Normal bundle of the two-dimensional sphere manifold embedded in $\mathbb R^4$

Question

Let $M \subset \mathbb R^4$ be a smooth manifold diffeomorphic to $S^2$. How can one prove that normal bundle of $M$ has at least one non-vanishing global section. I think that $M$ should be parametrizable and hence has one non-vanishing global section, however I can not prove rigorously that $M$ is parametrizable.

Answer 1

Since $M$ is simply-connected, its normal bundle $\nu$ in $R^4$ has to be orientable. Thus, to show its triviality, it suffices to show that $e(\nu)=0$. Since $e(\nu)$ is the self-intersection number of $M$ in $R^4$, it is zero. Hence, $\nu$ is trivial.