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EspañolUPPER HALF-PLANE
In mathematics, the 'upper half-plane' 'H' is the set of complex numbers
:
with positive imaginary part ''y''. Other names are hyperbolic plane, Poincaré plane and Lobachevsky plane, particularly in texts by Russian authors. Some authors prefer the symbol
It is the domain of many functions of interest in complex analysis, especially elliptic modular forms. The 'lower half-plane', defined by ''y'' < 0, is equally good, but less used by convention. The open unit disk 'D' is equivalent by a conformal mapping, meaning that it is usually possible to pass between 'H' and 'D'.
It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.
The uniformization theorem for surfaces states that the 'upper half-plane' is the universal covering space of surfaces with constant negative Gaussian curvature.
One natural generalization in differential geometry is hyperbolic ''n''-space 'H'''n'', the maximally symmetric, simply connected, ''n''-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is 'H'2 since it has real dimension 2.
In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product 'H'''n'' of ''n'' copies of the upper half-plane. Yet another space interesting to number theorists is the 'Siegel upper half-space H'''n'', which is the domain of Siegel modular forms.
Let
:
be the set of symmetric square matrices whose imaginary part is positive definite; that is the set of square matrices whose imaginary parts have positive eigenvalues. The set 'H'''n'' is called the ''Siegel upper half-space of genus n''.
★ Cusp neighborhood
★ Fuchsian group
★ Fundamental domain
★ Hyperbolic geometry
★ Kleinian group
★ Modular group
★ Poincaré metric
★ Riemann surface
★ Schwarz-Ahlfors-Pick theorem
★ Visual presentation of the upper half-plane
:
with positive imaginary part ''y''. Other names are hyperbolic plane, Poincaré plane and Lobachevsky plane, particularly in texts by Russian authors. Some authors prefer the symbol
It is the domain of many functions of interest in complex analysis, especially elliptic modular forms. The 'lower half-plane', defined by ''y'' < 0, is equally good, but less used by convention. The open unit disk 'D' is equivalent by a conformal mapping, meaning that it is usually possible to pass between 'H' and 'D'.
It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.
The uniformization theorem for surfaces states that the 'upper half-plane' is the universal covering space of surfaces with constant negative Gaussian curvature.
| Contents |
| Generalizations |
| See also |
| External links |
Generalizations
One natural generalization in differential geometry is hyperbolic ''n''-space 'H'''n'', the maximally symmetric, simply connected, ''n''-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is 'H'2 since it has real dimension 2.
In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product 'H'''n'' of ''n'' copies of the upper half-plane. Yet another space interesting to number theorists is the 'Siegel upper half-space H'''n'', which is the domain of Siegel modular forms.
Let
:
be the set of symmetric square matrices whose imaginary part is positive definite; that is the set of square matrices whose imaginary parts have positive eigenvalues. The set 'H'''n'' is called the ''Siegel upper half-space of genus n''.
See also
★ Cusp neighborhood
★ Fuchsian group
★ Fundamental domain
★ Hyperbolic geometry
★ Kleinian group
★ Modular group
★ Poincaré metric
★ Riemann surface
★ Schwarz-Ahlfors-Pick theorem
External links
★ Visual presentation of the upper half-plane
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