= Orientation of a smooth manifold
An orientation of a <smooth manifold> is a continuous choice of <orientation of a vector space> in every <tangent space>, equivalently an <orientation of a vector bundle> on its <tangent bundle>. It can be specified by an atlas with positive <Jacobian determinants> on all overlaps or by a nowhere-vanishing top-degree <differential form>. On a connected orientable manifold there are two choices, interchanged by reversing every tangent-space orientation.
// Target: differential-geometry.bigb
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