Current (mathematics) explained

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.

Definition

Let

\Omega ^{m(M)} _{c} denote the space of smooth m-forms with compact support on a smooth manifold M. A current is a linear functional on \Omega ^{m(M)} _{c} 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 \omega _{k} 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(\omega _{k)} tends to 0.

The space

l D _{m(M)} of m-dimensional currents on M is a real 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 \in l{D} _{m(M)} as the complement of the biggest open set U \subset M such that$$T(\backslash omega)\; =\; 0$$ whenever \omega \in \Omega ^{m(U)} _{c}

The linear subspace of

l D _{m(M)} consisting of currents with support (in the sense above) that is a compact subset of M is denoted l E _{m(M).}

Homological theory

Integration over a compact rectifiable 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:$$\backslash partial\; M(\backslash omega)\; =\; \backslash int\_\backslash omega\; =\; \backslash int\_M\; d\backslash omega\; =M(d\backslash omega).$$

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

\omega.

Certain subclasses of currents which are closed under

\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.

Topology and norms

T _{k} of currents, converges 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

\omega is an m-form, then define its comass by$$\backslash |\backslash omega\backslash |\; :=\; \backslash sup\backslash .$$

So if

\omega 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$$\backslash mathbf\; M\; (T)\; :=\; \backslash sup\backslash .$$

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$$\backslash mathbf\; F\; (T)\; :=\; \backslash inf\; \backslash .$$

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

\mu is a 0-current:$$T(f)\; =\; \backslash int\; f(x)\backslash ,\; d\backslash mu(x).$$

Let (x, y, z) be the coordinates in

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

See also

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

References

• Book: 0760450. Georges de Rham. de Rham. Georges. Differentiable manifolds. Forms, currents, harmonic forms. F. R.. Smith. With an introduction by S. S. Chern.. Translation of 1955 French original. Grundlehren der mathematischen Wissenschaften. 266. Springer-Verlag. Berlin. 1984. 3-540-13463-8. 0534.58003. 10.1007/978-3-642-61752-2.
• Book: Federer, Herbert. Herbert Federer. Geometric measure theory. Berlin–Heidelberg–New York. Springer-Verlag. Die Grundlehren der mathematischen Wissenschaften. 153. 1969. 978-3-540-60656-7. 0257325. 0176.00801 . 10.1007/978-3-642-62010-2.
• Book: Griffiths. Phillip. Harris. Joseph. Principles of algebraic geometry. Pure and Applied Mathematics. John Wiley & Sons. New York. 1978. 0-471-32792-1. 0507725. Phillip Griffiths. Joe Harris (mathematician). 0408.14001. 10.1002/9781118032527.
• Book: Simon, Leon . Leon Simon . Lectures on geometric measure theory . Canberra . . Proceedings of the Centre for Mathematical Analysis . 3 . 1983 . 0-86784-429-9 . 0756417 . 0546.49019 .
• Book: Whitney , Hassler . Hassler Whitney. 10.1515/9781400877577. Geometric integration theory. 9780691652900. Princeton Mathematical Series. 21. Princeton University Press and Oxford University Press. Princeton, NJ and London. 1957. 0087148. 0083.28204. .