Current (mathematics)


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.

Definition

Let denote the space of smooth m-forms with compact support on a smooth manifold. A current is a linear functional on which is continuous in the sense of distributions. Thus a linear functional
is an m-dimensional current if it is continuous in the following sense: If a sequence of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when tends to infinity, then tends to 0.
The space of m-dimensional currents on is a real vector space with operations defined by
Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current as the complement of the biggest open set such that
The linear subspace of consisting of currents with support that is a compact subset of is denoted.

Homological theory

over a compact rectifiable oriented submanifold M of dimension m defines an m-current, denoted by :
If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has:
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
via duality with the exterior derivative by
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
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
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
The mass of a current represents the weighted area of the generalized surface. A current such that M < ∞ 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
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
so that the following defines a 0-current:
In particular every signed regular measure is a 0-current:
Let be the coordinates in ℝ3. Then the following defines a 2-current :