What are some general conditions under which submanifolds of orientable manifolds will also be orientable. Of course this isn't true in general (for example, the Möbius band is a non-orientable submanifold of $\mathbb{R}^3$), but when is it true?
For example, what if you only consider closed (meaning compact and boundaryless, not topologically closed) submanifolds of closed, orientable manifolds?
EDIT: The conditions I listed above definitely aren't good enough. Consider $\mathbb{RP}^2\subset\mathbb{RP}^3$, I think .
A sufficient condition for $N\subset M$ to be orientable is the following:(Assume $M$ is orientable, $\dim M=m$, $\dim N=n)$:
$N$ is orientable if there are $m-n$ independent vector fields $V_{n+1} \ldots V_{m}$ along $N$ (tangent to $M$ but not tangent to $N$) such that for each $x\in N$, $T_{x} N \oplus \text{span} \{ V_{n+1}(x), \ldots V_{m}(x)\}= T_{x} M$.
The reason: Assume $\Omega$ is a volume form for $M$(orientability is equivalent to existence of volume form). Then $I_{V_{n+1}} \circ I_{V_{n+2}},\circ \ldots \circ I_{V_{m}}(\Omega)$ is a volume form for $N.$
Example:$S^{n} \subset \mathbb{R}^{n+1}$. The vector field $V_{n+1}=\frac{\partial}{\partial r}$ is the radial vector field, orthogonal to the sphere. Then $i_{\frac{\partial}{\partial r}} dx_{1}\wedge dx_{2}\wedge \ldots dx_{n}$ is the natural intrinsic volum form of the sphere.