It is a differential form of degree equal to the differentiable manifold dimension. A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.
A volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. Keep an eye: not every measure comes from a volume form (see here
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: