Let $M$ be a smooth, non-compact manifold.

a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ?

b) If not, are there some concrete general topological obstructions ?

c) What if we require $i$ only to be an immersion ?

Variants of these questions have thourougly been studied, for example, one can construct many examples of Riemannian manifolds $(M,g)$ for which there does not exist a conformal embedding/immersion into a compact Riemannian manifold of the same dimension.

Even in the non-Riemannian setting, but requiring additionally that $i(M) = N \setminus \partial N$, there are obstructions to the existence of such an embedding $i: M \to N$ (for example, this is never possible if $\pi_1(M)$ is infinitley generated).

But what about these slightly more general questions ?