Bundle Orientation

A real vector bundle pi:E->M has an orientation if there exists a covering by trivializations U_i×R^k such that the transition functions are vector space orientation-preserving. Alternatively, there exists a section of the projectivization of the top exterior power of the bundle, P_R( ^ ^kE). A bundle is called orientable if there exists an orientation. Hence a bundle E of bundle rank k is orientable iff  ^ ^kE is a trivial line bundle.

An orientation of the tangent bundle is equivalent to an orientation on the base manifold. Not all bundles are orientable, as can be seen by the tangent bundle of the Möbius strip. The nontrivial line bundle on the circle is also not orientable.

See also

Bundle, Manifold Orientation, Vector Space Orientation, Vector Bundle

This entry contributed by Todd Rowland

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Rowland, Todd. "Bundle Orientation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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