Conformal map

A conformal mapping is a conformal mapping, ie angles remain unchanged in the.

The conformal mappings of Minkowski space to itself include the Lorentz transformations and translations that leave the metric unchanged, the dilations that scale the metric by a smooth function, and the special conformal transformations, one of which is the inversion of a spherical surface.

As the Lorentz transformations and the Poincaré transformations form the conformal transformations of a Lie group, the conformal group.

If is an open subset of the complex plane, then the function

Compliant if and only if it is holomorphic or anti- holomorphic and its derivative zero is not equal to the whole. Thus, the conformal mappings form the geometric illustration of the complex differentiable ( analytic or holomorphic ) functions of a complex variable (see the illustration of real functions by plane curves ).

Physical applications

The figure 2 shows that complex curves can be mapped onto simpler. The illustrated example of a conformal mapping of the Joukowski function (also Zhukovsky function written ). In this figure, the Joukowski profile is mapped to a circle. The speed at which such flow around the air particles (two-dimensional ) the wing profile, it is easier to be calculated, if it comes to flow around a circular cylinder. This is plausible that the conformal mappings have an important meaning in the following areas, as long as you studied phenomena in the two-dimensional plane:

• Fluid mechanics (aerodynamics, hydrodynamics )
• Electrostatic (see the electrostatic field by analogy with flow fields )
• Heat conduction.

Invariance under conformal mappings

In the case of the -dimensional Minkowski space applies: The connected component is one of the group of orientation-preserving conformal transformations is isomorphic to the group, though. For this group is infinite dimensional. It is isomorphic to the infinite-dimensional group of orientation-preserving diffeomorphisms of called up.

In the case of the -dimensional Euclidean space, the corresponding group is isomorphic to. In case it is therefore isomorphic to the group of Möbius transformations.

Physical systems that are invariant under conformal mappings, have an important role in condensed matter physics and string theory, as well as in the conformal field theory.

Conformal Mapping on (semi-) Riemannian manifolds of

Let and be two Riemannian manifolds and semi - Riemannian manifolds. and denote the metric tensors. Two metrics on a manifold and hot compliant in Riemannian geometry equivalent if called with a positive function defined on the conformal factor. The class -compliant equivalent metrics is called conformal structure.

A diffeomorphism is called conforming if for all points and vectors of the tangent space. It expresses it well so that the pull-back metric is conformally equivalent to the metric of. The potency is to indicate that the factor is always greater than 0, that is to say on a conformal factor. An example of a conformal mapping of the stereographic projection of the sphere surface is the projective plane ( plane supplemented by the point at infinity ).

The conformal mappings of a manifold into itself generated by conformal Killing vector fields.