In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below.
Overview of metrics on Riemann surfaces
A metric on the complex plane may be generally expressed in the form
d s 2 = λ 2 ( z , z ¯ ) d z d z ¯ {\displaystyle ds^{2}=\lambda ^{2}(z,{\overline {z}})\,dz\,d{\overline {z}}}
where λ is a real, positive function of z {\displaystyle z} and z ¯ {\displaystyle {\overline {z}}} . The length of a curve γ in the complex plane is thus given by
l ( γ ) = ∫ γ λ ( z , z ¯ ) | d z | {\displaystyle l(\gamma )=\int _{\gamma }\lambda (z,{\overline {z}})\,|dz|}
The area of a subset of the complex plane is given by
Area ( M ) = ∫ M λ 2 ( z , z ¯ ) i 2 d z ∧ d z ¯ {\displaystyle {\text{Area}}(M)=\int _{M}\lambda ^{2}(z,{\overline {z}})\,{\frac {i}{2}}\,dz\wedge d{\overline {z}}}
where ∧ {\displaystyle \wedge } is the exterior product used to construct the volume form. The determinant of the metric is equal to λ 4 {\displaystyle \lambda ^{4}} , so the square root of the determinant is λ 2 {\displaystyle \lambda ^{2}} . The Euclidean volume form on the plane is d x ∧ d y {\displaystyle dx\wedge dy} and so one has
d z ∧ d z ¯ = ( d x + i d y ) ∧ ( d x − i d y ) = − 2 i d x ∧ d y . {\displaystyle dz\wedge d{\overline {z}}=(dx+i\,dy)\wedge (dx-i\,dy)=-2i\,dx\wedge dy.}
A function Φ ( z , z ¯ ) {\displaystyle \Phi (z,{\overline {z}})} is said to be the potential of the metric if
4 ∂ ∂ z ∂ ∂ z ¯ Φ ( z , z ¯ ) = λ 2 ( z , z ¯ ) . {\displaystyle 4{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\overline {z}}}}\Phi (z,{\overline {z}})=\lambda ^{2}(z,{\overline {z}}).}
The Laplace–Beltrami operator is given by
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