Definitions

beading plane

Upper half-plane

In mathematics, the upper half-plane H is the set of complex numbers

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.

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

mathbb{H}_n={Fin M_{n}(mathbb{C}) ; | F=F^T ;textrm{and}; Im (F) >0 }
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.

See also

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