 Current (mathematics)

In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a kcurrent in the sense of Georges de Rham is a functional on the space of compactly supported differential kforms, 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 denote the space of smooth mforms with compact support on . A current is a linear functional on which is continuous in the sense of distributions. Thus a linear functional
is an mcurrent if it is continuous in the following sense: If a sequence ω_{n} of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when n tends to infinity, then T(ω_{n}) tends to 0.
The space of mdimensional currents on ℝ^{n} is a real vector space with operations defined by
Multiplication by a constant scalar represents a change in the multiplicity of the surface^{[clarification needed]}. In particular multiplication by −1 represents the change of orientation of the surface.
Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current T as the complement of the biggest open set U such that T(ω) = 0 whenever the support of ω lies entirely in U.
The linear subspace of consisting of currents with compact support is denoted . It can be naturally identified with the dual space to the space of all smooth mforms on ℝ^{n}.
Homological theory
Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an mcurrent, denoted by [[M]]:
If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and one has Stokes' theorem:
This relates the exterior derivative d with the boundary operator ∂ on the homology of M.
More generally, a boundary operator can be defined on arbitrary currents
by dualizing the exterior derivative:
for all compactly supported (m−1)forms ω.
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, 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 mform, then define its comass by
So if ω is a simple mform, 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(T) < ∞ is representable by integration over a suitably weighted rectifiable submanifold. 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 0current:
In particular every signed regular measure μ is a 0current:
Let (x, y, z) be the coordinates in ℝ^{3}. Then the following defines a 2current (one of many):
See also
References
 de Rham, G. (1973) (in French), Variétés Différentiables, Actualites Scientifiques et Industrielles, 1222 (3rd ed.), Paris: Hermann, pp. X+198, Zbl 0284.58001.
 Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Berlin–Heidelberg–New York: SpringerVerlag, pp. xiv+676, ISBN 9783540606567, MR0257325, Zbl 0176.00801.
 Whitney, H. (1957), Geometric Integration Theory, Princeton Mathematical Series, 21, Princeton, NJ and London: Princeton University Press and Oxford University Press, pp. XV+387, MR0087148, Zbl 0083.28204.
 Lin, Fanghua; Yang, Xiaoping (2003), Geometric Measure Theory: An Introduction, Advanced Mathematics (Bejing/Boston), 1, Bejing/Boston: Science Press/International Press, pp. x+237, ISBN 9781571461254, MR2030862, Zbl 1074.49011
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Categories: Differential topology
 Generalized manifolds
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