Let $\Sigma^3 \subset \mathbb{R}^7$ be a non-compact, orientable, smooth $3$-dimensional submanifold. Its normal bundle $N\Sigma \to \Sigma$ has rank $4$, and therefore (I think) has a non-vanishing section.
I would be happy to learn that $N\Sigma$ must, in fact, be trivial, but I'm guessing that's not true. So, are there simple examples -- the simpler the better, really -- of non-compact, orientable, smooth $\Sigma^3 \subset \mathbb{R}^7$ with non-trivial normal bundle?
(I'm guessing there are easy examples, but frankly, even if I were to start listing simple embeddings $\Sigma^3 \hookrightarrow \mathbb{R}^7$, I wouldn't know how to quickly compute the relevant characteristic classes (in fact, what would those even be here?).)