 Upper halfplane

In mathematics, the upper halfplane H is the set of complex numbers with positive imaginary part y:
The term is associated with a common visualization of complex numbers with points in the plane endowed with Cartesian coordinates, with the Yaxis pointing upwards: the "upper halfplane" corresponds to the halfplane above the Xaxis.
It is the domain of many functions of interest in complex analysis, especially modular forms. The lower halfplane, defined by y < 0, is equally good, but less used by convention. The open unit disk D (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping (see "Poincaré metric"), meaning that it is usually possible to pass between H and D.
It also plays an important role in hyperbolic geometry, where the Poincaré halfplane 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 halfplane is the universal covering space of surfaces with constant negative Gaussian curvature.
Generalizations
One natural generalization in differential geometry is hyperbolic nspace H^{n}, the maximally symmetric, simply connected, ndimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper halfplane 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 halfplane. Yet another space interesting to number theorists is the Siegel upper halfspace H_{n}, which is the domain of Siegel modular forms.
See also
 Cusp neighborhood
 Extended complex upperhalf plane
 Fuchsian group
 Fundamental domain
 Hyperbolic geometry
 Kleinian group
 Modular group
 Riemann surface
 SchwarzAhlforsPick theorem
References
 Weisstein, Eric W., "Upper HalfPlane" from MathWorld.
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