A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a *twisted volume form* or *pseudo-volume form*. It also defines a measure, but exists on any differentiable manifold, orientable or not.

Kähler manifolds, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the *n*th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented pseudo-Riemannian manifolds have an associated canonical volume form.

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 (*X*_{1}, ..., *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

This page was last edited on 20 December 2017, at 13:31.

Reference: https://en.wikipedia.org/wiki/Volume_form under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Volume_form under CC BY-SA license.

- Mathematics
- Differentiable Manifold
- Differential Form
- Section
- Line Bundle
- Orientable Manifold
- Density
- Integral
- Function
- Measure
- Lebesgue Integral
- Volume Element
- KÃ¤hler Manifolds
- Complex Manifolds
- Exterior Power
- Symplectic Manifold
- Pseudo-Riemannian Manifolds
- Orientable
- Coordinate Atlas
- Jacobian Determinants
- Frames
- Acted Upon
- Group
- General Linear
- Principal GL+(n) Sub-bundle
- Linear Frame Bundle
- GL+(n)-structure

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