Hyperbolic space

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https://en.wikipedia.org/wiki/Hyperbolic_space 

 

A perspective projection of a dodecahedral tessellation in H³.
Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E³

In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define Euclidean geometry, and elliptic spaces that have a constant positive curvature.

When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially.

Formal definition

Hyperbolic n-space, denoted Hⁿ, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with a constant negative sectional curvature. Hyperbolic space is a space exhibiting hyperbolic geometry. It is the negative-curvature analogue of the n-sphere. Although hyperbolic space Hⁿ is diffeomorphic to Rⁿ, its negative-curvature metric gives it very different geometric properties.

Hyperbolic 2-space, H², is also called the hyperbolic plane.

Models of hyperbolic space

Hyperbolic space, developed independently by Nikolai Lobachevsky and János Bolyai, is a geometrical space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):

• Given any line L and point P not on L, there are at least two distinct lines passing through P which do not intersect L.

It is then a theorem that there are infinitely many such lines through P. This axiom still does not uniquely characterize the hyperbolic plane up to isometry; there is an extra constant, the curvature K < 0, which must be specified. However, it does uniquely characterize it up to homothety, meaning up to bijections which only change the notion of distance by an overall constant. By choosing an appropriate length scale, one can thus assume, without loss of generality, that K = −1.

Models of hyperbolic spaces that can be embedded in a flat (e.g. Euclidean) spaces may be constructed. In particular, the existence of model spaces implies that the parallel postulate is logically independent of the other axioms of Euclidean geometry.

There are several important models of hyperbolic space: the Klein model, the hyperboloid model, the Poincaré ball model and the Poincaré half space model. These all model the same geometry in the sense that any two of them can be related by a transformation that preserves all the geometrical properties of the space, including isometry (though not with respect to the metric of a Euclidean embedding).

Hyperboloid model

Main article: Hyperboloid model

The hyperboloid model realizes hyperbolic space as a hyperboloid in Rⁿ⁺¹ = {(x₀,...,x_n)|x_i∈R, i=0,1,...,n}. The hyperboloid is the set Hⁿ of points whose coordinates satisfy

${\displaystyle x_{0}^{2}-x_{1}^{2}-\cdots -x_{n}^{2}=1,\quad x_{0}>0.}$

In this model a line (or geodesic) is the curve formed by the intersection of Hⁿ with a plane through the origin in Rⁿ⁺¹.

The hyperboloid model is closely related to the geometry of Minkowski space. The quadratic form

${\displaystyle Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-\cdots -x_{n}^{2},}$

which defines the hyperboloid, polarizes to give the bilinear form

${\displaystyle B(x,y)=(Q(x+y)-Q(x)-Q(y))/2=x_{0}y_{0}-x_{1}y_{1}-\cdots -x_{n}y_{n}.}$

The space Rⁿ⁺¹, equipped with the bilinear form B, is an (n+1)-dimensional Minkowski space R^n,1.

One can associate a distance on the hyperboloid model by defining the distance between two points x and y on Hⁿ to be

${\displaystyle d(x,y)=\operatorname {arcosh} B(x,y).}$

This function satisfies the axioms of a metric space. It is preserved by the action of the Lorentz group on R^n,1. Hence the Lorentz group acts as a transformation group preserving isometry on Hⁿ.

Klein model

Main article: Klein model

An alternative model of hyperbolic geometry is on a certain domain in projective space. The Minkowski quadratic form Q defines a subset Uⁿ ⊂ RPⁿ given as the locus of points for which Q(x) > 0 in the homogeneous coordinates x. The domain Uⁿ is the Klein model of hyperbolic space.

The lines of this model are the open line segments of the ambient projective space which lie in Uⁿ. The distance between two points x and y in Uⁿ is defined by

${\displaystyle d(x,y)=\operatorname {arcosh} \left({\frac {B(x,y)}{\sqrt {Q(x)Q(y)}}}\right).}$

This is well-defined on projective space, since the ratio under the inverse hyperbolic cosine is homogeneous of degree 0.

This model is related to the hyperboloid model as follows. Each point x ∈ Uⁿ corresponds to a line L_x through the origin in Rⁿ⁺¹, by the definition of projective space. This line intersects the hyperboloid Hⁿ in a unique point. Conversely, through any point on Hⁿ, there passes a unique line through the origin (which is a point in the projective space). This correspondence defines a bijection between Uⁿ and Hⁿ. It is an isometry, since evaluating d(x,y) along Q(x) = Q(y) = 1 reproduces the definition of the distance given for the hyperboloid model.

Poincaré ball model

Main article: Poincaré disc model

A closely related pair of models of hyperbolic geometry are the Poincaré ball and Poincaré half-space models.

The ball model comes from a stereographic projection of the hyperboloid in Rⁿ⁺¹ onto the hyperplane {x₀ = 0}. In detail, let S be the point in Rⁿ⁺¹ with coordinates (−1,0,0,...,0): the South pole for the stereographic projection. For each point P on the hyperboloid Hⁿ, let P^∗ be the unique point of intersection of the line SP with the plane {x₀ = 0}.

This establishes a bijective mapping of Hⁿ into the unit ball

${\displaystyle B^{n}=\{(x_{1},\ldots ,x_{n})|x_{1}^{2}+\cdots +x_{n}^{2}<1\}}$

in the plane {x₀ = 0}.

The geodesics in this model are semicircles that are perpendicular to the boundary sphere of Bⁿ. Isometries of the ball are generated by spherical inversion in hyperspheres perpendicular to the boundary.

Poincaré half-space model

Main article: Poincaré half-plane model

The half-space model results from applying inversion in a circle with centre a boundary point of the Poincaré ball model Bⁿ above and a radius of twice the radius.

This sends circles to circles and lines, and is moreover a conformal transformation. Consequently, the geodesics of the half-space model are lines and circles perpendicular to the boundary hyperplane.

Hyperbolic manifolds

Every complete, connected, simply connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hⁿ. As a result, the universal cover of any closed manifold M of constant negative curvature −1, which is to say, a hyperbolic manifold, is Hⁿ. Thus, every such M can be written as Hⁿ/Γ where Γ is a torsion-free discrete group of isometries on Hⁿ. That is, Γ is a lattice in SO⁺(n,1).

Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group π₁=Γ; the groups that arise this way are known as Fuchsian groups. The quotient space H²/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is the universal cover of the other hyperbolic surfaces.

The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.

 

Non-Euclidean geometry

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Non-Euclidean_geometry 

 

 

Behavior of lines with a common perpendicular in each of the three types of geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either relaxing the metric requirement, or replacing the parallel postulate with an alternative. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l.

Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane):

• In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
• In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
• In elliptic geometry, the lines "curve toward" each other and intersect.

History

See also: Hyperbolic geometry § History

Background

Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.

The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote Elements. In the Elements, Euclid begins with a limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks to prove all the other results (propositions) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate", or simply the parallel postulate, which in Euclid's original formulation is:

If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears more complicated than Euclid's other postulates:

1. To draw a straight line from any point to any point.

2. To produce [extend] a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance [radius].

4. That all right angles are equal to one another.

For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham (Alhazen, 11th century), Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī (13th century), and Giovanni Girolamo Saccheri (18th century).

The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries". These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry, however, provided flawed proofs of the parallel postulate, containing assumptions that were essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries.

Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid, which he didn't realize was equivalent to his own postulate. Another example is al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented another hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements." His work was published in Rome in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis.

Giordano Vitale, in his book Euclide restituo (1680, 1686), used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.

In a work titled Euclides ab Omni Naevo Vindicatus (Euclid Freed from All Flaws), published in 1733, Saccheri quickly discarded elliptic geometry as a possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry.

He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it.

In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure now known as a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.

At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.

Discovery of non-Euclidean geometry

The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew Franz Taurinus did publish important results of hyperbolic trigonometry in two papers in 1825 and 1826, yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry.

Then, in 1829–1830 the Russian mathematician Nikolai Ivanovich Lobachevsky and in 1832 the Hungarian mathematician János Bolyai separately and independently published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.

Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines.

By formulating the geometry in terms of a curvature tensor, Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative curvature.

Terminology

It was Gauss who coined the term "non-Euclidean geometry". He was referring to his own work, which today we call hyperbolic geometry. Several modern authors still consider non-Euclidean geometry and hyperbolic geometry synonyms.

Arthur Cayley noted that distance between points inside a conic could be defined in terms of logarithm and the projective cross-ratio function. The method has become called the Cayley–Klein metric because Felix Klein exploited it to describe the non-Euclidean geometries in articles in 1871 and 1873 and later in book form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry.

Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry parabolic, a term that generally fell out of use). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.

There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways.

Axiomatic basis of non-Euclidean geometry

Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms. Hilbert's system consisting of 20 axioms most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff, for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry. As the first 28 propositions of Euclid (in The Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.

To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation. Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways:

• Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. In the first case, replacing the parallel postulate (or its equivalent) with the statement "In a plane, given a point P and a line l not passing through P, there exist two lines through P, which do not meet l" and keeping all the other axioms, yields hyperbolic geometry.
• The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, "In a plane, given a point P and a line l not passing through P, all the lines through P meet l", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry, but this statement says that there are no parallel lines. This problem was known (in a different guise) to Khayyam, Saccheri and Lambert and was the basis for their rejecting what was known as the "obtuse angle case". To obtain a consistent set of axioms that includes this axiom about having no parallel lines, some other axioms must be tweaked. These adjustments depend upon the axiom system used. Among others, these tweaks have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded. Riemann's elliptic geometry emerges as the most natural geometry satisfying this axiom.

Models of non-Euclidean geometry

Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions

 

Further information: Models of non-Euclidean geometry

 

On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.

Two dimensional Euclidean geometry is modelled by our notion of a "flat plane".

Elliptic geometry

Main article: Elliptic geometry

The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other (called antipodal points) are identified (considered the same). This is also one of the standard models of the real projective plane. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric.

In the elliptic model, for any given line l and a point A, which is not on l, all lines through A will intersect l.

Hyperbolic geometry

Main article: Hyperbolic geometry

Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for hyperbolic geometry?". The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model, which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.)

In the hyperbolic model, within a two-dimensional plane, for any given line l and a point A, which is not on l, there are infinitely many lines through A that do not intersect l.

In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of the way they are represented.

Three-dimensional non-Euclidean geometry

Main article: Thurston geometry

In three dimensions, there are eight models of geometries. There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry that is completely anisotropic (i.e. every direction behaves differently).

Uncommon properties

Lambert quadrilateral in hyperbolic geometry

Saccheri quadrilaterals in the three geometries

Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon the nature of parallelism. This commonality is the subject of absolute geometry (also called neutral geometry). However, the properties that distinguish one geometry from others have historically received the most attention.

Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following:

• A Lambert quadrilateral is a quadrilateral with three right angles. The fourth angle of a Lambert quadrilateral is acute if the geometry is hyperbolic, a right angle if the geometry is Euclidean or obtuse if the geometry is elliptic. Consequently, rectangles exist (a statement equivalent to the parallel postulate) only in Euclidean geometry.
• A Saccheri quadrilateral is a quadrilateral with two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic.
• The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic. The defect of a triangle is the numerical value (180° − sum of the measures of the angles of the triangle). This result may also be stated as: the defect of triangles in hyperbolic geometry is positive, the defect of triangles in Euclidean geometry is zero, and the defect of triangles in elliptic geometry is negative.

Importance

Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.

The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift.

Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed the way they viewed their subjects. Some geometers called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work.

The existence of non-Euclidean geometries impacted the intellectual life of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a book, Euclid and his Modern Rivals, written by Charles Lutwidge Dodgson (1832–1898) better known as Lewis Carroll, the author of Alice in Wonderland.

Planar algebras

In analytic geometry a plane is described with Cartesian coordinates : C = { (x,y) : x, y ∈ ℝ }. The points are sometimes identified with complex numbers z = x + y ε where ε² ∈ { –1, 0, 1}.

The Euclidean plane corresponds to the case ε² = −1 since the modulus of z is given by

${\displaystyle zz^{\ast }=(x+y\epsilon )(x-y\epsilon )=x^{2}+y^{2}}$

and this quantity is the square of the Euclidean distance between z and the origin. For instance, {z | z z* = 1} is the unit circle.

For planar algebra, non-Euclidean geometry arises in the other cases. When ε² = +1, then z is a split-complex number and conventionally j replaces epsilon. Then

${\displaystyle zz^{\ast }=(x+y\mathbf {j} )(x-y\mathbf {j} )=x^{2}-y^{2}\!}$

and {z | z z* = 1} is the unit hyperbola.

When ε² = 0, then z is a dual number.

This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of a complex number z.

Kinematic geometries

Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in 1908. Minkowski introduced terms like worldline and proper time into mathematical physics. He realized that the submanifold, of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions. Already in the 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions, though Macfarlane did not use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry.

The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number z = e^aj can represent a spacetime event one moment into the future of a frame of reference of rapidity a. Furthermore, multiplication by z amounts to a Lorentz boost mapping the frame with rapidity zero to that with rapidity a.

Kinematic study makes use of the dual numbers ${\displaystyle z=x+y\epsilon ,\quad \epsilon ^{2}=0,}$ to represent the classical description of motion in absolute time and space: The equations ${\displaystyle x^{\prime }=x+vt,\quad t^{\prime }=t}$ are equivalent to a shear mapping in linear algebra:

${\displaystyle {\begin{pmatrix}x'\\t'\end{pmatrix}}={\begin{pmatrix}1&v\\0&1\end{pmatrix}}{\begin{pmatrix}x\\t\end{pmatrix}}.}$

With dual numbers the mapping is ${\displaystyle t^{\prime }+x^{\prime }\epsilon =(1+v\epsilon )(t+x\epsilon )=t+(x+vt)\epsilon .}$

Another view of special relativity as a non-Euclidean geometry was advanced by E. B. Wilson and Gilbert Lewis in Proceedings of the American Academy of Arts and Sciences in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry of premises and deductions.

Fiction

Non-Euclidean geometry often makes appearances in works of science fiction and fantasy.

• In 1895, H. G. Wells published the short story The Remarkable Case of Davidson’s Eyes. To appreciate this story one should know how antipodal points on a sphere are identified in a model of the elliptic plane. In the story, in the midst of a thunderstorm, Sidney Davidson sees "Waves and a remarkably neat schooner" while working in an electrical laboratory at Harlow Technical College. At the story's close, Davidson proves to have witnessed H.M.S. Fulmar off Antipodes Island.
• Non-Euclidean geometry is sometimes connected with the influence of the 20th century horror fiction writer H. P. Lovecraft. In his works, many unnatural things follow their own unique laws of geometry: In Lovecraft's Cthulhu Mythos, the sunken city of R'lyeh is characterized by its non-Euclidean geometry. It is heavily implied this is achieved as a side effect of not following the natural laws of this universe rather than simply using an alternate geometric model, as the sheer innate wrongness of it is said to be capable of driving those who look upon it insane.
• The main character in Robert Pirsig's Zen and the Art of Motorcycle Maintenance mentioned Riemannian Geometry on multiple occasions.
• In The Brothers Karamazov, Dostoevsky discusses non-Euclidean geometry through his character Ivan.
• Christopher Priest's novel Inverted World describes the struggle of living on a planet with the form of a rotating pseudosphere.
• Robert Heinlein's The Number of the Beast utilizes non-Euclidean geometry to explain instantaneous transport through space and time and between parallel and fictional universes.
• Zeno Rogue's HyperRogue is a roguelike game set on the hyperbolic plane, allowing the player to experience many properties of this geometry. Many mechanics, quests, and locations are strongly dependent on the features of hyperbolic geometry.
• In the Renegade Legion science fiction setting for FASA's wargame, role-playing-game and fiction, faster-than-light travel and communications is possible through the use of Hsieh Ho's Polydimensional Non-Euclidean Geometry, published sometime in the middle of the 22nd century.
• In Ian Stewart's Flatterland the protagonist Victoria Line visits all kinds of non-Euclidean worlds.

Riemannian geometry

From Wikipedia, the free encyclopedia

Geometry
Projecting a sphere to a plane

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R³. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

Introduction

Bernhard Riemann

Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry.

Every smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which (in four dimensions) are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry.

There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature.

The following articles provide some useful introductory material:

• Metric tensor
• Riemannian manifold
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• Riemann curvature tensor
• List of differential geometry topics
• Glossary of Riemannian and metric geometry

Classical theorems

What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below).

The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.

General theorems

1. Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(M) where χ(M) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.
2. Nash embedding theorems. They state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rⁿ.

Geometry in large

In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances.

Pinched sectional curvature

1. Sphere theorem. If M is a simply connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a sphere.
2. Cheeger's finiteness theorem. Given constants C, D and V, there are only finitely many (up to diffeomorphism) compact n-dimensional Riemannian manifolds with sectional curvature |K| ≤ C, diameter ≤ D and volume ≥ V.
3. Gromov's almost flat manifolds. There is an ε_n > 0 such that if an n-dimensional Riemannian manifold has a metric with sectional curvature |K| ≤ ε_n and diameter ≤ 1 then its finite cover is diffeomorphic to a nil manifold.

Sectional curvature bounded below

1. Cheeger–Gromoll's soul theorem. If M is a non-compact complete non-negatively curved n-dimensional Riemannian manifold, then M contains a compact, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S (S is called the soul of M.) In particular, if M has strictly positive curvature everywhere, then it is diffeomorphic to Rⁿ. G. Perelman in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture: M is diffeomorphic to Rⁿ if it has positive curvature at only one point.
2. Gromov's Betti number theorem. There is a constant C = C(n) such that if M is a compact connected n-dimensional Riemannian manifold with positive sectional curvature then the sum of its Betti numbers is at most C.
3. Grove–Petersen's finiteness theorem. Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.

Sectional curvature bounded above

1. The Cartan–Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space Rⁿ with n = dim M via the exponential map at any point. It implies that any two points of a simply connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
2. The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic.
3. If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT(k) space. Consequently, its fundamental group Γ = π₁(M) is Gromov hyperbolic. This has many implications for the structure of the fundamental group:

• it is finitely presented;
• the word problem for Γ has a positive solution;
• the group Γ has finite virtual cohomological dimension;
• it contains only finitely many conjugacy classes of elements of finite order;
• the abelian subgroups of Γ are virtually cyclic, so that it does not contain a subgroup isomorphic to Z×Z.

Ricci curvature bounded below

1. Myers theorem. If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.
2. Bochner's formula. If a compact Riemannian n-manifold has non-negative Ricci curvature, then its first Betti number is at most n, with equality if and only if the Riemannian manifold is a flat torus.
3. Splitting theorem. If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic that minimizes distance on each interval) then it is isometric to a direct product of the real line and a complete (n-1)-dimensional Riemannian manifold that has nonnegative Ricci curvature.
4. Bishop–Gromov inequality. The volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space.
5. Gromov's compactness theorem. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric.

Negative Ricci curvature

1. The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.
2. Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature. (This is not true for surfaces.)

Positive scalar curvature

1. The n-dimensional torus does not admit a metric with positive scalar curvature.
2. If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n(n-1).