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

denote the space of smooth m-forms with compact support on a smooth manifold A current is a linear functional which is continuous in the sense of distributions is an m-dimensional current if it is continuous of smooth forms, all supported in the same compact set tends to 0.

of m is a real vector space

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support as the complement of the biggest open set

The linear subspace

Homological theory

Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m

If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem

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 for all compactly supported m

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 of currents, converges

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm is an m-form, then define its comass

is a simple m-form, then its mass norm is the usual L∞-norm of its coefficient. The mass

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

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

In particular every signed regular measure

Let (x, y, z

See also

• Georges de Rham

• Herbert Federer

• Differential geometry

• Varifold

Notes

References

• .