TITLE: Example of a nontrivial subbundle of a trivial vector bundle
QUESTION [8 upvotes]: It seems to me that you can have a nontrivial subbundle sitting inside a trivial vector bundle. Can anyone please give an example of this which one can visualize? Thanks!

REPLY [3 votes]: Take any compact manifold $M$ embedded in $\mathbb R^N$, with non-trivial tangent bundle (like $S^2\subset\mathbb R^3$). Then $TM$ sits inside the trivial vector bundle $M\times \mathbb R^N$.
As any compact manifold can be embedded into a suitable $\mathbb R^N$, this means that any tangent bundle of a compact manifold is subbundle of a trivial one. This is even more generally true for any finite-dimensional vector bundle over a compact manifold instead of the tangent bundle.