Hodge dual

The Hodge star operator or shortly Hodge operator is an object of differential geometry. He was introduced by the British mathematician William Vallance Douglas Hodge. The operator is an isomorphism, which operates on the external algebra of a finite Prähilbertraums or more generally on the space of differential forms.

Motivation

Be an n- dimensional smooth manifold and let the k-th exterior power of the Kotangentialraums. For all k with vector spaces have the same dimension and are therefore isomorphic. Has now additionally the structure of an oriented semi Riemannian manifold, then one can prove that this isomorphism can naturally construct. That is, there exists an isomorphism between the spaces of invariant semiriemannsche under the metric and the orientation -preserving diffeomorphisms is. The generalization of this isomorphism on the tangent bundle is called Hodge star operator.

Definition

Since the space from the above motivation is a finite vector space, starting here with the definition of the Hodge star operator on vector spaces.

Hodge star operator on vector spaces

Be one -dimensional oriented vector space with scalar product and its dual space. For the -th exterior power of the vector space of alternating multilinear forms of stage referred to above.

The Hodge star operator

Is uniquely defined by the following condition: If a positively oriented orthonormal basis of and the corresponding dual basis, then

It is not enough to require this condition for a single orthonormal basis. But one does not need to ask for any positively oriented orthonormal basis. It is sufficient to consider all even permutations of a single basis: if a positively oriented orthonormal basis of and the corresponding dual basis, the Hodge star operator is uniquely determined by the condition

For any even permutation of.

Global Hodge star operator

After this preliminary work can be transferred to the Hodge star operator on the exterior algebra of the cotangent bundle. As in the motivation was an orientable smooth Riemannian manifold again. In addition, define as the space of sections of the vector bundle. The space is therefore on the space of differential forms of degree. As a vector bundle, and thus in every point is a vector space, the Hodge star operator is defined pointwise.

The Hodge star operator is an isomorphism

So that for each point

Applies. The differential form, evaluated at the point, is again an element of a vector space, and thus taking the above definition for vector spaces. Has been implicated in this definition that the mold is a smooth differential form. This, however, is not clear and requires a proof.

Examples

Considering the three-dimensional Euclidean space as a Riemannian manifold with the Euclidean metric and the normal orientation, so you can apply the Hodge star operator under these conditions. Be the oriented standard basis of, and the corresponding dual basis. The elements can be understood as the differential shapes. Then for the Hodge star operator

Under these conditions, the Hodge star operator is implicit in the vector analysis in the cross product, and uses the derived rotation operator. This is explained in the article exterior algebra.

Properties of the Hodge star operator

Be an oriented, smooth Riemannian manifold, are, from, and is a Riemannian metric. Then the Hodge star operator has the following properties:

Riemannian volume form

Be a smooth, oriented, Riemannian manifold. You then regarded as constant one function on, the Riemannian volume form is defined as. This volume form is an important part of the integration with differential forms. This will be illustrated by a simple example. Be it a compact subset. For the volume of U. Summing up on as a manifold and as a compact subset contained therein, the volume is defined in this case as

The integration theory on manifolds thus also includes the integration of real subsets. According to this principle, one can also integrate functions on manifolds by these multiplied by the volume form.