Hodge theory

The Hodge decomposition or the set of Hodge is a central statement of the Hodge theory. This theory combines the mathematical subdivisions analysis, differential geometry and algebraic topology. Named are the Hodge decomposition and the Hodge theory after the mathematician William Vallance Douglas Hodge, who developed this as an extension to the De Rham cohomology in the 1930s.

Elliptic complex

With smooth cuts are referred to in a vector bundle. Be an oriented Riemannian manifold and a sequence of vector bundles. An elliptical complex is a sequence of partial differential operators of the first order

So that the following properties are valid.

  • The result is a Kokettenkomplex, it is applicable to all and
  • For each of the sequence of principal symbols

The rooms can be understood, for example, than that of differential forms rooms.

Set of Hodge

Let now a compact, oriented Riemannian manifold and the i-th cohomology group of the elliptic complex. In addition, defining a ( Laplace ) operator

By

This is an elliptical operator. Now, the following applies:

  • The -th cohomology group is isomorphic to all of the core, ie
  • The dimension of the -th cohomology group is for all finite
  • There is an orthogonal decomposition

Example: De Rham cohomology

The De Rham complex

Is an elliptical complex. The spaces are the spaces of differential forms of degree i and again is the exterior derivative. The corresponding sequence of the main symbols is the Koszul complex. The operator is the Hodge - Laplace operator. At the core of this operator is called the space of harmonic differential forms, as this is indeed defined analogously to the space of harmonic functions. By the theorem of Hodge now exists an isomorphism between the i-th de Rham cohomology group - and the space of harmonic differential forms of degree.

In addition,

Well-defined figures, as the De Rham cohomology groups have finite dimension. These numbers are called Betti numbers. The Hodge star operator also induces an isomorphism between the spaces and. This is the Poincaré duality and applies for the Betti numbers

Mathematical analysis Differentialoperator Differentialform Elliptic operator Lie algebra cohomology
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