In mathematics, the upper half-plane 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 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 $\backslash mathfrak\{h\}.$
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 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

• Riemann surface

• Schwarz-Ahlfors-Pick theorem