Let $M \subseteq N$ be an embedded submanifold. Then if both $M$, $N$ are oriented, it is claimed, pg87 line +17:
this induces an orientation on the normal bundle of $M$.
2026-03-26 11:07:16.1774523236
Orientation on normal bundle
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The obstruction of the orientability is the first Stiefel-Whitney class $w_1$. Suppose that $N$ is a submanifold of $M$, we denote by $TN$ the tangent bundle of $N$, $T(N,M)$ the restriction of $TM$ to $N$, and $LN$ the normal bundle of $TN$. Remark that by using a differentiable metric one can write $T(N,M)=TN \oplus LN$. This implies that $w_1(T(N,M))=w_1(TN)w_1(LN)$. We deduce that $w_1(T(N,M))=w_1(TN)=1$ implies that $w_1(LN)=1$.