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Notation and Definitions

For functions f(x1) or r(x1, x2) of one or multiple points respectively:
The ordinary derivative of f in the direction of coordinate number i.
The covariant derivative of f in direction i. It would differ from the ordinary derivative only if f had a composite value, i.e. vector or matrix valued rather than scalar.
r[1!i] or r[2!j]
The ordinary derivative of r in direction i of the first argument, or direction j of the second.
r[1/i] or r[2/j]
Similarly for covariant derivatives.
Consider a curve c(t) on the manifold. For any function f(p) on the manifold, you can get d/dt f(c(t)), the directional derivative. Focus on one point q at a time. Frequently different curves through q will produce the same directional derivative, no matter which function is differentiated -- if the curves are tangent. Thus make equivalence classes among curves (through q). These equivalence classes are called vectors (at q). They form a vector space.
A set of p (in 0 to n) vectors is referred to as a p-vector; it defines an element of area (p = n-1) or volume (p = n). The vectors in a p-vector are combined antisymmetrically with the Grassman product such that v^w = -w^v.
A linear functional on p-vectors is called a p-form. While the value of a form (when a vector is fed to it) is typically a scalar, a composite object (form or vector or matrix) could also be the value. Forms can also be combined with the Grassman product. Any function f (defined at point q) defines a 1-form at q, the gradient of f, wherein the argument vector takes the directional derivative of the function, and the coordinate functions define a basis of all the 1-forms, notated dx[i] (i = 1..n). For calculation, suppose the components of a form A are A[i] and a vector v is v[i]. Then to compute, the functional value A(v) = sum (i = 1..n) A[i]v[i]. This is often abbreviated A.v
Covariant and Contravariant
When a map y(x) from one manifold to another is represented by coordinates, the Jacobian J[i][j] = y[i][!j]. Given a function f(y) and define a new function g(x) = f(y(x)). The gradient df = f[!i], a 1-form. What is the gradient of g? By the chain rule dg[j] = sum (i = 1..n) f[!i] J[i][j]. 1-forms and any composite objects that transform similarly are called ``covariant''. Objects that transform through the inverse of the Jacobian, such as vectors, are called ``contravariant''.
Summation Convention
When covariant and contravariant objects appear in the same term they almost always are summed as in the above expression for dg[j]. When the same subscript letter is repeated in a covariant and a contravariant position, summation is assumed but the sum symbol is not actually written.
Metric Dual (represented by *)
For a p-form A, suppose you take its metric product with another p-form B. Point by point, this is a linear functional of p-forms, that is, it is a p-vector. Suppose you wished to integrate the function value relative to the volume element in the space. An alternative way to express the kernel of the integral would be to obtain a (n-p)-form *A and take its Grassman product with B. This is the metric dual of A. It has a simple form: it has the same components (some changing signs) multiplied by G, the positive square root of the determinant of the metric tensor. (It is unclear what the metric dual would be if the determinant were negative.) On an even dimension space, ** (metric dual twice) reverses the sign of odd degree forms but is an identity transformation for even degree forms. In odd dimensions it also either is the identity or reverses the sign, but it's more complicated to determine for which degrees this happens.
A linear functional on fields of forms. One example of a current is a shape (volume, area, etc.), the form being integrated over it. It's a theorem that every current T can be realized as the limit of a sequence of C-infinity forms A(j), where (B being the argument form field) T(B) = integral of *A(j) ^ B.
-*d* on a space of even dimension. When the dimension is odd, the sign is negative only for forms of odd degree.
(d + d). This is the "generalized Laplacian" and Green's function is the kernel of its inverse.
Harmonic form
A form for which H = 0.
For a set S to be compact means that every subset of S with infinitely many points has (at least one) limit point in S. (Corollary: every limit point is in S.) A subset of a metric space is compact if and only if it is closed and bounded (from the Heine-Borel theorem). A function has compact support if the set of points where it is nonzero is compact.

Kodaira's Theorem

A theory of Fredholm, extended (?) by De Rham (th. 22): On a compact Riemannian manifold, there are only a finite number of linearly independent harmonic forms. If a form B is prespecified, the equation U = B has a solution if and only if B is orthogonal to all the harmonic forms (specifically, to each basis form). If B has compact support, that's sufficient for this theorem to apply even on a noncompact manifold. U is not unique; you can add any harmonic form; but there is a unique component orthogonal to all harmonic forms.

Kodaira's theorem (th. 24): On a Riemannian manifold, any square integrable current A (a limit of a sequence of forms) can be uniquely decomposed as A = A1 + A2 + A3, where A1 is in the metric completion of the set of all forms compactly homologous to zero (i.e. there exists Z1 with compact support such that dZ1 = A1, or at least A1 is the limit of a sequence of such forms); A2 is in the metric completion of the set of forms compactly cohomologous to 0 (there exists Z2 with compact support such that Z2 = A2, or A2 is the limit of such); and A3 is a harmonic current. (If A3 has compact support, it's a theorem that dA3 = 0 and A3 = 0, but this isn't true if A3 doesn't have compact support. The square integrability of A may or may not preclude a harmonic component with noncompact support.)

A generally similar theorem (th. 25) holds for currents continuous in the mean at infinity. (Meaning: let f[i] be one of a set [i] of forms that are C-infinity and have compact support and (f[i],f[i]) (metric product) is bounded over [i]. A current A is "continuous in the mean at infinity" if A[f[i]] (the functional action or integral) is bounded over [i] for any such set of forms.)

For computing the decomposition: First take the metric product of A with each of a basis of harmonic forms, suitably normalized, and assemble a linear combination thereof having the same metric products; that's A3, and (A - A3) is orthogonal to all harmonic forms. Now integrate Green's function acting on (A - A3) to solve the Fredholm equation U = A. Then A1 = d(U) and A2 = (dU). On Rn, Green's function is 1/r(n-1) where r is the distance between the argument points.

Kodaira's theorem and friends were not proved for double currents, matrix-valued currents, etc; only for scalar-valued currents (electric, for example, which is why De Rham called them "currents"). This whole approach, however, assumes that an analog can be produced for matrix-valued currents.

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