Orientation of a smooth manifold - Imported Knowlpedia Content

Definition

Let $M$ be an $n$ -dimensional smooth manifold. An orientation of $M$ is an orientation of its tangent bundle $TM$ : at each $p\in M$ , one chooses one of the two orientation classes of ordered bases of $T_pM$ , and the choice must vary continuously. Equivalently, an orientation is a maximal smooth atlas whose coordinate changes have positive Jacobian determinant. A manifold together with such a choice is oriented; a manifold admitting a choice is orientable. On each connected component of a positive-dimensional orientable manifold, exactly two orientations exist.

Equivalent descriptions

For a smooth manifold, the following data determine the same orientation:

an orientation of $TM$ ;
an equivalence class of positively oriented atlases; or
a nowhere-vanishing smooth $n$ -form, where two such forms are equivalent when one is a positive smooth multiple of the other.

The equivalence uses smooth partitions of unity to construct a global positive top-degree form from compatible local choices; see Lee, Chapter 15.

Maps and boundaries

A local diffeomorphism between oriented $n$ -manifolds is orientation-preserving when its differential sends positive bases to positive bases, equivalently when its Jacobian in oriented charts is positive. If $M$ has boundary, the outward-normal-first convention orients $\partial M$ : a boundary basis $(v_1,\ldots,v_{n-1})$ is positive when $(\nu,v_1,\ldots,v_{n-1})$ is positive for an outward-pointing $\nu$ .

Examples and scope

The standard ordered coordinates orient $\mathbb{R}^n$ . Every complex manifold has a canonical orientation because a complex basis gives a real basis $(v_1,iv_1,\ldots,v_m,iv_m)$ . The Möbius band is not orientable. Orientability is a property; an orientation is one of the possible choices, so an orientable manifold is not automatically an oriented manifold.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. DOI record. Relevant: Chapter 15.
L. W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: §21.3.