the way I used it above.
