Differential geometry MOC

Volume form

Let be an -dimensional -manifold. A volume form is simply a choice of a top form, #m/def/geo/diff i.e. a differential form of rank . In certain cases there exists a canonical volume form

• Riemannian volume form for an oriented semi-Riemannian manifold.

Properties

1. A nonvanishing volume form induces an orientation on .
2. is orientable iff it admits a nonvanishing volume form .

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