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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 ...
Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
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]
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.
In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and ...
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
Then a connection on may be specified by a Lie algebra-valued differential form on the total space of the principal bundle. This connection has a curvature form F A {\displaystyle F_{A}} , which is a two-form on X {\displaystyle X} with values in the adjoint bundle ad ( P ) {\displaystyle \operatorname {ad} (P)} of P {\displaystyle P} .
This shows how a minimal submanifold can be characterized either by the critical point theory of the volume functional or by an explicit partial differential equation for the immersion. The special case of the first variation formula arising when S is an interval on the real number line is particularly well-known.
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