Everything about Volume Form totally explained
In
mathematics, a
volume form is a nowhere zero
differential n-form on an
n-
manifold. Every volume form defines a
measure on the manifold, and thus a means to calculate volumes in a generalized sense.
A manifold has a volume form if and only if it's orientable, and orientable manifolds have infinitely many volume forms (
details below).
There is a generalized notion of
pseudo-volume form which exists on any manifold, orientable or not.
Many classes of manifolds come with canonical (pseudo-)volume forms, that is, they've extra structure which allows the choice of a preferred volume form.
In the complex setting, a
Kähler manifold with a
holomorphic volume form is a
Calabi–Yau manifold.
Definition
A
volume form is a nowhere vanishing
differential form of top degree (
n-form on an
n-manifold).
In the language of
line bundles,
n-forms are sections of the line bundle
of top exterior powers, called the
determinant line bundle.
For nonorientable manifolds, a volume
pseudo-form, also called
odd or
twisted volume form, may be defined as a nowhere vanishing section of the
orientation bundle; this definition also applies for orientable manifolds. In this context (untwisted) differential forms are specified as
even n-forms; unless one is specifically discussing twisted forms, the adjective "even" is omitted for simplicity.
Twisted differential forms were apparently first introduced by
de Rham.
Orientation
A manifold has a volume form if and only if it's orientable; this can be taken as a definition of orientability.
In the language of
G-structures, a volume form is an SL-structure,
As
), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.
Further Information
Get more info on 'Volume Form'.
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