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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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...
On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...
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 physical sizes). ...
A manifold has a volume form if and only if it is orientable, and orientable manifolds have infinitely many volume forms (details below).
The definitions above refer more precisely to even differential n-forms. There exist also so-called odd or twisted or pseudo- n-forms which can be defined and be non-vanishing even on non-orientable manifolds (some details below). The following discussion concerns mostly even n-forms, and the adjective "even" will be omitted for simplicity.
Many classes of manifolds come with canonical volume forms, that is, they have extra structure which allows the choice of a preferred volume form.
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  is a deformation retract (since  , 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 a n-manifold M, for a given structure group G (which is a Lie subgroup of the general linear group GL(n)) is a G-subbundle of the frame bundle on M. The notion of G-structures includes many other structures on manifolds...
In topology, a retraction, as the name suggests, retracts an entire space into a subspace. ...
In the language of line bundles, n-forms are the line bundle Ωn(M) = Λn(T * M) of top exterior powers. Triviality of this bundle 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. ...
Relation to measures For nonorientable manifolds, a volume pseudo-form, also called odd or twisted volume form, may be defined as a section of the orientation bundle; this definition also applies for orientable manifolds. 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 In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ...
 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 single variable calculus, writing  considers dx as a volume form, not simple a measure, and  indicates "integrate over the cell [a,b] with the opposite orientation, sometimes denoted ![overline{[a,b]}](http://upload.wikimedia.org/math/0/8/8/088c92f6797a6e776d3327bd5cf9d83a.png) ". For other uses, see Calculus (disambiguation). ...
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,Σ), if a measure ν on (X,Σ) is absolutely continuous with respect to a sigma-finite measure μ on (X,Σ), then there is a measurable function f on X and taking values in...
Absolute continuity of real functions In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n...
Twisted differential forms were apparently first introduced by de Rham. Georges de Rham (10 September 1903-9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. ...
Examples
Lie groups For any Lie group, a natural volume form may be defined by translation. That is, if ωe is an element of  , then a left-invariant form may be defined by  , where Lg 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 mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...
In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...
Symplectic manifolds Any symplectic manifold (or indeed any almost 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. In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...
In differential geometry, an almost symplectic structure on a differentiable manifold M is a two-form ω on M which is everywhere non-singular. ...
In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...
In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that the map from V to V* (the dual space of V) given by v f(-,v) is not a bijection. ...
In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ...
Riemannian volume form Any Riemannian or pseudo-Riemannian manifold has a natural volume pseudo-form. In local coordinates, it can be expressed as In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...
Local coordinates are measurement indices into a local coordinate system or a local coordinate space. ...
 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 dxi are the 1-forms providing a basis for the cotangent bundle of the manifold. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
(Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...
In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...
A number of different notations are in use for the volume form. These include
 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 signficant linear map introduced in general by W. V. D. Hodge. ...
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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
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  and a mapping function An open surface with X-, Y-, and Z-contours shown. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
 thus defining a surface embedded in  . The Jacobian matrix of the mapping is In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
 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 = λTλ on the set U, with matrix elements In mathematics a metric or distance function is a function which defines a distance between elements of a set. ...
 The determinant of the metric is given by In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
 where  is the wedge product. For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...
Now consider a change of coordinates on U, given by a diffeomorphism In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
 so that the coordinates (u1,u2) are given in terms of (v1,v2) by (u1,u2) = f(v1,v2). The Jacobian matrix of this transformation is given by  In the new coordinates, we have  and so the metric transforms as  where  is the metric in the v coordinate system. The determinant is  . 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  is given by the integral  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. This is a geometric form of the Radon–Nikodym theorem. In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ...
In mathematics, the Radon–Nikodym theorem is a result in functional analysis that states that, given a measurable space (X,Σ), if a measure ν on (X,Σ) is absolutely continuous with respect to a sigma-finite measure μ on (X,Σ), then there is a measurable function f on X and taking values in...
Given a non-vanishing function f on M, and a volume form ω, fω 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. Intrinsically, it is the Radon–Nikodym derivative of ω' with respect to ω. In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...
In mathematics, the Radon–Nikodym theorem is a result in functional analysis that states that, given a measurable space (X,Σ), if a measure ν on (X,Σ) is absolutely continuous with respect to a sigma-finite measure μ on (X,Σ), then there is a measurable function f on X and taking values in...
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 ωM,ωN, for any points  , there is a map  (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):  . 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  , define
 Then the standard Lebesgue measure dx pulls back to ω under f: ω = f * dx. Concretely,  . In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is...
In higher dimensions, given any point  , it has a neighborhood locally homeomorphic to  , 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 . On a disconnected manifold, the volume of each connected component is the invariant.
In symbol, if  is a homeomorphism of manifolds that pulls back ωN to ωM, then
| μ(N) = | ∫ | ωN = | ∫ | ωN = | ∫ | f * ωN = | ∫ | ωM = 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 the case of an infinite sheeted cover (such as  ), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...
See also In mathematics, the Poincaré metric is the natural metric tensor for Poincaré half-plane model of hyperbolic geometry. ...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
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 physical sizes). ...
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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