|
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing top-dimensional form (i.e., a differential form of top degree). Thus on a manifold ''M'' of dimension ''n'', a volume form is an ''n''-form, a section of the line bundle , that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing 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 ''n''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 Riemannian manifolds and pseudo-Riemannian manifolds have an associated canonical volume form. == Orientation == 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 ''M''. A volume form ''ω'' on ''M'' gives rise to an orientation in a natural way as the atlas of coordinate charts on ''M'' that send ''ω'' to a positive multiple of the Euclidean volume form . A volume form also allows for the specification of a preferred class of frames on ''M''. Call a basis of tangent vectors (''X''1, ..., ''X''''n'') right-handed if : The collection of all right-handed frames is acted upon by the group GL+(''n'') of general linear mappings in ''n'' dimensions with positive determinant. They form a principal GL+(''n'') sub-bundle of the linear frame bundle of ''M'', and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of ''M'' to a sub-bundle with structure group GL+(''n''). That is to say that a volume form gives rise to GL+(''n'')-structure on ''M''. More reduction is clearly possible by considering frames that have Thus a volume form gives rise to an SL(''n'')-structure as well. Conversely, given an SL(''n'')-structure, one can recover a volume form by imposing () for the special linear frames and then solving for the required ''n''-form ''ω'' by requiring homogeneity in its arguments. A manifold is orientable if and only if it has a volume form. Indeed, is a deformation retract since , where the positive reals are embedded as scalar matrices. Thus every GL+(''n'')-structure is reducible to an SL(''n'')-structure, and GL+(''n'')-structures coincide with orientations on ''M''. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「volume form」の詳細全文を読む スポンサード リンク
|