I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$.
Does the same hold for a general immersion $f : M^n \to N^{n+1}$? That is,
$M$ is orientable $\iff$ $\exists \, \eta : M \to TN$ with $\eta(p) \perp Df(p)(T_p M)$ and $\|\eta(p)\|=1$, $\, \forall p \in M$?