Volume form

Last updated

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold of dimension , a volume form is an -form. It is an element of the space of sections of the line bundle , denoted as . A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.

Contents

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 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.

Orientation

The following will only be about orientability of differentiable manifolds (it's a more general notion defined on any topological manifold).

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 A volume form on gives rise to an orientation in a natural way as the atlas of coordinate charts on that send to a positive multiple of the Euclidean volume form

A volume form also allows for the specification of a preferred class of frames on Call a basis of tangent vectors right-handed if

The collection of all right-handed frames is acted upon by the group of general linear mappings in dimensions with positive determinant. They form a principal sub-bundle of the linear frame bundle of and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of to a sub-bundle with structure group That is to say that a volume form gives rise to -structure on More reduction is clearly possible by considering frames that have

Thus a volume form gives rise to an -structure as well. Conversely, given an -structure, one can recover a volume form by imposing ( 1 ) for the special linear frames and then solving for the required -form by requiring homogeneity in its arguments.

A manifold is orientable if and only if it has a nowhere-vanishing volume form. Indeed, is a deformation retract since where the positive reals are embedded as scalar matrices. Thus every -structure is reducible to an -structure, and -structures coincide with orientations on More concretely, triviality of the determinant bundle is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus, the existence of a volume form is equivalent to orientability.

Relation to measures

Given a volume form on an oriented manifold, the density is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds.

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

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 as a volume form, not simply a measure, and indicates "integrate over the cell with the opposite orientation, sometimes denoted ".

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 need not be absolutely continuous.

Divergence

Given a volume form on one can define the divergence of a vector field as the unique scalar-valued function, denoted by satisfying where denotes the Lie derivative along and denotes the interior product or the left contraction of along If is a compactly supported vector field and is a manifold with boundary, then Stokes' theorem implies which is a generalization of the divergence theorem.

The solenoidal vector fields are those with It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, in fluid mechanics where the divergence of a velocity field measures the compressibility of a fluid, which in turn represents the extent to which volume is preserved along flows of the fluid.

Special cases

Lie groups

For any Lie group, a natural volume form may be defined by translation. That is, if is an element of then a left-invariant form may be defined by where 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.

Symplectic manifolds

Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. If is a -dimensional manifold with symplectic form then 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.

Riemannian volume form

Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as where the are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.

The volume form is denoted variously by

Here, the is the Hodge star, thus the last form, emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita tensor

Although the Greek letter is frequently used to denote the volume form, this notation is not universal; the symbol often carries many other meanings in differential geometry (such as a symplectic form).

Invariants of a volume form

Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. Given a non-vanishing function on and a volume form is a volume form on Conversely, given two volume forms 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 with respect to On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of the Radon–Nikodym theorem.

No local structure

A volume form on a manifold has no local structure in the sense that it is not possible on small open sets to distinguish between the given volume form and the volume form on Euclidean space ( Kobayashi 1972 ). That is, for every point in there is an open neighborhood of and a diffeomorphism of onto an open set in such that the volume form on is the pullback of along

As a corollary, if and are two manifolds, each with volume forms then for any points there are open neighborhoods of and of and a map such that the volume form on restricted to the neighborhood pulls back to volume form on restricted to the neighborhood :

In one dimension, one can prove it thus: given a volume form on define Then the standard Lebesgue measure pulls back to under : Concretely, 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 has a single global invariant, namely the (overall) volume, denoted 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 symbols, if is a homeomorphism of manifolds that pulls back to then 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.

See also

Related Research Articles

<span class="mw-page-title-main">Curl (mathematics)</span> Circulation density in a vector field

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

In vector calculus and differential geometry the generalized Stokes theorem, also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or and the divergence theorem is the case of a volume in Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by Shiing-Shen Chern.

<span class="mw-page-title-main">Frame bundle</span> Functions in mathematics

In mathematics, a frame bundle is a principal fiber bundle associated with any vector bundle . The fiber of over a point is the set of all ordered bases, or frames, for . The general linear group acts naturally on via a change of basis, giving the frame bundle the structure of a principal -bundle.

In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.

In differential geometry, a field in mathematics, Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem.

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x.

In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten, using the Seiberg–Witten theory studied by Nathan Seiberg and Witten during their investigations of Seiberg–Witten gauge theory.

In differential geometry, the integration along fibers of a k-form yields a -form where m is the dimension of the fiber, via "integration". It is also called the fiber integration.

In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces. It was introduced by Raymond D. Holmes and Anthony Charles Thompson.

In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory that admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.

In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear in algebraic geometry, which guarantees the existence of a solution to the special Lagrangian equation inside a Hamiltonian isotopy class of Lagrangian submanifolds. In particular the conjecture contains two difficulties: first it asks what a suitable stability condition might be, and secondly if one can prove stability of an isotopy class if and only if it contains a special Lagrangian representative.

References