Current (mathematics)

Short description: Distributions on spaces of differential forms

In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M.

## Contents

* 1 Definition
* 2 Homological theory
* 3 Topology and norms
* 4 Examples
* 5 See also
* 6 Notes
* 7 References

## Definition

Let Ωcm(M) denote the space of smooth m-forms with compact support on a smooth manifold M. A current is a linear functional on Ωcm(M) which is continuous in the sense of distributions. Thus a linear functional T:Ωcm(M)->ℝ is an m-dimensional current if it is continuous in the following sense: If a sequence ωk of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when k tends to infinity, then T(ωk) tends to 0.

The space 𝒟m(M) of m-dimensional currents on M is a real vector space with operations defined by (T+S)(ω):=T(ω)+S(ω),(λT)(ω):=λT(ω).

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current T∈𝒟m(M) as the complement of the biggest open set U⊂M such that T(ω)=0 whenever ω∈Ωcm(U)

The linear subspace of 𝒟m(M) consisting of currents with support (in the sense above) that is a compact subset of M is denoted ℰm(M).

## Homological theory

Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by M: M(ω)=∫Mω.

If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has: ∂M(ω)=∫∂Mω=∫Mdω=M(dω).

This relates the exterior derivative d with the boundary operator ∂ on the homology of M.

In view of this formula we can define a boundary operator on arbitrary currents ∂:𝒟m+1->𝒟m via duality with the exterior derivative by (∂T)(ω):=T(dω) for all compactly supported m-forms ω.

Certain subclasses of currents which are closed under ∂ can be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.

## Topology and norms

The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence Tk of currents, converges to a current T if Tk(ω)->T(ω),∀ω.

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If ω is an m-form, then define its comass by ‖ω‖:=sup⁡{|⟨ω,ξ⟩|:ξ is a unit, simple, m-vector}.

So if ω is a simple m-form, then its mass norm is the usual L∞-norm of its coefficient. The mass of a current T is then defined as 𝐌(T):=sup⁡{T(ω):supx||ω(x)||≤1}.

The mass of a current represents the weighted area of the generalized surface. A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting point of homological integration.

An intermediate norm is Whitney's flat norm, defined by 𝐅(T):=inf⁡{𝐌(T−∂A)+𝐌(A):A∈ℰm+1}.

Two currents are close in the mass norm if they coincide away from a small part. On the other hand, they are close in the flat norm if they coincide up to a small deformation.

## Examples

Recall that Ωc0(ℝn)≡Cc∞(ℝn) so that the following defines a 0-current: T(f)=f(0).

In particular every signed regular measure μ is a 0-current: T(f)=∫f(x)dμ(x).

Let (x, y, z) be the coordinates in ℝ3. Then the following defines a 2-current (one of many): T(adx∧dy+bdy∧dz+cdx∧dz):=∫01∫01b(x,y,0)dxdy.

## See also

* Georges de Rham
* Herbert Federer
* Differential geometry
* Varifold

## Notes

## References

* de Rham, Georges (1984). Differentiable manifolds. Forms, currents, harmonic forms. Grundlehren der mathematischen Wissenschaften. 266. With an introduction by S. S. Chern. (Translation of 1955 French original ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-642-61752-2. ISBN 3-540-13463-8.
* Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7.
* Griffiths, Phillip; Harris, Joseph (1978). Principles of algebraic geometry. Pure and Applied Mathematics. New York: John Wiley & Sons. doi:10.1002/9781118032527. ISBN 0-471-32792-1.
* Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis. 3. Canberra: Centre for Mathematical Analysis at Australian National University. 1983. ISBN 0-86784-429-9. https://projecteuclid.org/proceedings/proceedings-of-the-centre-for-mathematics-and-its-applications/Lectures-on-Geometric-Measure-Theory/toc/pcma/1416406261.
* Whitney, Hassler (1957). Geometric integration theory. Princeton Mathematical Series. 21. Princeton, NJ and London: Princeton University Press and Oxford University Press. doi:10.1515/9781400877577. ISBN 9780691652900. .
* Lin, Fanghua; Yang, Xiaoping (2003), Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), 1, Beijing/Boston: Science Press/International Press, pp. x+237, ISBN 978-1-57146-125-4

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