Upper half-plane

Template:Short description Template:Refimprove In mathematics, the upper half-plane, Template:Tmath is the set of points Template:Tmath in the Cartesian plane with Template:Tmath The lower half-plane is the set of points Template:Tmath with Template:Tmath instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants.

Affine geometry

The affine transformations of the upper half-plane include

1. shifts <math>(x,y)\mapsto (x+c,y)</math>, <math>c\in\mathbb{R}</math>, and
2. dilations <math>(x,y)\mapsto (\lambda x,\lambda y)</math>, <math>\lambda > 0.</math>

Proposition: Let Template:Tmath and Template:Tmath be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes <math>A</math> to <math>B</math>.

Proof: First shift the center of Template:Tmath to Template:Tmath Then take <math>\lambda=(\text{diameter of}\ B)/(\text{diameter of}\ A)</math>

and dilate. Then shift Template:Tmath to the center of Template:Tmath

Inversive geometry

Definition: <math>\mathcal{Z} := \left\{\left( \cos^2\theta,\tfrac12 \sin 2\theta \right) \mid 0 < \theta < \pi \right\} </math>.

Template:Tmath can be recognized as the circle of radius Template:Tmath centered at Template:Tmath and as the polar plot of <math>\rho(\theta) = \cos \theta.</math>

Proposition: Template:Tmath Template:Tmath in Template:Tmath and Template:Tmath are collinear points.

In fact, <math>\mathcal{Z}</math> is the inversion of the line <math>\bigl\{(1, y) \mid y > 0 \bigr\}</math> in the unit circle. Indeed, the diagonal from Template:Tmath to Template:Tmath has squared length <math>1 + \tan^2 \theta = \sec^2 \theta </math>, so that <math>\rho(\theta) = \cos \theta</math> is the reciprocal of that length.

Metric geometry

The distance between any two points Template:Tmath and Template:Tmath in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from Template:Tmath to Template:Tmath either intersects the boundary or is parallel to it. In the latter case Template:Tmath and Template:Tmath lie on a ray perpendicular to the boundary and logarithmic measure can be used to define a distance that is invariant under dilation. In the former case Template:Tmath and Template:Tmath lie on a circle centered at the intersection of their perpendicular bisector and the boundary. By the above proposition this circle can be moved by affine motion to Template:Tmath Distances on Template:Tmath can be defined using the correspondence with points on <math>\bigl\{(1, y) \mid y > 0 \bigr\}</math> and logarithmic measure on this ray. In consequence, the upper half-plane becomes a metric space. The generic name of this metric space is the hyperbolic plane. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model.

Complex plane

Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part:

<math>\mathcal{H} := \{x + iy \mid y > 0;\ x, y \in \mathbb{R} \} .</math>

The term arises from a common visualization of the complex number <math>x+iy</math> as the point <math>(x,y)</math> in the plane endowed with Cartesian coordinates. When the <math>y</math> axis is oriented vertically, the "upper half-plane" corresponds to the region above the <math>x</math> axis and thus complex numbers for which <math>y > 0</math>.

It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by Template:Tmath is equally good, but less used by convention. The open unit disk Template:Tmath (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to Template:Tmath (see "Poincaré metric"), meaning that it is usually possible to pass between Template:Tmath and Template:Tmath

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.

The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.

Generalizations

One natural generalization in differential geometry is hyperbolic <math>n</math>-space Template:Tmath the maximally symmetric, simply connected, Template:Tmath-dimensional Riemannian manifold with constant sectional curvature <math>-1</math>. In this terminology, the upper half-plane is Template:Tmath since it has real dimension Template:Tmath

In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product Template:Tmath of Template:Tmath copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space Template:Tmath which is the domain of Siegel modular forms.

See also

• Cusp neighborhood
• Extended complex upper-half plane
• Fuchsian group
• Fundamental domain
• Half-space
• Kleinian group
• Modular group
• Moduli stack of elliptic curves
• Riemann surface
• Schwarz–Ahlfors–Pick theorem

References

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