Manifold Orientation

An orientation on an n-dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an orientation exists on M, then M is called orientable.


Not all manifolds are orientable, as exemplified by the Möbius strip and the Klein bottle, illustrated above.


However, an (n-1)-dimensional submanifold of R^n is orientable iff it has a unit normal vector field. The choice of unit determines the orientation of the submanifold. For example, the sphere S^2 is orientable.

Some types of manifolds are always orientable. For instance, complex manifolds, including varieties, and also symplectic manifolds are orientable. Also, any unoriented manifold has a double cover which is oriented.

A map f:M->N between oriented manifolds of the same dimension is called orientation preserving if the volume form on N pulls back to a positive volume form on M. Equivalently, the differential df maps an oriented basis in TM to an oriented basis in TN.

See also

Bundle Orientation, Differential k-Form, Orientable Manifold, Vector Space Orientation, Volume Form

This entry contributed by Todd Rowland

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

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