Hyperbolic geometry studies spaces of constant negative curvature.
A six-sided polygon in the hyperbolic plane whose sides are geodesic segments. The especially useful right-angled case is a right-angled hyperbolic hexagon.
A geodesic hexagon in the hyperbolic plane with all six interior angles . Any three positive alternating side lengths determine it up to isometry. If those sides are , the opposite seam satisfies . Two congruent such hexagons glued along the other three sides form a pair of pants with boundary lengths .
The Beltrami-Klein model realizes the hyperbolic plane inside an open Euclidean disk, with geodesics given by straight chords. Its metric differs from the Euclidean metric, and angles are generally distorted. The straight chords make geodesic convexity especially easy to see.
A surface with a complete Riemannian metric of constant Gaussian curvature . Its local model is the Poincare half-plane model; the orientable case is a quotient by a free properly discontinuous group action of orientation-preserving isometries.
For a marked closed hyperbolic surface, the representation of its fundamental group whose image acts as the deck group on the hyperbolic plane. It is discrete and faithful, determined up to conjugation by the marking and the metric, and gives . For a closed orientable surface it admits determinant-one lifts. It is different from the tangent-space action of the Riemannian holonomy group.
The unmarked length spectrum of a compact hyperbolic surface is the multiset of lengths of its closed geodesics, with a consistent convention for primitive curves and orientation. The marked version labels lengths by free homotopy classes. The Selberg trace formula relates the unmarked multiset to the Laplace-Beltrami operator spectrum.
In a compact smooth family of marked closed hyperbolic surfaces, only finitely many free homotopy classes of closed geodesics have length below a fixed bound somewhere in the family. A uniform bilipschitz equivalence with a fixed metric reduces this to local finiteness of its length spectrum.
For fixed genus and systole lower bound , a uniform finite length cutoff determines the entire unmarked length spectrum of a closed hyperbolic surface in the thick part, including multiplicities. Equality up to forces equality everywhere. This is a finite-determination statement for spectra and does not claim that all isospectral surfaces are isometric.
For a compact hyperbolic surface, the Selberg trace formula equates a sum over Laplace-Beltrami operator eigenvalues to an area term and a sum over primitive closed geodesics and their iterates. It proves that the Laplacian spectrum and the unmarked length spectrum, with multiplicities, determine each other.
A collar around a simple closed geodesic of length is an embedded annulus with metric , where . The collar lemma gives half-width . As tends to zero the collar gets wide, forcing crossing geodesics to become long.
In curvature , its length is . The level-two principal congruence subgroup has traces congruent to two modulo four, so the smallest hyperbolic absolute trace is six, achieved by . The hyperbolic translation length formula gives the result; the shortest closed geodesic is nonsimple.
A collection of disjoint essential simple closed curves that cuts a closed genus- surface into surfaces of type pair of pants. There are curves. Geodesic representatives are used for Fenchel–Nielsen coordinates.
An elementary move replaces one cuff inside the four-holed sphere or one-holed torus obtained by uniting its adjacent pants. The replacement is an essential curve meeting the old cuff minimally. It changes the pants decomposition graph without changing the underlying surface; the identity surface map identifies the two descriptions.
The graph of a pants decomposition has one vertex per pair of pants and one edge per glued cuff. It is a connected trivalent multigraph, possibly with loops. A closed surface of genus gives vertices and edges, so its first Betti number is . The graph records a decomposition rather than the homeomorphism class alone, and omits cuff lengths and twists.
For fixed , a constant bounds all cuff lengths in some pants decomposition of every closed genus- hyperbolic surface. Together with Fenchel–Nielsen coordinates this implies Mumford's compactness theorem.
For a fixed pants decomposition, the positive cuff lengths and real twist parameters give coordinates on Teichmüller space. With twists measured in length units, a full Dehn twist changes by ; with angle units it changes the twist by .
A simple closed curve that is a geodesic of length on a hyperbolic surface has an embedded collar of half-width with . In circumference-one coordinates its metric is , and its conformal modulus of an annulus is .
The infimum of nonconstant closed geodesic lengths in curvature , with value if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus , a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.
The metric area of a hyperbolic surface. In the Poincare half-plane model the density is . A finite-area torsion-free quotient of genus with cusps has area by the Gauss-Bonnet theorem.
A hyperbolic metric is a complete Riemannian metric of constant sectional curvature . On a surface it supplies local charts modeled on the hyperbolic plane.
The hyperbolic plane is the simply connected two-dimensional geometry of constant curvature minus one.
A hyperbolic disc consists of points within a specified hyperbolic distance of its centre. In the Poincare disc model, the disc centred at zero with radius has Euclidean radius ; an isometry transports this description to any other centre.
A horocycle in the Poincare disc model is a Euclidean circle internally tangent to the boundary circle. In the upper half-plane model, a horocycle is a Euclidean circle tangent to the real axis from above, or a horizontal line . The common geometric feature is a single ideal centre of a horocycle.
Two horocycles have a unique common orthogonal hyperbolic line when their ideal centres of horocycles differ: it is the line with those two ideal endpoints. If the centres agree, an isometry puts both horocycles on horizontal lines; every vertical hyperbolic line meets both orthogonally, so there are infinitely many. This classification remains valid for intersecting or tangent horocycles.
The ideal centre of a horocycle is its point of tangency to the boundary at infinity; a horizontal horocycle in the upper half-plane model has centre infinity. Sending the centre to infinity by an isometry makes the horocycle horizontal. Orthogonal hyperbolic lines then become vertical, proving that a hyperbolic line meets a horocycle orthogonally exactly when it has the centre as an ideal endpoint.
A hyperbolic circle of centre and radius is the level set for the hyperbolic distance. Equally spaced directions from its centre give vertices of a regular geodesic polygon.
In the Poincare half-plane model, a hyperbolic circle of radius centred at is the Euclidean circle . Its Euclidean centre is shifted upward from its hyperbolic centre, and its Euclidean radius is . The lower height is positive. The formula follows by mapping the centre to zero in the Poincare disc model and using radial distance .
A hyperbolic line is a complete geodesic in the hyperbolic plane. In the Poincare half-plane model, hyperbolic lines are vertical rays and semicircles orthogonal to the boundary.
An ideal endpoint of a hyperbolic line is one of its two limiting points on the boundary at infinity. In the Poincare disc model these lie on the unit circle; in the upper half-plane model they lie on . These are boundary points, not points of the hyperbolic plane. Any two distinct ideal endpoints determine a unique hyperbolic line.
Two distinct hyperbolic lines are parallel, in the limiting-parallel convention, when they do not intersect in the hyperbolic plane and have exactly one common ideal endpoint. In the upper half-plane model, distinct vertical lines are parallel, sharing infinity. Disjoint lines with no common ideal endpoint are instead ultraparallel hyperbolic lines; only these have a common perpendicular of ultraparallel hyperbolic lines.
Two hyperbolic lines are ultraparallel when they are disjoint and have no common ideal endpoint. They possess a unique common perpendicular, which realizes the distance between them.
A horoball in the hyperbolic plane is a region bounded by a horocycle. In the upper-half-plane model, a horoball based at infinity is a Euclidean half-plane ; quotients by parabolic isometries give cusp neighborhoods.
A noncentral finite-order orientation-preserving hyperbolic isometry fixes an interior point and is elliptic.
For , the Möbius transformation is an isometry of the Poincare disc model sending to zero. Rotations combined with these transformations describe its orientation-preserving isometries.
In curvature minus one, a hyperbolic disc of radius has areaIn the Poincare disc this follows by integrating up to Euclidean radius .
Hyperbolic discs of area have radius . Starting at the origin and placing their centers successively along one radial geodesic gives center distance and Euclidean coordinate
An isometry sends the centers of a collinear family of equal hyperbolic discs to points on one hyperbolic geodesic. In the upper-half-plane model no geodesic contains three distinct points on a horizontal Euclidean line, since geodesics are vertical lines or semicircles orthogonal to the boundary.
The geodesics are diameters and arcs of circles orthogonal to the unit circle. A radial segment is length minimizing becausewith equality for constant angle and monotone radius. Hence the distance from the origin to is .
If is at hyperbolic distance from the origin, the Poincare-disc geodesic through perpendicular to is represented by a Euclidean circle of radius whose centre is at Euclidean distance from the origin.
For a hyperbolic Möbius transformation represented by , the infimum of is attained on its axis and equals . It is the length of the corresponding closed geodesic in a quotient.
For a determinant-one lift of a hyperbolic holonomy representation, the squared matrix trace of each fixed group word is a polynomial in the finitely many generator-matrix entries. The displayed identity identifies equal positive lengths with equal squared traces. Lift signs have no effect. Finite choices of length matchings therefore give finite unions of affine algebraic sets.
The geodesics in the upper half-plane are vertical lines and semicircles centred on the boundary axis. Indeed, the length integrand has the Beltrami identitywhich integrates to . Swapping the coordinate axes gives circles centred on the vertical boundary for the metric .
Two disjoint hyperbolic lines with four distinct ideal endpoints have a unique common perpendicular, and the segment of that perpendicular between them realizes their minimum distance. An upper-half-plane isometry can make the two lines concentric semicircles; their common perpendicular then becomes the imaginary axis.
Three pairwise ultraparallel hyperbolic lines may have disjoint common perpendiculars or coincident ones. In the upper-half-plane model, three semicircles of radius centred at have mutually disjoint common perpendiculars. Three concentric semicircles instead have the same vertical common perpendicular. Thus pairwise ultraparallelism alone does not determine the incidence of their perpendiculars.
The hyperbolic metric in the half-plane model is conformal, so hyperbolic and Euclidean angles agree. The line meeting the imaginary axis at , , at signed angle is represented by the Euclidean circle with centre on the real axis and radius .
PSL2(R) is . It acts on the upper half-plane by the Möbius transformationsand is the group of orientation-preserving hyperbolic isometries.
A real Möbius transformation represented by with . It has two fixed ideal boundary points and translates along the geodesic joining them; its hyperbolic translation length is . Conjugating by a real Möbius transformation puts it in the form with ; its axis is then the imaginary axis.
A real Möbius transformation represented by with . It fixes a point of the Poincare half-plane model; an integral matrix of this type has trace or . In the Poincare disc model, moving its fixed point to zero makes it a rotation about zero.
A nonidentity real Möbius transformation represented by with . It has one ideal boundary fixed point and is conjugate to a nonzero translation on the Poincare half-plane model.
Every orientation-preserving isometry of the Poincare half-plane model lies in . Every orientation-reversing isometry is the composition of one of these with the hyperbolic reflection in the imaginary axis. Equivalently, the full isometry group consists of real Möbius and anti-Möbius transformations, modulo nonzero scalar matrices.
The group acts transitively on pairs consisting of a hyperbolic line and a point on it. Send the two ideal endpoints of to and , then use a positive dilation to move the resulting point on the imaginary axis to .
Reflection in a hyperbolic line is the unique orientation-reversing hyperbolic isometry fixing that line pointwise. The product of reflections in intersecting lines is a rotation through twice their angle.
A hyperbolic half-turn is the unique orientation-preserving isometry of order two fixing a given point. It exchanges every pair of points lying at equal opposite distances along a geodesic through its centre.
The hyperboloid model of the curvature-minus-one hyperbolic plane isIts metric is the restriction of the Minkowski bilinear form to tangent planes, and its geodesics are intersections with two-dimensional linear subspaces through the origin.
A Lambert quadrilateral is a hyperbolic quadrilateral with three right angles. If its remaining angle is and the two sides incident with that angle have lengths , then
A hyperbolic triangle is bounded by three hyperbolic geodesics and may have vertices on the ideal boundary.
Two corresponding side lengths and an equal nonincluded angle need not determine a hyperbolic triangle. Set , , . The hyperbolic law of cosines has two distinct positive solutions . The two triangles share but have different third sides. An included angle instead determines the third side by the cosine rule.
Three corresponding side lengths determine a hyperbolic triangle up to an isometry of the hyperbolic plane. First match an ordered pair of vertices by an isometry. The third vertex lies at the intersection of two prescribed hyperbolic distance circles. There are at most two choices, interchanged by reflection in the geodesic through the matched pair; either choice can therefore be matched.
For a geodesic hyperbolic -gon of curvature with interior angles , triangulation or the Gauss--Bonnet theorem gives
In curvature , a regular -gon of circumradius has interior angle , by the right-triangle angle form of the hyperbolic law of cosines. Thus its hyperbolic polygon area is , continuously and strictly increasing from to . Every area in that open interval is therefore attained at exactly one circumradius.
For opposite side , .
A hyperbolic triangle is determined up to isometry by its three angles. Indeed, the angle form of the hyperbolic law of cosines determines every side length from the angles, after which hyperbolic side-side-side congruence determines the triangle. This differs from Euclidean geometry because hyperbolic space has an intrinsic length scale.
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Hyperbolic geometry is a non-Euclidean geometry that arises from altering Euclid's fifth postulate, the parallel postulate. In hyperbolic geometry, the essential distinction is that, given a line and a point not on that line, there are infinitely many lines through that point that do not intersect the original line. This contrasts with Euclidean geometry, where there is exactly one parallel line that can be drawn through a point not on a line.