Non-orientable submanifolds Let $M$ be a $n$-manifold and let $S \subset M$ be a non-orientable $n$-dimensional submanifold possibly with boundary. Under what conditions can I conclude that $M$ is also non-orientable? Is compactness sufficient?
In particular consider the case where $M$ is a surface and $S$ is a Mobius strip.
 A: No conditions needed.
Proof. Suppose that that $M$ is orientable, hence there is non-vanishing $n$-form $\omega$ defined on $M.$ As a result $\omega|_S$ is non-vanishing $n$-form on $S.$ Thus $S$ is orientable. Contradition. So $M$ must be non-orientable.

Other approach.
Here and here you have alternative approach to non-orientability. Namely.

Theorem $\star$. Let $N$ be a smooth manifold. $N$ is non-orientable, if and only if there are two charts $(U_a,\phi_a),(U_b,\phi_b),$ such that $U_a,U_b$ are connected, $U_a\cap U_b\neq\emptyset$ and transformation function $\phi_{ab}$ neither preserves nor reverses the orientation.

We just need to use this theorem $\star$ twice.
Proof. Since $S$ is non-orientable, we get by $\star$ that there are two charts $(U_a,\phi_a),(U_b,\phi_b)$ of $S$ such as in $\star$. $S$ is open, hence $(U_a,\phi_a),(U_b,\phi_b)$ are charts of $M$ as well. Again by $\star$ we get that $M$ is non-orientable. 

Remark. This equivalent condition is very convinient in proving that something is non-orientable. Generally you would have to prove that something (non-vanishing form, oriented atlas,...) doesn't exist. Here you just need to indicate two charts with some properties.

A: Another approach: you can read non-orientability on loops, via the first Stiefel-Whitney class: $M$ is orientable if and only if $w_1(TM)=0$. 
If $$\iota : S\hookrightarrow M$$
is your inclusion map, then $$w_1(TS)=\iota^\star(w_1(TM))$$
so that if $M$ is orientable, then so is $S$.
Concretely, a manifold is non-orientable if and only if there is an embedded loop along which the orientation is reversed $(\star)$. The first Stiefel-Whitney class takes the non-trivial value $1\in \Bbb Z/2\Bbb Z$ on such a loop. 
$(\star)$ equivalently, a loop that cannot be lifted into a loop in the $2$-fold orientation covering.
