Upper half-plane

In mathematics, the upper half-plane, ⁠${\mathscr{H}},$⁠ is the set of points ⁠$(x,y)$⁠ in the Cartesian plane with ⁠$y>0.$⁠ The lower half-plane is the set of points ⁠$(x,y)$⁠ with ⁠$y<0$⁠ instead. Each is an example of two-dimensional half-space.

Affine geometry

The affine transformations of the upper half-plane include

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

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

Proof: First shift the center of ⁠$A$⁠ to ⁠$(0,0).$⁠ Then take $$\begin{gathered} \lambda =(\text{diameter of} \\ B)/(\text{diameter of} \\ A) \end{gathered}$$

and dilate. Then shift ⁠$(0,0)$⁠ to the center of ⁠$B.$⁠

Inversive geometry

Definition: ${𝒵}:=\{({\cos }^2\theta ,\frac{1}{2}\sin 2\theta )\mid 0<\theta <\pi \}$. ⁠${𝒵}$⁠ can be recognized as the circle of radius ⁠$\frac{1}{2}$⁠ centered at ⁠$(\frac{1}{2},0),$⁠ and as the polar plot of $\rho (\theta )=\cos \theta .$ Proposition: ⁠$(0,0),$⁠ ⁠$\rho (\theta )$⁠ in ⁠${𝒵},$⁠ and ⁠$(1,\tan \theta )$⁠ are collinear points. In fact, ${𝒵}$ is the inversion of the line $\{(1,y)\mid y>0\}$ in the unit circle. Indeed, the diagonal from ⁠$(0,0)$⁠ to ⁠$(1,\tan \theta )$⁠ has squared length $1+{\tan }^2\theta ={\sec }^2\theta$, so that $\rho (\theta )=\cos \theta$ is the reciprocal of that length.

Metric geometry

The distance between any two points ⁠$p$⁠ and ⁠$q$⁠ in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from ⁠$p$⁠ to ⁠$q$⁠ either intersects the boundary or is parallel to it. In the latter case ⁠$p$⁠ and ⁠$q$⁠ 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 ⁠$p$⁠ and ⁠$q$⁠ 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 ⁠${𝒵}.$⁠ Distances on ⁠${𝒵}$⁠ can be defined using the correspondence with points on $\{(1,y)\mid y>0\}$ 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:

$$\begin{gathered} {\mathscr{H}}:=\{x+iy\mid y>0; \\ x,y\in \mathbb{ℝ}\}. \end{gathered}$$

The term arises from a common visualization of the complex number $x+iy$ as the point $(x,y)$ in the plane endowed with Cartesian coordinates. When the $y$ axis is oriented vertically, the "upper half-plane" corresponds to the region above the $x$ axis and thus complex numbers for which $y>0$. It is the domain of many functions of interest in complex analysis, especially 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 to ⁠${\mathscr{H}}$⁠ (see "Poincaré metric"), meaning that it is usually possible to pass between ⁠${\mathscr{H}}$⁠ 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. 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 $n$-space ⁠${{\mathscr{H}}}^n,$⁠ the maximally symmetric, simply connected, ⁠$n$⁠-dimensional Riemannian manifold with constant sectional curvature $-1$. In this terminology, the upper half-plane is ⁠${{\mathscr{H}}}^2$⁠ 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 ⁠${{\mathscr{H}}}^n$⁠ of ⁠$n$⁠ copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space ⁠${{\mathscr{H}}}_n,$⁠ 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

• Weisstein, Eric W. "Upper Half-Plane". MathWorld.