Upper half-plane
From Free net encyclopedia
In mathematics, the upper half-plane H is the set of complex numbers
- <math>\mathbb{H} = \{x + iy \;| y > 0; x, y \in \mathbb{R} \}</math>
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 <math>\mathfrak{h}.</math>
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 sectional curvature.
Generalizations
One natural generalization in differential geometry is hyperbolic n-space Hn, the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is H2 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 Hn of n copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space Hn, which is the domain of Siegel modular forms.
Let
- <math>\mathbb{H}_n=\{F\in M_{n}(\mathbb{C}) \; | F=F^T \;\textrm{and}\; \Im (F) >0 \}</math>
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 Hn is called the Siegel upper half-space of genus n.