I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of compact manifolds.
From the classification of compact surfaces I can see that the closed ones cannot be embedded in the plane, and that the ones that can be embedded are disks and annuli, which have non empty boundary.
I'd like to know if this intuition is still sound in dimension $n\geq 3$, so my question is: if a compact manifold of dimension $n$ embeds in $\mathbb{R}^n$, is it forced to have a non empty boundary?
I've read the wiki article about Whitney embedding and it's section about "sharper results", but there they give general estimates for the whole class of compact $n$-manifolds, whereas I would be interested in just a single example of a compact boundaryless $n$-manifold embedded in $\mathbb{R}^n$ (possibly in low dimensions), or a proof/reference if this can never happen. Notice I'm not interested in distinction between orientable/non-orientable manifolds, but in the presence of a boundary.
Thanks in advance!