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

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.

• What is true in that case is that the normal bundle is always orientable. For one-dimensional real vector bundles (such as the normal bundle of a hypersurface), orientability is equivalent to triviality. Aug 2 '16 at 21:58

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.
• The simplest example would be the tangent bundle $TX=E$ of $X =\mathbb S^2$. The tangent bundle is always orientable, and $\mathbb S^2$ is also orientable. But it's tangent bundle is nontrivial. @gj255