Orientable smooth manifold

A smooth manifold is orientable when it admits an atlas whose transition maps have positive Jacobian determinant wherever they are defined.

This atlas criterion expresses orientability for a differentiable 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

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.

The boundary of an oriented manifold with boundary is oriented by the outward-normal-first convention: an ordered basis $(v_2,\dots ,v_n)$ of $T_p\partial X$ is positive when $(v_1,v_2,\dots ,v_n)$ is positive in $T_{p}X$ for an outward-pointing normal $v_1$.

For a smooth $n$-manifold, the following are equivalent: an atlas with positive transition Jacobians, a smooth choice of orientation of every tangent space, and a nowhere-vanishing smooth $n$-form. Positive charts orient their coordinate frames; oriented frames select the positive local coordinate volume forms, which a partition of unity glues; and a nonzero top form declares a frame positive when its value on that frame is positive.

If $M\times N$ is orientable, fix a point of $N$ and an ordered basis of its tangent space. Contracting a product orientation with that basis gives a smooth orientation of $M$. Interchanging the factors gives an orientation of $N$.

An orientation-reversing diffeomorphism reverses the chosen orientation. In oriented local coordinates its derivative has negative determinant.