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 ?