Volume form - Citizendia

In mathematics, a volume form is a nowhere zero differential n-form on an n-manifold. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Every volume form defines a measure on the manifold, and thus a means to calculate volumes in a generalized sense. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

A manifold has a volume form if and only if it is 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 have 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. In Mathematics, a Kähler manifold is a Manifold with unitary structure (a ''U''(''n''-structure) satisfying an Integrability condition Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial

1 Definition
2 Orientation
3 Relation to measures
4 Examples
5 Invariants of a volume form
- 5.1 No local structure
- 5.2 Global structure: volume
6 See also
7 References

Definition

A volume form is a nowhere vanishing differential form of top degree (n-form on an n-manifold). In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is

In the language of line bundles, n-forms are sections of the line bundle $Ω n (M) = Λ n (T * M)$ of top exterior powers, called the determinant line bundle. In Mathematics, a line bundle expresses the concept of a line that varies from point to point of a space

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. Georges de Rham ( 10 September 1903 &ndash 9 October 1990) was a Swiss Mathematician, known for his contributions to

Orientation

A manifold has a volume form if and only if it is orientable; this can be taken as a definition of orientability.

In the language of G-structures, a volume form is an SL-structure, As $\mbox{SL} \to \mbox{GL}^+$ is a deformation retract (since $\mbox{GL}^+ = \mbox{SL} \times \mathbf{R}^+$ , where the positive reals are embedded as scalar matrices), a manifold admits an SL-structure if and only if it admits a $GL +$ -structure, which is an orientation. In Differential geometry, a G -structure on an n - Manifold M, for a given Structure group G, is a G

In the language of line bundles, triviality of the determinant bundle $Ω n (M)$ is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere vanishing section, so again, the existence of a volume form is equivalent to orientability. In Mathematics, a line bundle expresses the concept of a line that varies from point to point of a space

For pseudo-volume forms, a pseudo-volume form is an $\mbox{SL}^\pm$ -structure, and since $\mbox{SL}^\pm \to \mbox{GL}$ is a homotopy equivalence (indeed, a deformation retract), every manifold admits a pseudo-volume form. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical Similarly, the orientation bundle is always trivial, so every manifold admits a pseudo-volume form.

Relation to measures

Any manifold admits a volume pseudo-form, as the orientation bundle is trivial (as a line bundle). Given a volume form ω on an oriented manifolds, the density |ω| is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation.

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

$\mu_\omega(U)=\int_U\omega. \,\!$

The difference is that while a measure can be integrated over a (Borel) subset, a volume form can only be integrated over an oriented cell. In Mathematics, the Borel algebra (or Borel &sigma-algebra) on a Topological space X is a &sigma-algebra of Subsets of In single variable calculus, writing $\int_b^a f\,dx = -\int_a^b f\,dx$ considers $d x$ as a volume form, not simple a measure, and $\int_b^a$ indicates "integrate over the cell $[a, b]$ with the opposite orientation, sometimes denoted $\overline{[a,b]}$ ". Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form needn't be absolutely continuous. In Mathematics, the Radon–Nikodym theorem is a result in Functional analysis that states that given a measurable space ( X,&Sigma if a In Mathematics, one may talk about absolute continuity of functions and absolute continuity of measures, and these two notions are closely connected

Examples

Lie groups

For any Lie group, a natural volume form may be defined by translation. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group 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. In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define

Symplectic manifolds

Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Differential geometry, an almost symplectic structure on a Differentiable manifold M is a two-form ω on M which is everywhere 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. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that 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. In Mathematics, a Kähler manifold is a Manifold with unitary structure (a ''U''(''n''-structure) satisfying an Integrability condition

Riemannian volume form

Any Riemannian (or pseudo-Riemannian) manifold has a natural volume (or pseudo volume) form. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In local coordinates, it can be expressed as

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

where the manifold is an n dimensional manifold. Local coordinates are measurement indices into a local Coordinate system or a local Coordinate space. Here, $| g |$ is the absolute value of the determinant of the metric tensor on the manifold. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space The $d x i$ are the 1-forms providing a basis for the cotangent bundle of the manifold. In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces

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

$\omega = \mathrm{vol}_n = \epsilon = *(1) . \,\!$

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. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W

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 (such as a symplectic form); thus, the appearance of ω in a formula does not necessarily mean that it is the volume form. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry

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. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Consider a subset $U \subset \mathbf{R}^2$ and a mapping function

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

thus defining a surface embedded in $\mathbf{R}^n$ . 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. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. 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

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

where $\wedge$ is the wedge product. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n 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

$f \colon U\to U , \,\!$

so that the coordinates $(u 1, u 2)$ are given in terms of $(v 1, v 2)$ by $(u 1, u 2) = f (v 1, v 2)$ . In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable 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

$\begin{align} \mbox{Area}(B) &= \iint_B \sqrt{\det g}\; du_1 du_2 \\ &= \iint_B \sqrt{\det g} \;\det F \;dv_1 dv_2 \\ &= \iint_B \sqrt{\det \tilde{g}} \;dv_1 dv_2 \end{align}$

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.

Invariants of a volume form

Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. In Mathematics, a principal homogeneous space, or torsor, for a group G is a set X on which G acts freely and This is a geometric form of the Radon–Nikodym theorem. In Mathematics, the Radon–Nikodym theorem is a result in Functional analysis that states that given a measurable space ( X,&Sigma if a

Given a non-vanishing function f on M, and a volume form $ω$ , $f ω$ is a volume form on M. Conversely, given two volume forms $ω,ω'$ , their ratio is a non-vanishing function (positive if they define the same orientation, negative if they define opposite orientations).

In coordinates, they are both simply a non-zero function times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to Intrinsically, it is the Radon–Nikodym derivative of $ω'$ with respect to $ω$ . In Mathematics, the Radon–Nikodym theorem is a result in Functional analysis that states that given a measurable space ( X,&Sigma if a

No local structure

A volume form has no local structure: any two volume forms (on manifolds of the same dimension) are locally isomorphic.

Formally, this statement means that given two manifolds of the same dimension $M, N$ with volume forms $ω M,ω N$ , for any points $m\in M, n\in N$ , there is a map $f\colon U \to V$ (where $U$ is a neighborhood of $m$ and $V$ is a neighborhood of $n$ ) such that the volume form on $N$ (restricted to the neighborhood $V$ ) pulls back to volume form on $M$ (restricted to the neighborhood $U$ ): $f^*\omega_N\vert_V = \omega_M\vert_U$ . Differentiable manifolds of a given dimension are locally diffeomorphic; the added criterion is that the volume form pulls back to the volume form.

In one dimension, one can prove it thus: given a volume form $ω$ on $\mathbf{R}$ , define

$f(x) := \int_0^x \omega$

Then the standard Lebesgue measure $d x$ pulls back to $ω$ under f: $ω = f * d x$ . In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from Concretely, $\omega = f\,dx$ .

In higher dimensions, given any point $m \in M$ , it has a neighborhood locally homeomorphic to $\mathbf{R}\times\mathbf{R}^{n-1}$ , and one can apply the same procedure.

Global structure: volume

A volume form on a connected manifold M has a single global invariant, namely the (overall) volume (denoted $μ(M)$ ), which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on $\mathbf{R}^n$ . On a disconnected manifold, the volume of each connected component is the invariant.

In symbol, if $f\colon M \to N$ is a homeomorphism of manifolds that pulls back $ω N$ to $ω M$ , then

μ(N) =	∫	ω_N =	∫	ω_N =	∫	f ^* ω_N =	∫	ω_M = μ(M)
	N		f(M)		M		M

and the manifolds have the same volume.

Volume forms can also be pulled back under covering maps, in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber). In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In the case of an infinite sheeted cover (such as $\mathbf{R} \to S^1$ ), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.

References

Michael Spivak, Calculus on Manifolds, (1965) W. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Michael David Spivak (*1940 in Queens, New York) is a Mathematician specializing in Differential geometry, an expositor of Mathematics 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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