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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. Formally currents behave like Schwartz distributions on a space of differential forms. 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

Definition

Let Ω 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 Ω c m ( M ) which is continuous in the sense of distributions. Thus a linear functional

T : Ω c m ( M ) → R

is an m-current if it is continuous in the following sense: If a sequence ω 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 ( ω k ) tends to 0.

The space D m ( M ) of m-dimensional currents on M is a real vector space with operations defined by

( T + S ) ( ω ) := T ( ω ) + S ( ω ) , ( λ T ) ( ω ) := λ T ( 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 ∈ D m ( M ) as the complement of the biggest open set U ⊂ M such that

T ( ω ) = 0 whenever ω ∈ Ω c m ( U )

The linear subspace of D m ( M ) consisting of currents with support (in the sense above) that is a compact subset of M is denoted 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 ] ] ( ω ) = ∫ M ω .

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 ) = ∫ ∂ M ω = ∫ M d ω = [ [ M ] ] ( d ω ) .

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

∂ : D m + 1 → D m

via duality with the exterior derivative by

( ∂ T ) ( ω ) := T ( d ω )

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

T k ( 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 ) : sup x | | 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 ∈ E m + 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

Ω c 0 ( R n ) ≡ C c ∞ ( R n )

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 ) d μ ( x ) .

Let (x, y, z) be the coordinates in ℝ3. Then the following defines a 2-current (one of many):

T ( a d x ∧ d y + b d y ∧ d z + c d x ∧ d z ) = ∫ 0 1 ∫ 0 1 b ( x , y , 0 ) d x d y .

References

Current (mathematics) Wikipedia
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