I was trying to find an explicit example of a trivial tangent bundle, i.e. $TM = M\times \mathbb{R}^n$, with $M$ a smooth manifold without boundary of dimension $n\in\mathbb{N}$, which is not diffeomorphic to any open subset $U\subseteq\mathbb{R}^{2n}$.
It seems, but maybe I'm wrong, that all the easy classical example of trivial tangent bundle are actually diffeomorphic to some open $2n$-dimensional set. For instance, take the circle $S^1$, its tangent bundle is a cylinder $S^1\times\mathbb{R}$. We can parametrize it as $$ \\\{ (x,y,z)\in\mathbb{R}^3 , x^2+y^2=1 \}, $$ and by dilating/schrinking it a bit and projecting it on the $z=0$ plane we obtain that it is diffeomorphic to an open annulus: take $$ C = \{ (x,y,z), \; x^2+y^2 = \frac{2}{\pi}\arctan z + 2 \} $$ The projection of $C$ on the plane $z=0$ is one-to-one and its image is the open annulus $A=\{ 1 < x^2+y^2 < 3 \}$.
The 2D-torus doesn't work either, because it's diffeomorphic to $S^1\times S^1$ and adapting the previous argument one obtain again that the tangent bundle is diffeomorphic to an open set of $\mathbb{R}^4$.
I have also considered some examples of Lie groups, for instance $SL(2,\mathbb{R})$, but also this fails since it can be seen as $S^1\times\mathbb{R}^2$ and again one can apply the same argument.
Does anyone know a nice example?
EDIT: I'd rather appreciate a solution that takes into account the case of $M$ a non-compact manifold (even though I don't know if it changes anything).
Since every smooth manifold embeds into $\mathbb{R}^{2n}$ by $\iota: M\to \mathbb{R}^{2n}$, the tubular neighborhood theorem tells us that if $M$ is compact there is a diffeomorphism $\Phi: N(M)\to U$ for some open set $U\subset \mathbb{R}^{2n}$, where $N(M)$ is the normal bundle of $M$ corresponding to this embedding. Because in this case $\iota^* T\mathbb{R}^{2n}\cong TM\oplus N(M)\cong (\mathbb{R}^{n}\times M)\oplus (\mathbb{R}^n\times M)$, we see that by applying any bundle isomorphism $TM\to N(M)$, we yield a diffeomorphism $TM\to N(M)\to U$. Things may be somewhat more subtle in the noncompact case.