The <equivalent formulations of orientability of a smooth manifold> are:

* an atlas whose transition maps have positive <Jacobian determinant>;
* a smooth choice of one of the two orientations of every <tangent space>;
* a nowhere-zero smooth top-degree <differential form>.

A positive chart orients its coordinate frame. Conversely, smoothly oriented frames determine local positive coordinate volume forms. A <partition of unity> subordinate to an oriented atlas glues these positive forms: at each point they are positive multiples of one another, so their weighted sum cannot vanish. Finally, a nowhere-zero top form $\omega$ declares $(v_1,\ldots,v_n)$ positive exactly when $\omega(v_1,\ldots,v_n)>0$. These constructions are inverse at the level of orientations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 1 / a / Solution

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.

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 Incoming links: Equivalent formulations of orientability of a smooth manifold