Poincaré half-plane model

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Stellated Regular Eptagonal Tiling of the model.
Stellated Regular Eptagonal Tiling of the model.

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

It is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model (due to Riemann), to show that hyperbolic geometry was equiconsistent with Euclidean geometry. The disk model and the half-plane model are isomorphic under a conformal mapping.

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[edit] Symmetry groups

The projective linear group PGL(2,C) acts on the Riemann sphere by the Möbius transformations. The subgroup that stabilizes the upper half-plane is PGL(2,R), the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space.

There are four closely related Lie groups that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance.

  • 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)/{±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)/{±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 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.

[edit] Isometric symmetry

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

\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 z_1,z_2\in\mathbb{H}, there exists a g\in {\rm PSL}(2,\mathbb{R}) such that gz1 = z2. It is also faithful, in that if 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 g\in{\rm PSL}(2,\mathbb{R}) leave z unchanged: gz=z. The stabilizer of i is the rotation group

{\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 bundle of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle, is isomorphic to PSL(2,R).

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

[edit] 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

\gamma(t) = \left(\begin{matrix}e^{t/2}&0\\ 0&e^{-t/2}\\ \end{matrix}\right) \cdot i = ie^t.

Because PSL(2,R) acts by isometries 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

\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.

[edit] See also

[edit] References

  • Eugenio Beltrami, Theoria fondamentale delgi spazil di curvatura constanta, Annali. di Mat., ser II 2 (1868), 232-255
  • 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).
  • Saul Stahl, The Poincaré Half-Plane, Jones and Bartlett, 1993, ISBN 0-86720-298-X.
  • John Stillwell (1998) Numbers and Geometry,pp.100-104, Springer-Verlag,NY ISBN 0-387-98289-2 .An elementary introduction to the Poincaré half-plane model of the hyperbolic plane.
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