 Orientation line bundle (source code)

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= Orientation line bundle
{title2=$\det(TM)$}

The orientation line bundle of a smooth $n$-manifold is the real line bundle $\bigwedge^nTM$. Its transition functions have the signs of the tangent-coordinate transition determinants. A nonzero continuous section specifies an <orientation> of the manifold. Its orientation double cover records the two possible local orientations.

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