Volume form
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 formula_1 of dimension formula_2, a volume form is an formula_2-form. It is an element of the space of sections of the line bundle formula_4, denoted as formula_5. 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 function yields another volume form. On non-orientable manifolds, one may instead define the weaker  notion of a density. 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 formula_2th 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 formula_7  A volume form formula_8 on formula_1 gives rise to an orientation in a natural way as the atlas of coordinate charts on formula_1 that send formula_8 to a positive multiple of the Euclidean volume form formula_12 A volume form also allows for the specification of a preferred class of frames on formula_7 Call a basis of tangent vectors formula_14 right-handed if formula_15 The collection of all right-handed frames is acted upon by the group formula_16 of general linear mappings in formula_2 dimensions with positive determinant. They form a principal formula_16 sub-bundle of the linear frame bundle of  formula_19 and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of formula_1 to a sub-bundle with structure group formula_21 That is to say that a volume form gives rise to formula_16-structure on formula_7 More reduction is clearly possible by considering frames that have Thus a volume form gives rise to an formula_24-structure as well. Conversely, given an formula_24-structure, one can recover a volume form by imposing () for the special linear frames and then solving for the required formula_2-form formula_8 by requiring homogeneity in its arguments. A manifold  is orientable if and only if it has a nowhere-vanishing volume form. Indeed, formula_28 is a deformation retract since formula_29 where the positive reals are embedded as scalar matrices. Thus every formula_16-structure is reducible to an formula_24-structure, and formula_16-structures coincide with orientations on formula_7 More concretely, triviality of the determinant bundle formula_34 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 formula_8 on an oriented manifold, the  density formula_36 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 formula_8 (and therefore also any volume form) defines a measure on the Borel sets by formula_38 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 formula_39 considers formula_40 as a volume form, not simply a measure, and formula_41 indicates "integrate over the cell formula_42 with the opposite orientation , sometimes denoted formula_43". 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 formula_8 on formula_19 one can define the divergence of a vector field formula_46 as the unique scalar-valued function, denoted by formula_47 satisfying formula_48 where formula_49 denotes the Lie derivative along formula_46 and formula_51 denotes the interior product or the left contraction of formula_8 along formula_53 If formula_46 is a compactly supported vector field and  formula_1 is a manifold with boundary, then Stokes' theorem implies formula_56 which is a generalization of the divergence theorem. The solenoidal vector fields are those with formula_57 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 formula_58 is an element of formula_59 then a left-invariant form may be defined by formula_60 where formula_61 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 formula_1 is a formula_63-dimensional manifold with symplectic form formula_64 then formula_65 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 formula_66 where the formula_67 are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, formula_68 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 formula_69 Here, the formula_70 is the Hodge star, thus the last form, formula_71 emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita "tensor" formula_72 Although the Greek letter formula_8 is frequently used to denote the volume form, this notation is not universal; the symbol formula_8 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 formula_75 on formula_19 and a volume form formula_64 formula_78 is a volume form on formula_7 Conversely, given two volume forms formula_80 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 formula_81 with respect to formula_82 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 . That is, for every point formula_83 in formula_19 there is an open neighborhood formula_85 of formula_83 and a diffeomorphism formula_87 of formula_85 onto an open set in formula_89 such that the volume form on formula_85 is the pullback of formula_91 along formula_92 As a corollary, if  formula_1 and formula_94 are two manifolds, each with volume forms formula_95 then for any points formula_96 there are open neighborhoods formula_85 of formula_98 and formula_99 of formula_2 and a map formula_101 such that the volume form on formula_94 restricted to the neighborhood formula_99 pulls back to volume form on formula_1 restricted to the neighborhood formula_85: formula_106 In one dimension, one can prove it thus: given a volume form formula_8 on formula_108 define formula_109 Then the standard Lebesgue measure formula_40 pulls back to formula_8 under formula_75: formula_113 Concretely, formula_114 In higher dimensions, given any point formula_115 it has a neighborhood locally  homeomorphic to formula_116 and one can apply the same procedure. Global structure: volume. A volume form on a connected manifold formula_1 has a single global invariant, namely the (overall) volume, denoted formula_118 which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on formula_119 On a disconnected manifold, the volume of each connected component is the invariant. In symbols, if formula_120 is a homeomorphism of manifolds that pulls back formula_121 to formula_122 then formula_123 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 formula_124), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.