submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Question

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if $\nu(M)$ is an oriented vector bundle.

The hint is as follows:

Using $\Lambda(V\oplus W)=\Lambda(V)\otimes \Lambda(W)$, prove $\Lambda^nTM\otimes \Lambda^k\nu(M)$ is a trivial bundle. I have proved this. But I do not know how to use this to solve the original question. Thank you a lot.

Is the condition that $M$ is a submanifold of $\mathbb{R}^n$ essential? How about a submanifold of an oriented riemannian manifold?

Answer 1

The result should extend to submanifolds of an orientable manifold.

The key point is that a bundle is orientable precisely if its top wedge power is trivial. (And, by definition, a manifold is orientable iff its tangent bundle is orientable.) You should be able to combine this with what you've already proved to finish.