= Orientability of a vector bundle total space
{title2=$\det TE\cong\pi^*(\det E\otimes\det TM)$}
The total space of a real <vector bundle> $E\to M$ has the exact sequence $0\to\pi^*E\to TE\to\pi^*TM\to0$. Taking <determinant line bundles> gives the displayed identity. If $M$ is an <orientable smooth manifold>, the total space is orientable exactly when $\det E$ is trivial. Sufficiency follows by pullback; necessity follows by restricting the total-space determinant identity to the <zero section>. For the <cotangent bundle>, base and fibre coordinate determinants cancel, giving an orientation even when the base is not orientable.
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