Möbius transformation

This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps.

Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2:

Firstly, the projective linear group PGL(2, K) is sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially dimension counting, as the group is 3-dimensional).

This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more.

Möbius transformations are also sometimes written in terms of their fixed points in so-called normal form.

The natural action of PGL(2, C) on the complex projective line CP1 is exactly the natural action of the Möbius group on the Riemann sphere Here, the projective line CP1 and the Riemann sphere are identified as follows:

By using homogeneous coordinates, many calculations involving Möbius transformations can be simplified, since no case distinctions dealing with ∞ are required.

This is the group of those Möbius transformations that map the upper half-plane H = {x + iy : y > 0} to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps H → H. If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H2, the Poincaré half-plane model, and PSL(2, R) is the group of all orientation-preserving isometries of H2 in this model.

[6] If we require the coefficients a, b, c, d of a Möbius transformation to be integers with ad − bc = 1, we obtain the modular group PSL(2, Z), a discrete subgroup of PSL(2, R) important in the study of lattices in the complex plane, elliptic functions and elliptic curves.

The discrete subgroups of PSL(2, R) are known as Fuchsian groups; they are important in the study of Riemann surfaces.

A Möbius transform is parabolic if and only if it has exactly one fixed point in the extended complex plane

Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes.

The appearance of the night sky is now transformed continuously in exactly the manner described by the one-parameter subgroup of elliptic transformations sharing the fixed points 0, ∞, and with the number α corresponding to the constant angular velocity of our observer.

Here are some figures illustrating the effect of an elliptic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):

The one-parameter subgroup which it generates continuously moves points along the family of circular arcs suggested by the pictures.

Then the appearance of the night sky is transformed in exactly the manner described by the one-parameter subgroup of hyperbolic transformations sharing the fixed points 0, ∞, with the real number ρ corresponding to the magnitude of his acceleration vector.

Here are some figures illustrating the effect of a hyperbolic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):

The angle that the loxodrome subtends relative to the lines of longitude (i.e. its slope, the "tightness" of the spiral) is the argument of k. Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles.

You can probably guess the physical interpretation in the case when the two fixed points are 0, ∞: an observer who is both rotating (with constant angular velocity) about some axis and moving along the same axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and with ρ, α determined respectively by the magnitude of the actual linear and angular velocities.

These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants.

In dimension n = 2, the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here.

[10] An isomorphism of the Möbius group with the Lorentz group was noted by several authors: Based on previous work of Felix Klein (1893, 1897)[11] on automorphic functions related to hyperbolic geometry and Möbius geometry, Gustav Herglotz (1909)[12] showed that hyperbolic motions (i.e. isometric automorphisms of a hyperbolic space) transforming the unit sphere into itself correspond to Lorentz transformations, by which Herglotz was able to classify the one-parameter Lorentz transformations into loxodromic, elliptic, hyperbolic, and parabolic groups.

Other authors include Emil Artin (1957),[13] H. S. M. Coxeter (1965),[14] and Roger Penrose, Wolfgang Rindler (1984),[15] Tristan Needham (1997)[16] and W. M. Olivia (2002).

Furthermore, since the kernel of the action (1) is the subgroup {±I}, then passing to the quotient group gives the group isomorphism Focusing now attention on the case when (x0, x1, x2, x3) is null, the matrix X has zero determinant, and therefore splits as the outer product of a complex two-vector ξ with its complex conjugate: The two-component vector ξ is acted upon by SL(2, C) in a manner compatible with (1).

The stereographic projection from the north pole (1, 0, 0, 1) of this sphere onto the plane x3 = 0 takes a point with coordinates (1, x1, x2, x3) with

In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere.

By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space R1,n+1, there is an isomorphism of Möb(n) with the restricted Lorentz group SO+(1,n+1) of Lorentz transformations with positive determinant, preserving the direction of time.

Then he interpreted the x's as homogeneous coordinates and {x | Q(x) = 0}, the null cone, as the Cayley absolute for a hyperbolic space of points {x | Q(x) < 0}.

Coxeter notes that Felix Klein also wrote of this correspondence, applying stereographic projection from (0, 0, 1) to the complex plane

Every orientation-preserving isometry of H3 gives rise to a Möbius transformation on the Riemann sphere and vice versa.