De Rham cohomology

In differential geometry, differential forms on the differentiable manifold that are exterior derivatives are called exact; and forms such that their exterior derivatives are 0 are called closed (see closed and exact differential forms).

Exact forms are closed, so the vector spaces of k-forms together with the outer derivative are a complex of costrings. The vector spaces of the closed forms modulo the exact forms are called De Rham cohomology groups. The wedge product endows the direct sum of these groups with a ring structure.

De Rham's theorem, proved by Georges de Rham in 1931, states that for an orientable compact differentiable manifold M, these groups are isomorphic as real vector spaces with the singular cohomology groups H^p(M; R). Furthermore, the two cohomology rings are isomorphic (as a graded ring). The generalized Stokes theorem is an expression of the duality between de Rham cohomology and complex string homology.

Harmonic shapes

For the differentiable manifold M, we can equip it with some auxiliary Riemann metric. Then the Laplacian Δ, defined by

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using the exterior derivative and the Hodge dual defines a homogeneous linear differential operator (in grading) that acts on the exterior algebra formed by the differential forms: we can look at its action on each component of degree p separately.

If M is compact and oriented, the dimension of its kernel acting on the space of p-forms is then equal (by Hodge's theory) to that of the cohomology group de Rham of degree p: the Laplacian selects a unique harmonic form in each cohomology class of closed forms, in particular the space of all forms p-harmonics in M is isomorphic to H^p (M; R).