Is the normal bundle of an orientable submanifold of an orientable manifold always trivial?

Question

There is another question on this site which confirms that for an orientable hypersurface of an orientable manifold, the normal bundle is trivial. However, I was wondering if this result generalises to arbitrary orientable submanifolds.

Answer 1

It is not true. For example, take a nontrivial vector bundle $E \to X$, then the zero section $Z \subset E$ has normal bundle isomorphic to $E$.

One can also find example when the ambient space is $\mathbb R^M$, since there are orientable manifolds which cannot be embedded in the euclidean space with trivial normal bundle. Together with Whitney's embedding theorem, there are orientable submanifolds in euclidean spaces with nontrivial normal bundle.