Volume Form - Orientation

Orientation

A manifold is orientable if it has a coordinate atlas all of whose transition functions have positive Jacobian determinants. A selection of a maximal such atlas is an orientation on M. A volume form ω on M gives rise to an orientation in a natural way as the atlas of coordinate charts on M that send ω to a positive multiple of the Euclidean volume form .

A volume form also allows for the specification of a preferred class of frames on M. Call a basis of tangent vectors (X₁,...,X_n) right-handed if

The collection of all right-handed frames is acted upon by the group GL+(n) of general linear mappings in n dimensions with positive determinant. They form a principal GL+(n) sub-bundle of the linear frame bundle of M, and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of M to a sub-bundle with structure group GL+(n). That is to say that a volume form gives rise to GL+(n)-structure on M. More reduction is clearly possible by considering frames that have

(1)

Thus a volume form gives rise to an SL(n)-structure as well. Conversely, given an SL(n)-structure, one can recover a volume form by imposing (1) for the special linear frames and then solving for the required n-form ω by requiring homogeneity in its arguments.

A manifold is orientable if and only if it has a volume form. Indeed, SL(n) → GL+(n) is a deformation retract since GL+ = SL × R+, where the positive reals are embedded as scalar matrices. Thus every GL+(n)-structure is reducible to an SL(n)-structure, and GL+(n)-structures coincide with orientations on M. More concretely, triviality of the determinant bundle is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus the existence of a volume form is equivalent to orientability.