Orientable manifold

Orientation of vector bundles

A real vector bundle, which a priori has a ${\displaystyle \operatorname {GL} (n)}$structure group, is called orientable when the structure group may be reduced to ${\displaystyle \operatorname {GL} ^{+}(n)}$, the group of matrices with positive determinant. For the tangent bundle, this reduction is always possible if the underlying base manifold is orientable and in fact this provides a convenient way to define the orientability of a smooth real manifold: a smooth manifold is defined to be orientable if its tangent bundle is orientable (as a vector bundle). Note that as a manifold in its own right, the tangent bundle is always orientable, even over nonorientable manifolds.