normal bundle on a submanifold Can you give me an example of a nontrivial normal bundle of a submanifold (of any manifold)?
There is standard example of the core circle of a mobius band, but can you give an example of a submanifold of an orientable manifold? 

Proposition: Let $M$ be a submanifold of $N$, where $N$ is orientable and $M$ is not. Then the corresponding normal bundle is non-trivial.
Proof: Suppose the normal bundle is trivial, then it has a global frame. This frame, together with an orientation of $N$, induces an orientation of $M$.
In view of the above proposition, you can take any non-orientable submanifold of $\mathbb{R}^n$, for example an open Moebius band inside $\mathbb{R}^3$ or a Klein bottle inside $\mathbb{R}^4$.
