# Compliant mapping

Fig. 1: A rectangular mesh (above) and its image (below) after a conformal mapping  . Line pairs that intersect at 90 ° are mapped to line pairs that also intersect at 90 °.${\ displaystyle f}$

A conformal mapping is a conformal mapping.

This means, among other things, that from a right-angled coordinate network through a conformal mapping, an i. General The result is a curvilinear coordinate network, but the right-angled network structure is completely retained “on a small scale”, in particular the intermediate angles and the length ratios of any two vectors.

Such images find multiple applications in theoretical physics , u. a. in the theory of complicated electrostatic potentials and the associated electrostatic fields as well as in fluid mechanics .

## definition

A linear mapping is called conformal if it applies to all and its determinant is positive ( if it is negative, it is called anti-conformal). Here is the standard scalar product and the Euclidean norm . In other words, (linear) conformal or anti-conformal mappings get the magnitude of the angle between any two vectors; while a conformal maintains the orientation of the angle, it reverses an anti-conformist. ${\ displaystyle L \ colon \ mathbb {R} ^ {m} \ to \ mathbb {R} ^ {n}}$${\ displaystyle {\ frac {\ langle Lv, Lw \ rangle} {\ | Lv \ | _ {2} \ | Lw \ | _ {2}}} = {\ frac {\ langle v, w \ rangle} { \ | v \ | _ {2} \ | w \ | _ {2}}}}$${\ displaystyle v, w \ in \ mathbb {R} ^ {m}}$${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$${\ displaystyle \ | \ cdot \ | _ {2}}$

Furthermore is a differentiable map compliant in when their differential in is compliant. ${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle x}$

## properties

• If there is an open subset of the complex plane , then the function is conformal if and only if it is holomorphic or anti-holomorphic and its derivative is not equal to zero to whole . The conformal mappings thus form the geometric illustration of the complex differentiable ( analytical or holomorphic ) functions of a complex variable (cf. the illustration of real functions using plane curves). Real or imaginary part of such a function or its locally rectangular coordinate networks can be, for. B. be interpreted as potentials of an electrostatic field or a flow field. Also meromorphic functions are useful because the pole the dipoles , quadrupoles , etc., in general: the multipoles generate these potentials.${\ displaystyle U}$ ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle f \ colon U \ to \ mathbb {C}}$${\ displaystyle U}$

## Physical applications

Fig. 2: The wing and the circle are connected by a conformal mapping.

The adjacent Figure 2 shows, using an example from “aircraft construction”, that the conformal mapping enables complicated curves to be mapped in a much simpler way. The example shown of a conformal mapping is the Joukowski function (also written as the “Schukowski function”). In this illustration, the Joukowski profile is mapped onto a circle. The speed at which air particles flow around the (two-dimensional) airfoil profile is easier to calculate when it comes to flow around a circular cylinder. This makes it plausible that the conformal mappings have an important meaning in the following areas, as long as one investigates phenomena in the two-dimensional plane:

### Invariance among conformal mappings

In the case of the -dimensional Minkowski space, the following applies: The connected component of the 1 of the group of orientational conformal transformations is isomorphic to the group if the following applies. For this group is infinitely dimensional. It is isomorphic to , where the infinitely dimensional group of orientational diffeomorphisms denotes to itself. ${\ displaystyle d}$${\ displaystyle SO (d, 2)}$${\ displaystyle d> 2}$${\ displaystyle d = 2}$${\ displaystyle \ mathrm {Diff} _ {+} (\ mathbb {R}) \ times \ mathrm {Diff} _ {+} (\ mathbb {R})}$${\ displaystyle \ mathrm {Diff} _ {+} (\ mathbb {R})}$${\ displaystyle \ mathbb {R}}$

In the case of n-dimensional Euclidean space, the corresponding group is isomorphic to , . In the case it is therefore isomorphic to the group of Möbius transformations. ${\ displaystyle d}$${\ displaystyle SO (d + 1,1)}$${\ displaystyle d \ geq 2}$${\ displaystyle d = 2}$

Physical systems that are invariable under conformal mappings have great importance in solid state physics , string theory, and conformal field theory .

## Conformal mappings on (semi-) Riemannian manifolds

Let and be two Riemannian manifolds or semi-Riemannian manifolds . and denote the metric tensors . Two metrics and on a manifold are called "conformally equivalent" in Riemannian geometry , if with a positive function defined on which is called the conformal factor . The class of conformal equivalent metrics on is called conformal structure.${\ displaystyle (M, g)}$${\ displaystyle (N, h)}$${\ displaystyle g}$${\ displaystyle h}$${\ displaystyle g}$${\ displaystyle h}$${\ displaystyle M}$${\ displaystyle g = uh}$${\ displaystyle M}$${\ displaystyle u}$${\ displaystyle M}$

A diffeomorphism is called conformal if it holds for all points and vectors of the tangent space . This is also expressed in such a way that the pullback metric on conformal is equivalent to the metric of . The power is intended to indicate that the factor is always greater than 0, i.e. that it is a conforming factor. An example of a conformal mapping is the stereographic projection of the spherical surface onto the projective plane (plane supplemented by a point at infinity). ${\ displaystyle f \ colon M \ to N}$${\ displaystyle h_ {f (x)} (\ mathrm {d} f_ {x} (v), \ mathrm {d} f_ {x} (w)) = e ^ {\ sigma (x)} \ cdot g_ {x} (v, w)}$${\ displaystyle x \ in M}$${\ displaystyle v, w \ in T_ {x} M}$${\ displaystyle M}$${\ displaystyle M}$${\ displaystyle e ^ {\ sigma (x)}}$

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