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Upper half-plane
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In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrarily 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
[edit]The affine transformations of the upper half-plane include
- shifts , , and
- dilations ,
Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes to .
- Proof: First shift the center of to Then take
and dilate. Then shift to the center of
Inversive geometry
[edit]Definition: .
can be recognized as the circle of radius centered at and as the polar plot of
Proposition: in and are collinear points.
In fact, is the inversion of the line in the unit circle. Indeed, the diagonal from to has squared length , so that is the reciprocal of that length.
Metric geometry
[edit]The distance between any two points and in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from to either intersects the boundary or is parallel to it. In the latter case and 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 and 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 Distances on can be defined using the correspondence with points on 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
[edit]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:
The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the axis is oriented vertically, the "upper half-plane" corresponds to the region above the axis and thus complex numbers for which .
It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by is equally good, but less used by convention. The open unit disk (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to (see "Poincaré metric"), meaning that it is usually possible to pass between and
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
[edit]One natural generalization in differential geometry is hyperbolic -space the maximally symmetric, simply connected, -dimensional Riemannian manifold with constant sectional curvature . In this terminology, the upper half-plane is since it has real dimension
In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product of copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space which is the domain of Siegel modular forms.
See also
[edit]References
[edit]Upper half-plane
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Definition and Fundamentals
Definition
The upper half-plane is formally defined as the set of points in the Euclidean plane such that .[6] Equivalently, in the context of the complex plane, it consists of all complex numbers where and the imaginary part .[6][7] The boundary of the upper half-plane is the real axis, which serves as a natural "horizon" or limit set separating it from the lower half-plane.[6][8] It is conventionally denoted by , , , or .[6] This defines an open half-plane, excluding the boundary real axis, in contrast to the closed upper half-plane which includes points where ; the concept generalizes to open or closed half-spaces in higher-dimensional Euclidean spaces for .[9][6]Basic Properties
The upper half-plane, denoted $ H = { z = x + iy \in \mathbb{C} \mid y > 0 } $, is an open subset of the complex plane , as the condition defines an open set in the standard topology of .[6] Its boundary is the real axis, characterized by the equation .[10] Topologically, $ H $ is simply connected, meaning every closed curve in $ H $ can be continuously contracted to a point within $ H $, and contractible, as it admits a continuous deformation retraction to any point inside it.[10] This follows from its convexity: for any two points $ z_1, z_2 \in H $ with and , the line segment connecting them lies entirely in $ H $, since the imaginary part varies linearly and remains positive.[10] Furthermore, $ H $ is homeomorphic to the open unit disk $ \mathbb{D} = { w \in \mathbb{C} \mid |w| < 1 } $ via the Cayley transform $ F(z) = \frac{z - i}{z + i} $, which is a continuous bijection with continuous inverse.[10] Algebraically, viewing as a vector space over , $ H $ is a proper open convex subset but not a subspace, as it fails closure under scalar multiplication by all real numbers (e.g., multiplication by maps points in $ H $ to the lower half-plane).[11] It is also not closed under complex multiplication; for instance, multiplying $ i \in H $ by itself yields $ i \cdot i = -1 $, which lies on the boundary . With respect to the standard Lebesgue measure on (identifying ), $ H $ has infinite area, as the integral diverges due to unbounded extent in both real and imaginary directions.[12]Geometric Contexts
Euclidean and Affine Geometry
The upper half-plane, denoted , is embedded in the Euclidean plane as an open half-space bounded below by the real axis. In this embedding, it inherits the standard structure of Euclidean geometry, where points are equipped with coordinates and the geometry is flat, with parallel lines remaining parallel and the Pythagorean theorem holding locally. The Euclidean distance between two points and in is given by
This metric induces the usual notions of length, angle, and area, with straight line segments serving as shortest paths (geodesics) within the space.[4]
Affine transformations that preserve the upper half-plane consist primarily of horizontal translations where , which shift points parallel to the real axis without altering the imaginary part, and positive dilations (scalings) where , which expand or contract distances radially from the origin while maintaining the sign of the imaginary part. These form a subgroup of the projective special linear group , specifically the affine subgroup generated by upper-triangular matrices, acting via Möbius transformations of the form with and . This subgroup acts transitively on in a way that respects its Euclidean structure, preserving parallelism and ratios of distances along parallel lines.[13]
A key property of these affine transformations is that they map semicircles in that are orthogonal to the real axis—namely, those with diameters lying on the real axis—to other such semicircles. Since translations and dilations are similarities (preserving angles up to orientation), they maintain the right-angle condition at the endpoints on the real axis, transforming the curves into scaled and shifted versions while keeping them perpendicular to the boundary. For instance, the affine transformation maps the semicircle with diameter from to (centered at with radius ) to the semicircle with diameter from to (centered at with radius ). This preservation highlights how the affine group stabilizes the family of these semicircles, which serve as "generalized lines" in certain contexts but are not geodesics in the Euclidean (or affine) sense, contrasting with their role as geodesics in the hyperbolic geometry of .[2][13]
Inversive Geometry
In inversive geometry, the upper half-plane is studied through transformations that preserve the family of circles and lines, known as generalized circles or clines. Inversion with respect to a circle is a fundamental operation, mapping points P to P' such that the center O, P, and P' are collinear and OP · OP' = r², where r is the radius of the inversion circle. For a circle C centered at o with radius r, the image of a point z under inversion is given by
This transformation maps clines to clines: the inverse image of a circle or line under inversion is another circle or line, unless the original passes through the center o, in which case it maps to itself.[14][15]
A specific case is inversion in the unit circle, where o = 0 and r = 1, given by the map $ z \mapsto 1 / \bar{z} $. This transformation is an involution (its own inverse) and conformal, preserving angles up to orientation. For z in the upper half-plane (Im z > 0), the image satisfies Im(1 / \bar{z}) = (Im z) / |z|^2 > 0, so it maps the upper half-plane to itself. To relate the upper half-plane to the unit disk, one composes inversion with reflection or uses inversion in a suitable non-unit circle; for instance, inversion in the circle centered at i with radius (passing through -1 and 1 on the real axis) maps the open unit disk bijectively to the upper half-plane, with the unit circle mapping to the real axis. The inverse transformation then maps the upper half-plane to the unit disk, facilitating the equivalence of the two domains in inversive contexts.[14][16][17]
In inversive geometry, the family of clines orthogonal to the real axis—vertical lines and semicircles with diameters on the real axis—plays a key role, as inversions preserve orthogonality relations. A key proposition concerns collinearity under inversion: three points are collinear if and only if their images under inversion lie on a cline passing through the center of inversion. If the original line avoids the center, the images lie on a circle through the center; if it passes through the center, the images remain collinear. This property underscores how inversion distorts Euclidean collinearity while preserving incidence relations among clines.[14][18]
Complex Analysis and Transformations
Representation in the Complex Plane
The upper half-plane, denoted , is the set of all complex numbers , where and . This region represents the portion of the complex plane above the real axis, excluding the boundary.[6] As an open subset of the complex plane , serves as a domain for holomorphic functions, which are complex differentiable everywhere within it. For example, the exponential function is entire, meaning it is holomorphic on all of and thus restricted to . Similarly, the principal branch of the complex logarithm, where , is holomorphic on minus the non-positive real axis (the branch cut), a region that contains .[19][20] A fundamental feature of in complex analysis is its conformal equivalence to the open unit disk via the Cayley transform , which is a biholomorphic mapping from onto . This transformation sends the upper half-plane to the interior of the unit disk, the real axis (boundary of ) to the unit circle, and the point to the origin.[21] The upper half-plane is conformally equivalent to one component of the Riemann sphere minus the extended real line. The Riemann sphere is the one-point compactification of the complex plane, and the extended real line embeds as a Jordan curve (homeomorphic to a circle) on it; removing this curve leaves two connected components—the open upper and lower half-planes—each of which is simply connected and conformally equivalent to the unit disk .[22]Möbius Transformations and Group Actions
Möbius transformations that preserve the upper half-plane are those of the form $ z \mapsto \frac{az + b}{cz + d} $, where $ a, b, c, d \in \mathbb{R} $ and $ ad - bc = 1 $.[23] These transformations correspond to matrices in the special linear group $ \mathrm{SL}(2, \mathbb{R}) $, and the action factors through the projective special linear group $ \mathrm{PSL}(2, \mathbb{R}) = \mathrm{SL}(2, \mathbb{R}) / { \pm I } $, since $ -\begin{pmatrix} a & b \ c & d \end{pmatrix} $ induces the same map as $ \begin{pmatrix} a & b \ c & d \end{pmatrix} $.[24] The group $ \mathrm{PSL}(2, \mathbb{R}) $ acts as the full group of biholomorphic automorphisms of $ \mathbb{H} $, preserving its complex structure and acting transitively on $ \mathbb{H} $.[25] This action is faithful and consists of orientation-preserving isometries of $ \mathbb{H} $, with fixed points of non-identity elements lying on the boundary $ \mathbb{R} \cup { \infty } $.[24] Specific examples include translations $ z \mapsto z + b $ for $ b \in \mathbb{R} $, corresponding to matrices $ \begin{pmatrix} 1 & b \ 0 & 1 \end{pmatrix} $; dilations (or scalings) $ z \mapsto k^2 z $ for $ k > 0 $, given by $ \begin{pmatrix} k & 0 \ 0 & 1/k \end{pmatrix} $; and inversions such as $ z \mapsto -1/z $, represented by $ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} $.[23] These generate $ \mathrm{PSL}(2, \mathbb{R}) $ under composition, and each preserves $ \mathbb{H} $ while mapping the boundary to itself.[25] A discrete subgroup of interest is the modular group $ \mathrm{PSL}(2, \mathbb{Z}) $, which acts on $ \mathbb{H} $ via the same Möbius transformations but with integer entries in the matrices.[26] This action is properly discontinuous, and a fundamental domain for $ \mathrm{PSL}(2, \mathbb{Z}) $ is the region $ D = { z \in \mathbb{H} : |z| > 1, |\operatorname{Re}(z)| < 1/2 } $, which tiles $ \mathbb{H} $ under the group action up to boundary identifications.[26] The upper half-plane is conformally equivalent to the unit disk via a Möbius transformation, allowing similar group actions to be studied there.[23]Hyperbolic Geometry
Poincaré Half-Plane Model
The Poincaré half-plane model realizes the upper half-plane as a geometric model of the hyperbolic plane, where geodesics are either vertical lines extending to the real axis or semicircles centered on the real axis that intersect it orthogonally.[27] These semicircles arise naturally from properties of inversive geometry, preserving orthogonality under inversion with respect to circles.[28] This model provides an isomorphism between equipped with the hyperbolic metric and the two-dimensional hyperbolic space , which has constant Gaussian curvature .[29] The isometries of this model are generated by the projective special linear group , acting via Möbius transformations that map to itself and preserve the hyperbolic structure.[30] For instance, the geodesic connecting the points and is the vertical line segment , which extends infinitely in the hyperbolic sense toward the boundary.[29] The model is conformal, meaning it preserves angles as measured in the Euclidean plane, though Euclidean distances are distorted to reflect the hyperbolic geometry.[27] This conformality facilitates visualizations of hyperbolic figures, such as triangles with angle sums less than , directly within the familiar complex plane.[31]Hyperbolic Metric and Distance
The upper half-plane is endowed with the Poincaré hyperbolic metric, a Riemannian metric of constant curvature that defines the geometry of the hyperbolic plane. This metric is expressed in real coordinates as
where the infinitesimal arc length element measures distances conformally scaled by the imaginary part .[32] In complex notation, identifying , the metric takes the form , which highlights its invariance under the action of the special linear group .[17]
The hyperbolic distance between two points is the infimum of the lengths of all piecewise smooth paths connecting them, where the length of a path is given by the integral
This length is realized uniquely along the geodesic connecting and , which in the Poincaré model consists of semicircles orthogonal to the real axis or vertical rays. The closed-form distance formula is
derived by integrating the metric along the geodesic and using hyperbolic trigonometric identities.[33] An equivalent expression is , obtained via the identity .[17]
A key feature of this metric is its behavior near the boundary : as , the factor causes local distances to expand dramatically, effectively modeling an infinite expanse approaching the geodesic boundary at infinity. This expansion is evident in the length integral, where paths near the real axis accrue large hyperbolic lengths despite short Euclidean distances. Horocycles, which are curves of constant distance from a fixed ideal point on the boundary, appear in the upper half-plane as Euclidean circles tangent to the real axis or as horizontal lines parallel to it.[34]
Applications
In Number Theory and Modular Forms
The modular group, denoted PSL(2, ℤ), consists of 2×2 matrices with integer entries and determinant 1, modulo the center {±I}, and acts on the upper half-plane ℍ via Möbius transformations of the form τ ↦ (aτ + b)/(cτ + d) for γ = \begin{pmatrix} a & b \ c & d \end{pmatrix} ∈ PSL(2, ℤ). This action is properly discontinuous, allowing the construction of a fundamental domain, typically the region D = {τ ∈ ℍ : |Re(τ)| ≤ 1/2, |τ| ≥ 1}, which tiles ℍ under the group action, with identifications along its boundaries and a cusp at infinity corresponding to the rational projective line ℙ¹(ℚ) ∪ {∞}.[35] Modular forms are central to this framework, defined as holomorphic functions f: ℍ → ℂ of weight k ∈ 2ℤ_{≥0} satisfying the transformation property (f|_k γ)(τ) = f(τ) for all γ ∈ PSL(2, ℤ), where the slash operator is (f|k γ)(τ) = (cz + d)^{-k} f(γτ), along with holomorphy at the cusps, meaning the Fourier expansion q ↦ ∑{n=0}^∞ a_n q^n (with q = e^{2πiτ}) at infinity has no negative powers after suitable scaling by cusp width. This invariance encodes arithmetic data, with the space of such forms M_k(PSL(2, ℤ)) forming a finite-dimensional vector space over ℂ, graded by weight. Cusp forms, the kernel of the constant term map, further refine this to functions vanishing at cusps. Prominent examples include the Eisenstein series E_k(τ) for even k ≥ 4, defined as the normalized sum E_k(τ) = \frac{1}{2ζ(k)} ∑{(m,n) ≠ (0,0)} (mτ + n)^{-k} over the integer lattice ℤ² \ {0}, which are non-constant holomorphic modular forms generating the Eisenstein subspace. Their Fourier expansions are $E_k(\tau) = 1 - \frac{2k}{B_k} \sum{n=1}^\infty \sigma_{k-1}(n) q^n $, where $ q = e^{2\pi i \tau} $ and $ \sigma_{k-1}(n) = \sum_{d \mid n} d^{k-1} $ is the sum of the (k-1)th powers of the positive divisors of n; the constant term relates to Bernoulli numbers $ B_k $ via the normalization. These series converge absolutely for Im(τ) > 0 and transform correctly under the group. Another key example is the j-invariant, j(τ) = 1728 \frac{E_4(τ)^3}{E_4(τ)^3 - E_6(τ)^2} (up to the discriminant Δ = η^{24}), a weight-zero modular function (hauptmodul) that maps the fundamental domain bijectively onto ℂ, parameterizing isomorphism classes of elliptic curves over ℂ.[36] The quotient ℍ / PSL(2, ℤ), compactified by adding cusps to form the modular curve X(1), is biholomorphic to the Riemann sphere ℙ¹(ℂ), with the j-invariant inducing this isomorphism and serving as a coordinate on the curve. This structure plays a pivotal role in class number problems in algebraic number theory, where the values of j at quadratic imaginary points τ (singular moduli) generate ring class fields of imaginary quadratic orders, and the minimal polynomials of these values relate directly to class numbers; for instance, the class number one problem for imaginary quadratic fields was resolved using properties of Heegner points on X(1), confirming only nine such fields exist.[37]In Riemann Surfaces and Uniformization
The uniformization theorem, a cornerstone of complex geometry, classifies simply connected Riemann surfaces up to biholomorphic equivalence as one of three canonical models: the Riemann sphere , the complex plane , or the open unit disk (which is biholomorphically equivalent to the upper half-plane via the Cayley transform). Riemann surfaces of hyperbolic type—those admitting a non-constant bounded holomorphic function or equivalently those with universal cover of infinite area—have (or ) as their universal cover. This result, proved independently by Henri Poincaré and Paul Koebe in 1907, underscores the upper half-plane's role as the prototypical simply connected hyperbolic Riemann surface, enabling the global study of conformal structures through its rich automorphism group .[38] For multiply connected Riemann surfaces, particularly compact ones of genus , the uniformization theorem extends via quotient constructions: such a surface is biholomorphic to , where is a discrete, torsion-free Fuchsian group acting freely and properly discontinuously on , with isomorphic to the fundamental group of . These quotients inherit the hyperbolic metric from , yielding a complete Riemann surface of finite area whose Euler characteristic is . Seminal work by Poincaré (1882) and later developments by Felix Klein and Henri Poincaré formalized this representation, highlighting how Fuchsian groups tile to produce the topology and complex structure of . This framework not only uniformizes higher-genus surfaces but also facilitates the computation of invariants like the Gauss-Bonnet theorem's total curvature, fixed at .[39] Teichmüller space for a surface of genus parameterizes the distinct hyperbolic structures up to isotopy, modeled as the space of marked Fuchsian representations modulo conjugation, where each defines a quotient biholomorphic to with a marked hyperbolic metric. Oswald Teichmüller's 1940 work established that is a contractible complex manifold of dimension , equipped with the Teichmüller metric measuring quasiconformal deformations, and it serves as the "universal deformation space" near the basepoint corresponding to a fixed embedding of . This structure captures the local rigidity and global flexibility of hyperbolic geometries on surfaces, with Bers' simultaneous uniformization theorem (1958) embedding into spaces of quadratic differentials on .[40] In the parabolic case of genus (tori), uniformization uses for a lattice , but the moduli space of isomorphism classes of such tori is parameterized by the upper half-plane via the modulus , with elliptic functions like the Weierstrass -function providing the explicit biholomorphism from the torus to its projective model in . The modular group acts on to identify equivalent moduli, yielding the orbifold as the coarse moduli space, compactified by the -invariant map. This connection, developed by Karl Weierstrass and others in the 19th century, links the upper half-plane directly to the analytic uniformization of elliptic curves.[41] The upper half-plane's uniformization role extends to boundary value problems: for a Riemann surface of hyperbolic type, the Dirichlet problem—solving with prescribed continuous boundary values—reduces via the covering map to solving the analogous problem on , where the Poisson integral formula provides the explicit harmonic solution for and boundary data on . Pulling back via yields the solution on , with -invariance ensuring well-definedness; this method, rooted in Poincaré's integral representations, leverages the explicit geometry of for computational and theoretical advances in potential theory on surfaces.[42]Generalizations and Extensions
Higher-Dimensional Half-Spaces
The upper half-space in dimensions, denoted , is defined as the set .[43] This generalizes the two-dimensional upper half-plane by extending the domain to higher Euclidean dimensions while preserving the boundary at .[43] In coordinates with , it can be expressed equivalently as .[43] The Poincaré half-space model endows with a hyperbolic metric of constant sectional curvature , given by
[43] This metric is conformal to the Euclidean metric and induces the hyperbolic distance function , where denotes the Euclidean norm in and are the final coordinates of and . Geodesics in this model are the intersections of with Euclidean spheres (or hyperplanes) that meet the boundary orthogonally; these include vertical lines parallel to the -axis and semicircles (or hemispherical arcs) centered on the boundary.[44]
A key geometric property of is the exponential growth of ball volumes: the volume of a hyperbolic ball of radius centered at any point satisfies as , reflecting the rapid expansion characteristic of negative curvature spaces.[45] The full isometry group of is the orthogonal group , acting transitively and preserving the metric; the orientation-preserving subgroup is .[43] This group is generated by scalings, translations in the first coordinates, rotations via , and inversions in spheres orthogonal to the boundary.[43]