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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 ...
6.7 Volume form. 6.8 Hodge operator on p-forms. 6.9 Codifferential on p-forms. 6.10 Laplacian on functions. ... is also a Riemannian metric on . We say that ~ is ...
An oriented -dimensional Riemannian manifold (,) has a unique -form called the Riemannian volume form. [7] The Riemannian volume form is preserved by orientation-preserving isometries. [8] The volume form gives rise to a measure on which allows measurable functions to be integrated. [citation needed] If is compact, the volume of is . [7]
On an oriented Riemannian manifold of dimension n, the volume element is a volume form equal to the Hodge dual of the unit constant function, () =: = Equivalently, the volume element is precisely the Levi-Civita tensor ϵ {\displaystyle \epsilon } . [ 1 ]
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. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume.
For each individual value of the parameter t, the immersion f t induces a Riemannian metric on S, which itself induces a differential form on S known as the Riemannian volume form ω t. The first variation of area refers to the computation
Every almost Hermitian manifold M has a canonical volume form which is just the Riemannian volume form determined by g. This form is given in terms of the associated (1,1)-form ω by =!, where ω n is the wedge product of ω with itself n times.