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? 

If you consider the the manifold $X=\mathbb P^2(\mathbb R)$ with coordinates $x:y:z$, its submanifold $Y$ given by $z=0$ is isomorphic to $X=\mathbb P^1(\mathbb R)$ and the normal bundle $N(Y/X)$ corresponds in that isomorphism to the non-trivial  tautological line bundle $\xi$ of $\mathbb P^1(\mathbb R)$.
Thus the normal bundle $N(Y/X)$ is non-trivial.
