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.
Contents
Definition
Let
Ω
c
m
(
M
)
{\displaystyle \Omega _{c}^{m}(M)}
denote the space of smooth m-forms with compact support on a smooth manifold
M
.
{\displaystyle M.}
A current is a linear functional on
Ω
c
m
(
M
)
{\displaystyle \Omega _{c}^{m}(M)}
which is continuous in the sense of distributions. Thus a linear functional
T
:
Ω
c
m
(
M
)
→
R
{\displaystyle T:\Omega _{c}^{m}(M)\to \mathbb {R} }
is an m-dimensional current if it is continuous in the following sense: If a sequence
ω
k
{\displaystyle \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
{\displaystyle k}
tends to infinity, then
T
(
ω
k
)
{\displaystyle T(\omega _{k})}
tends to 0.
The space
D
m
(
M
)
{\displaystyle {\mathcal {D}}_{m}(M)}
of m-dimensional currents on
M
{\displaystyle M}
is a real vector space with operations defined by
(
T
+
S
)
(
ω
)
:=
T
(
ω
)
+
S
(
ω
)
,
(
λ
T
)
(
ω
)
:=
λ
T
(
ω
)
.
{\displaystyle (T+S)(\omega ):=T(\omega )+S(\omega ),\qquad (\lambda T)(\omega ):=\lambda T(\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
∈
D
m
(
M
)
{\displaystyle T\in {\mathcal {D}}_{m}(M)}
as the complement of the biggest open set
U
⊂
M
{\displaystyle U\subset M}
such that
T
(
ω
)
=
0
{\displaystyle T(\omega )=0}
whenever
ω
∈
Ω
c
m
(
U
)
{\displaystyle \omega \in \Omega _{c}^{m}(U)}
The linear subspace of
D
m
(
M
)
{\displaystyle {\mathcal {D}}_{m}(M)}
consisting of currents with support (in the sense above) that is a compact subset of
M
{\displaystyle M}
is denoted
E
m
(
M
)
.
{\displaystyle {\mathcal {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
]
]
{\displaystyle [[M]]}
:
[
[
M
]
]
(
ω
)
=
∫
M
ω
.
{\displaystyle [[M]](\omega )=\int _{M}\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:
[
[
∂
M
]
]
(
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
{\displaystyle T_{k}}
of currents, converges to a current
T
{\displaystyle T}
if
T
k
(
ω
)
→
T
(
ω
)
,
∀
ω
.
{\displaystyle T_{k}(\omega )\to T(\omega ),\qquad \forall \omega .}
It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If
ω
{\displaystyle \omega }
is an m-form, then define its comass by
‖
Examples
Recall that
Ω
c
0
(
R
n
)
≡
C
c
∞
(
R
n
)
{\displaystyle \Omega _{c}^{0}(\mathbb {R} ^{n})\equiv C_{c}^{\infty }(\mathbb {R} ^{n})}
so that the following defines a 0-current:
T
(
f
)
=
f
(
0
)
.
{\displaystyle T(f)=f(0).}
In particular every signed regular measure
μ
{\displaystyle \mu }
is a 0-current: