Poincaré halfplane model

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In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. The model is commonly expressed in terms of the complex upper half-plane or the unit disc, both of which are related through a conformal mapping.

The Poincaré metric is a metric tensor on the disk or plane, expressing the hyperbolic nature.

Symmetry groups

A variety of different groups appear in the discussion of the upper half-plane. One is the linear group GL(2,C), called the Möbius group. The projective linear group PGL(2,C) is isomorphic to the group of all orientation-preserving conformal maps of the upper half-plane. Important subgroups of PGL(2,C) are called Kleinian groups. However, conformal maps do not in general preserve the Poincaré metric and thus are not an isometry of the Poincaré model.

There are four groups that do preserve the metric tensor. These are:

  • The special linear group SL(2,R) which consists of the set of 2×2 matrices with real entries whose determinant equals +1. Note that many texts (including Wikipedia) often say SL(2,R) when they really mean PSL(2,R).
  • The group S*L(2,R) consisting of the set of 2×2 matrices with real entries whose determinant equals +1 or −1. Note that SL(2,R) is a subgroup of this group.
  • The projective linear group PSL(2,R) = SL(2,R)/{+ or −I}, consisting of the matrices in SL(2,R) modulo plus or minus the identity matrix.
  • The group PS*L(2,R) = S*L(2,R)/{+ or −I} is again a projective group, and again, modulo plus or minus the identity matrix.

The relationship of these groups to the Poincaré model is as follows:

  • The group of all isometries of H, sometimes denoted as Isom(H), is isomorphic to PS*L(2,R). This includes both the orientation preserving and the orientation-reversing isometries. The orientation-reversing map (the mirror map) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\rightarrow -\overline{z}} .
  • The group of orientation-preserving isometries of H, sometimes denoted as Isom+(H), is isomorphic to PSL(2,R).

Important subgroups of the isometry group are the Fuchsian groups.

One also frequently sees the modular group SL(2,Z). This group is important in two ways. First, it is a symmetry group of the square 2x2 lattice of points. Thus, functions that are periodic on a square grid, such as modular forms and elliptic functions, will thus inherit an SL(2,Z) symmetry from the grid. Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area.

Isometric symmetry

The group action of the special linear group PSL(2,R) on H is defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{matrix}a&b\\ c&d\\ \end{matrix}\right) \cdot z = \frac{az+b}{cz+d} = {(ac|z|^2+bd+(ad+bc)\Re(z)) + i\Im(z)\over|cz+d|^2}.}

Note that the action is transitive, in that for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1,z_2\in\mathbb{H}} , there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in {\rm PSL}(2,\mathbb{R})} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gz_1=z_2} . It is also faithful, in that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gz=z} for all z in H, then g=e.

The stabilizer or isotropy subroup of an element z in H is the set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in{\rm PSL}(2,\mathbb{R})} leave z unchanged: gz=z. The stabilizer of i is the rotation group

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm SO}(2) = \left\{ \left(\begin{matrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{matrix}\right)\,:\,\theta\in{\mathbf R}\right\}.}

Since any element z in H is mapped to i by an element of PSL(2,R), this means that the isotropy subgroup of any z is isomorphic to SO(2). Thus, H = PSL(2,R)/SO(2). Alternately, the vector bundle of unit-length tangent vectors on the upper half-plane is isomorphic to PSL(2,R).

The upper half-plane is tessellated into free regular sets by the modular group SL(2,Z).

Geodesics

The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis.

The unit-speed geodesic going up vertically, through the point i is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t) = \left(\begin{matrix}e^{t/2}&0\\ 0&e^{-t/2}\\ \end{matrix}\right) \cdot i = ie^t.}

Because PSL(2,R) is an isometry of the upper half-plane, this geodesic is mapped into the other geodesics through the action of PSL(2,R). Thus, the general unit-speed geodesic is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t) = \left(\begin{matrix}a&b\\ c&d\\ \end{matrix}\right) \left(\begin{matrix}e^{t/2}&0\\ 0&e^{-t/2}\\ \end{matrix}\right) \cdot i = \frac {aie^t +b} {cie^t +d} }

This provides the complete description of the geodesic flow on the unit-length tangent bundle (complex line bundle) on the upper half-plane.

See also

References

  • Henri Poincaré (1882) "Théorie des Groupes Fuchsiens", Acta Mathematica v.1,p.1.First article in a legendary series exploiting half-plane model.On page 52 one can see an example of the semicircle diagrams so characteristic of the model.
  • Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4.
  • Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 2.3).

fr:Demi-plan de Poincaré