Current (mathematics)

 ( Differential Topology ) 

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 () spread out along subsets of M.

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Definition Let $\backslash Omega\_c^m(M)$ denote the space of smooth m-forms with compact support on a smooth manifold $M.$ A current is a linear functional on $\backslash Omega\_c^m(M)$ which is continuous in the sense of distributions. Thus a linear functional $$T\; :\; \backslash Omega\_c^m(M)\backslash to\; \backslash R$$ is an m-dimensional current if it is continuous in the following sense: If a sequence $\backslash omega\_k$ of smooth forms, all supported in the same Compact space, is such that all derivatives of all their coefficients tend uniformly to 0 when $k$ tends to infinity, then $T(\backslash omega\_k)$ tends to 0.

The space $\backslash mathcal\; D\_m(M)$ of m-dimensional currents on $M$ is a Real number vector space with operations defined by $$(T+S)(\backslash omega)\; :=\; T(\backslash omega)+S(\backslash omega),\backslash qquad\; (\backslash lambda\; T)(\backslash omega)\; :=\; \backslash lambda\; T(\backslash omega).$$

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current $T\; \backslash in\; \backslash mathcal\{D\}\_m(M)$ as the complement of the biggest open set $U\; \backslash subset\; M$ such that $$T(\backslash omega)\; =\; 0$$ whenever $\backslash omega\; \backslash in\; \backslash Omega\_c^m(U)$

The linear subspace of $\backslash mathcal\; D\_m(M)$ consisting of currents with support (in the sense above) that is a compact subset of $M$ is denoted $\backslash mathcal\; E\_m(M).$

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Homological theory Integral over a compact Rectifiable set oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by $M$: $$M(\backslash omega)=\backslash int\_M\; \backslash omega.$$

If the boundary ∂ M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has: $$\backslash partial\; M(\backslash omega)\; =\; \backslash int\_\{\backslash partial\; M\}\backslash omega\; =\; \backslash int\_M\; d\backslash omega\; =\; M(d\backslash omega).$$

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 $$\backslash partial\; :\; \backslash mathcal\; D\_\{m+1\}\; \backslash to\; \backslash mathcal\; D\_m$$ via duality with the exterior derivative by $$(\backslash partial\; T)(\backslash omega)\; :=\; T(d\backslash omega)$$ for all compactly supported m-forms $\backslash omega.$

Certain subclasses of currents which are closed under $\backslash partial$ 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.

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Topology and norms The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence $T\_k$ of currents, Limit point to a current $T$ if $$T\_k(\backslash omega)\; \backslash to\; T(\backslash omega),\backslash qquad\; \backslash forall\; \backslash omega.$$

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If $\backslash omega$ is an m-form, then define its comass by $$\backslash |\backslash omega\backslash |\; :=\; \backslash sup\backslash \{\backslash left|\backslash langle\; \backslash omega,\backslash xi\backslash rangle\backslash right|\; :\; \backslash xi\; \backslash mbox\{\; is\; a\; unit,\; simple,\; \}m\backslash mbox\{-vector\}\backslash \}.$$

So if $\backslash omega$ is a Tensor rank m-form, then its mass norm is the usual L^∞-norm of its coefficient. The mass of a current $T$ is then defined as $$\backslash mathbf\; M\; (T)\; :=\; \backslash sup\backslash \{\; T(\backslash omega)\; :\; \backslash sup\_x\; |\backslash vert\backslash omega(x)|\backslash vert\backslash le\; 1\backslash \}.$$

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 $$\backslash mathbf\; F\; (T)\; :=\; \backslash inf\; \backslash \{\backslash mathbf\; M(T\; -\; \backslash partial\; A)\; +\; \backslash mathbf\; M(A)\; :\; A\backslash in\backslash mathcal\; E\_\{m+1\}\backslash \}.$$

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.

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Examples Recall that $$\backslash Omega\_c^0(\backslash R^n)\backslash equiv\; C^\backslash infty\_c(\backslash R^n)$$ so that the following defines a 0-current: $$T(f)\; =\; f(0).$$

In particular every Signed measure regular measure $\backslash mu$ is a 0-current: $$T(f)\; =\; \backslash int\; f(x)\backslash ,\; d\backslash mu(x).$$

Let ( x, y, z) be the coordinates in $\backslash R^3.$ Then the following defines a 2-current (one of many): $$T(a\backslash ,dx\backslash wedge\; dy\; +\; b\backslash ,dy\backslash wedge\; dz\; +\; c\backslash ,dx\backslash wedge\; dz)\; :=\; \backslash int\_0^1\; \backslash int\_0^1\; b(x,y,0)\backslash ,\; dx\; \backslash ,\; dy.$$

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See also

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

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Notes

• (1984). 9783540134633, Springer-Verlag. ISBN 9783540134633
• Herbert Federer (1969). 9783540606567, Springer-Verlag. ISBN 9783540606567
• (1978). 9780471327929, John Wiley & Sons. ISBN 9780471327929
• Leon Simon (1983). 9780867844290, Centre for Mathematical Analysis at Australian National University. . ISBN 9780867844290
• Hassler Whitney (2025). 9780691652900, Princeton University Press and Oxford University Press. ISBN 9780691652900
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