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In mathematics, the upper half-plane H is the set of complex numbers ## 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._{n} is called the Siegel upper half-space of genus n.
## See also

## External links

- $mathbb\{H\}\; =\; \{x\; +\; iy\; ;|\; y\; >\; 0;\; x,\; y\; in\; mathbb\{R\}\; \}$

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 Y-axis pointing upwards: the "upper half-plane" corresponds to the half-plane above the X-axis.

When endowed with a particular metric, the upper half-plane may be called the hyperbolic plane, Poincaré half-plane, or Lobachevsky plane, particularly in texts by Russian authors. Some authors prefer the symbol $mathfrak\{h\}.$

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 (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 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.

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

- $mathbb\{H\}\_n=\{Fin\; M\_\{n\}(mathbb\{C\})\; ;\; |\; F=F^T\; ;textrm\{and\};\; Im\; (F)\; >0\; \}$

- Cusp neighborhood
- Fuchsian group
- Fundamental domain
- Hyperbolic geometry
- Kleinian group
- Modular group
- Riemann surface
- Schwarz-Ahlfors-Pick theorem

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Last updated on Thursday October 02, 2008 at 17:22:49 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday October 02, 2008 at 17:22:49 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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