Orientability of a smooth manifold - Imported Knowlpedia Content

Definition

An $n$ -dimensional smooth manifold $M$ is orientable if its tangent bundle $TM$ admits an orientation. Equivalently, $M$ has an atlas whose transition maps all have positive Jacobian determinant. Orientability is an existence property: an orientable manifold need not come with a selected orientation. A manifold together with such a selection is oriented. If no orientation exists, the manifold is nonorientable. These terms apply componentwise when the dimension is locally constant rather than globally fixed.

Equivalent tests

For an ordinary second-countable smooth $n$ -manifold, the following are equivalent:

$TM$ is orientable;
$M$ has an atlas with positive transition determinants; and
$M$ admits a nowhere-vanishing smooth $n$ -form.

Equivalently, the top exterior-power line bundle $\bigwedge^nT^*M$ is trivial. The equivalence between compatible local choices and a global top form uses smooth partitions of unity Lee, chapter on orientations.

Choices on connected components

Every connected orientable manifold of positive dimension has exactly two orientations. For a disconnected manifold, orientations may be chosen independently on its connected components. A zero-dimensional manifold is orientable and has the canonical orientation determined by the unique ordered basis of each zero-dimensional tangent space.

Examples and obstruction

Euclidean space and every sphere are orientable. Every complex manifold is canonically orientable after forgetting its complex structure. The Möbius band is nonorientable, and real projective $n$ -space is orientable exactly when $n$ is odd. The first Stiefel–Whitney class of $TM$ is the standard cohomological obstruction to orientability Milnor and Stasheff, §§4 and 9.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: the chapter on orientations.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: §§4 and 9, Stiefel–Whitney classes and oriented vector bundles.