= Normal bundle of a regular fibre is trivial
{title2=$TM|_F/TF\cong F\times T_qN$}
For a <regular value> $q$ of a <smooth map> $f:M\to N$, let $F=f^{-1}(q)$. The <differential of a smooth map> identifies the quotient <vector bundle> $TM|_F/TF$ with the fixed target <tangent space> $T_qN$ over every point of $F$. After choosing a <Riemannian metric>, that quotient identifies with the <normal bundle>. Hence the normal bundle is trivial even when the target manifold has no global orientation.
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