= Oriented atlas
{title2=$\det D(x_\beta\circ x_\alpha^{-1})>0$}
An oriented atlas is a <smooth atlas> whose <smooth transition maps> have positive <Jacobian determinants> on every overlap. The coordinate <bases> then give compatible choices of <orientation of a vector space> on all <tangent spaces>. A nowhere-vanishing top-degree <differential form> gives such an atlas: restrict each chart until its coefficient has constant sign, reverse one coordinate if necessary to make that sign positive, and compare the positive coefficients on overlaps. Conversely, a <partition of unity> glues positive local coordinate top forms to a global <volume form>.
Back to article page
