Volume form

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In mathematics, a volume form is a nowhere zero differential n-form on an oriented manifold. Every volume form defines a measure on the manifold, and thus a means to calculate volumes in a generalized sense.

In general a manifold may have no volume form, or infinitely many volume forms. Many classes of manifolds do come with canonical volume forms, that is, they have extra structure which allows the choice of a preferred volume form.

1 Orientation
2 Lie groups
3 Symplectic manifolds
4 Riemannian volume form
- 4.1 Example: Volume form of a surface
5 See also
6 References

[edit] Orientation

A nonoriented manifold may not have a volume form. In fact, a manifold has a volume form if and only if it is orientable. This is often taken as the definition of orientability. For nonoriented manifolds, a volume pseudo-form may be defined as a section of the orientation bundle. Any manifold admits a volume pseudo-form. If the nonoriented manifold is orientable, then for any volume form ω of one of the two corresponding oriented manifolds, the density |ω| is a volume pseudo-form on the nonoriented manifold.

Any volume pseudo-form ω (and therefore also any volume form) defines a measure on the Borel sets by

μ_ω(U) =	∫	ω.
	U

[edit] Lie groups

For any Lie group, a natural volume form may be defined by translation. That is, if ω_e is an element of $\bigwedge^n T_e^*G$ , then a left-invariant form may be defined by $\omega_g=L_g^*\omega_e$ , where L_g is left-translation. As a corollary, every Lie group is orientable. This volume form is unique up to a scalar, and the corresponding measure is known as the Haar measure.

[edit] Symplectic manifolds

Any symplectic manifold has a natural volume form. If M is a 2n-dimensional manifold with symplectic form ω, then ωⁿ is nowhere zero as a consequence of the nondegeneracy of the symplectic form. As a corollary, any symplectic manifold is orientable (indeed, oriented). If the manifold is both symplectic and Riemannian, then the two volume forms agree if the manifold is Kähler.

[edit] Riemannian volume form

Any Riemannian or pseudo-Riemannian manifold has a natural volume pseudo-form. In local coordinates, it can be expressed as

$\omega = \sqrt{|g|} dx^1\wedge ... \wedge dx^n$

where the manifold is an n dimensional manifold. Here, $| g |$ is the absolute value of the determinant of the metric tensor on the manifold. The $d x i$ are the 1-forms providing a basis for the cotangent bundle of the manifold.

A number of different notations are in use for the volume form. These include

ω = vol n = ε = * (1)

Here, the * is the Hodge dual, thus the last form, *(1), emphasizes that the volume form is the Hodge dual of the constant map on the manifold.

Although the Greek letter ω is frequently used to denote the volume form, this notation is hardly universal; the symbol ω often carries many other meanings in differential geometry; thus, the appearance of ω in a formula does not necessarily mean that it is the volume form.

[edit] Example: Volume form of a surface

A simple example of a volume form can be explored by considering a two-dimensional surface embedded in n-dimensional Euclidean space. Consider a subset $U \subset \mathbb{R}^2$ and a mapping function

$\phi:U\to \mathbb{R}^n$

thus defining a surface embedded in $\mathbb{R}^n$ . The Jacobian matrix of the mapping is

$\lambda_{ij}=\frac{\partial \phi_i} {\partial u_j}$

with index i running from 1 to n, and j running from 1 to 2. The Euclidean metric in the n-dimensional space induces a metric $g = λ T λ$ on the set U, with matrix elements

$g_{ij}=\sum_{k=1}^n \lambda_{ki} \lambda_{kj} = \sum_{k=1}^n \frac{\partial \phi_k} {\partial u_i} \frac{\partial \phi_k} {\partial u_j}$

The determinant of the metric is given by

$\det g = \left| \frac{\partial \phi} {\partial u_1} \wedge \frac{\partial \phi} {\partial u_2} \right|^2$

where $\wedge$ is the wedge product. For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.

Now consider a change of coordinates on U, given by a diffeomorphism

$f:U\to U$

so that the coordinates $(u 1, u 2)$ are given in terms of $(v 1, v 2)$ by $(u 1, u 2) = f (v 1, v 2)$ . The Jacobian matrix of this transformation is given by

$F_{ij}= \frac{\partial f_i} {\partial v_j}$

In the new coordinates, we have

$\frac{\partial \phi_i} {\partial v_j} = \sum_{k=1}^2 \frac{\partial \phi_i} {\partial u_k} \frac{\partial f_k} {\partial v_j}$

and so the metric transforms as

$\tilde{g} = F^T g F$

where $\tilde{g}$ is the metric in the v coordinate system. The determinant is

$\det \tilde{g} = \det g (\det F)^2$ .

Given the above construction, it should now be straightforward to understand how the volume form is invariant under a change of coordinates. In two dimensions, the volume is just the area. The area of a subset $B\subset U$ is given by the integral

$\mbox{Area}(B) = \int \int_B \sqrt{\det g}\; du_1 du_2 = \int \int_B \sqrt{\det g} \;\det F \;dv_1 dv_2 = \int \int_B \sqrt{\det \tilde{g}} \;dv_1 dv_2$

Thus, in either coordinate system, the volume form takes the same expression: the expression of the volume form is invariant under a change of coordinates.

Note that there was nothing particular to two dimensions in the above presentation; the above trivially generalizes to arbitrary dimensions.

[edit] See also

Poincaré metric provides a review of the volume form on the complex plane.
measure (mathematics)

[edit] References

Michael Spivak, Calculus on Manifolds, (1965) W.A. Benjamin, Inc. Reading, Massachusetts ISBN 0-8053-9021-9 (Provides an elementary introduction to the modern notation of differential geometry, assuming only a general calculus background)

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Categories: Differential forms | Determinants