Laplace operators in differential geometry

Generalized Laplacians are mathematical objects, which will be examined in differential geometry, in particular in the global analysis. As the name suggests, the treated here are generalizations of the operators known from real analysis Laplacian. These generalizations are necessary to define the Laplace operator on Riemannian manifolds can. An important role of these operators in the proofs for the Atiyah-Singer index theorem and the Atiyah - Bott fixed-point theorem.

• 2.4.1 Definition
• 2.4.2 Local representation

Definition

Be an n-dimensional Riemannian manifold, a Hermitian vector bundle and a geometric differential operator of second order. This is called a generalized Laplace operator, if for its main symbol

Applies to and. The norm induced by the Riemannian metric and therefore is dependent on the definition of the metric.

Examples

The following are some well-known examples are presented generalized Laplace operators. In this regard it again as in the definition an n-dimensional compact Riemannian manifold and a vector bundle.

Laplace -Beltrami operator

Definition

The Laplace -Beltrami operator is defined by

For two continuous functions. It refers to the gradient of the function, a vector field. The divergence of a vector field on at the point is defined as the trace of the linear map, where the Levi- Civita connection is on. If one has a domain but not a real manifold is an open subset of, it is the context, the ordinary directional derivative and the known from real analysis divergence. In this case, we obtain the well-known Laplace operator.

Local coordinates

There are local coordinates on the associated base and fields of the tangent bundle. With the components of the Riemannian metric may be referred to with respect to this basis.

The representation of the gradient in local coordinates is then

Here, the inverse matrix of the matrix.

The presentation of the divergence of a vector field

Where is the determinant of the matrix.

Putting these equations together, we obtain the local representation

Of the Laplace -Beltrami operator with respect to the metric. Substituting in this formula for the Laplace -Beltrami operator, the representation of the Euclidean metric tensor in polar, cylindrical or spherical coordinates, we obtain the representation of the usual Laplace operator in these coordinate systems.

Hodge Laplacian

Be the space of differential forms over and the exterior derivative. The adjoint exterior derivative is denoted by. Then called the operator

Hodge Laplace or Laplace de Rham operator and is a generalized Laplace operator. Therefore, the names are that this operator is used in the classical Hodge theory and the closely associated De Rham complex.

Dirac - Laplace operator

A Dirac operator

Is precisely defined so that it induced by squaring a generalized Laplace operator. That is, is a generalized Laplace operator, and is called Dirac Laplace operator. This Laplace operators play an important role in the proof of the index theorem.

Bochner Laplacian

Definition

The Bochner - Laplace operator is defined with the metric connection on the vector bundle. Be also the Levi- Civita connection and by and induced connection on the bundle

Then Bochner Laplace operator by

Defined. The figure is the Tensorverjüngung with respect to the Riemannian metric.

Is an equivalent definition of Bochner Laplacian

Here is the adjoint operator with respect to the Riemannian metric.

Local representation

If you choose the connection the Levi- Civita connection is obtained in local coordinates with the orthonormal frame representation

Properties

• A generalized Laplace operator is a geometric differential operator of order two.
• Each second-order positive definitem main symbol differential operator is a generalized Laplace operator with respect to a suitable Riemannian metric.
• Are smooth cuts, the following applies

• The operator is non-negative and essentially self- respect. The definition of on manifolds can be found in the article on density bundles.
• Each generalized Laplace operator is uniquely determined on the vector a connection beam and a sectional view, such that, the Bochner Laplace operator. Thus, each generalized Laplace operator coincides with the Bochner Laplacian to a malfunction of the zero order.

Swell

• Isaac Chavel: Eigenvalues ​​in Riemannian Geometry ( Pure and Applied Mathematics = 115). Academic Press, Orlando, FL, inter alia, 1984, ISBN 0-12-170640-0.
• Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific Pub Co., Singapore, among others 2007, ISBN 978-981-270853-3.
• Martin Schottenloher: Geometry and symmetry in physics. Leitmotif of Mathematical Physics ( Vieweg = textbook Mathematical Physics ). Vieweg, Braunschweig, inter alia, 1995, ISBN 3-528-06565-6.