Riemann geometry

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elliptic geometry

One of the non-Euclidean geometries, i.e. a geometrical theory based on axioms whose requirements are different from the requirements of the axioms of Euclidean geometry. Unlike Euclidean geometry, elliptic geometry has one of the two possible negations of the axiom of parallelism in Euclidean geometry: In the plane, through a point that is not incident to a given straight line there is no straight line that does not intersect the given line; the other negation of the Euclidean axiom on parallelism occurs in Lobachevskii geometry: In the plane, through a given point that is not incident to a given straight line there are at least two straight lines that do not intersect the given line. From now on "line" is understood to correspond to the concept "straight line" .

A system of axioms of three-dimensional elliptic geometry can be constructed using the same concepts as in the Hilbert system of axioms of Euclidean geometry: the basic concepts are "point" , "line" , "plane" . "Line" and "plane" are regarded as certain classes of "points" , and "space" is taken to be the set of all objects: "points" , "lines" and "planes" .

The system of axioms is composed of four groups:

Group I. Axioms of incidence.

This group contains all the axioms comprising group 1 of Hilbert's system, plus one further axiom: Any two distinct lines in a plane have one and only one common point.

Group II. Axioms of order or of position of points on a line.

The axioms of this group describe the concept of "separation of two pairs of points on a line" , with the aid of which the order of points on a line is determined.

$ \textrm{ II } _ {1} $. Given three distinct points $ A, B, C $ of an arbitrary line, there is on this line a point $ D $ such that the pair $ A, B $ separates the pair $ C, D $( designated by $ AB \div CD $). If $ AB \div CD $, then all four points $ A $, $ B $, $ C $, and $ D $ are distinct.

$ \textrm{ II } _ {2} $. If $ AB \div CD $, then $ BA \div CD $ and $ CD \div AB $.

$ \textrm{ II } _ {3} $. Given four distinct points $ A, B, C, D $ on a line, two separated pairs can always be constructed from them.

$ \textrm{ II } _ {4} $. Let the points $ A $, $ B $, $ C $, $ D $, and $ E $ lie on a line; if $ CD \div AB $ and $ CE \div AB $, then the pair $ DE $ does not separate the pair $ AB $.

$ \textrm{ II } _ {5} $. If the pairs $ CD $ and $ CE $ do not separate the pair $ AB $, then the pair $ DE $ also does not separate the pair $ AB $( see $ \textrm{ II } _ {4} $).

$ \textrm{ II } _ {6} $. If four distinct lines of a certain pencil are intersected by two distinct lines at points $ A, B, C, D $ and $ A _ {1} , B _ {1} , C _ {1} , D _ {1} $, respectively, then $ AB \div CD $ implies $ A _ {1} B _ {1} \div C _ {1} D _ {1} $.

Group III. Axioms of congruence.

These describe the relationship "congruence" of segments, angles, etc. A segment is taken to mean the set of points of a line determined by a pair of distinct points $ A, B $ of this line in the following way. According to the axioms of group II, there is on the line a pair of points $ M, N $ such that $ AB \div MN $; the set of points $ X $ that satisfy the relation $ AB \div MX $ form the class of interior points of the segment determined by the points $ A $ and $ B $; this is written as $ [ AB] _ {M} $. The points of the line exterior to $ [ AB] _ {M} $ form the mutually complementary segment $ [ AB] _ {N} $, the points $ A $ and $ B $ being called the ends of the segments $ [ AB] _ {M} $ and $ [ AB] _ {N} $.

$ \textrm{ III } _ {1} $. Each segment is congruent to itself.

$ \textrm{ III } _ {2} $. If the first segment is congruent to a second, then the second segment is congruent to the first.

$ \textrm{ III } _ {3} $. If the first segment is congruent to a second, and the second is congruent to a third, then the first is congruent to the third.

$ \textrm{ III } _ {4} $. If two segments are congruent, then their mutually complementary segments are also congruent.

$ \textrm{ III } _ {5} $. A segment is not congruent to a part of it. Congruent mutually complementary segments of a line are called half-lines; the ends of such segments are called orthogonal points of the line.

$ \textrm{ III } _ {6} $. Each point on a line has an orthogonal point.

$ \textrm{ III } _ {7} $. All half-lines are congruent to each other.

$ \textrm{ III } _ {8} $. If a segment $ [ AB] $ is congruent to a segment $ [ A _ {1} B _ {1} ] $ and a point $ C $ is an interior point of the first segment, then inside the second segment there is a point $ C _ {1} $ such that the segment $ [ C _ {1} B _ {1} ] $ is congruent to the segment $ [ CB] $.

On the sides of the angle formed by two lines there are orthogonal points relative to the vertex of the angle; the segment joining these two points and located inside the given angle is known as the segment associated with (measuring) the angle. Two angles are called congruent if the segments associated with them are congruent.

$ \textrm{ III } _ {9} $. If in two triangles $ ABC $ and $ A _ {1} B _ {1} C _ {1} $ the side $ AB $ is congruent to the side $ A _ {1} B _ {1} $ and the side $ AC $ is congruent to the side $ A _ {1} C _ {1} $, then the angle $ A $ is congruent to the angle $ A _ {1} $ if and only if the sides $ BC $ and $ B _ {1} C _ {1} $ are congruent.

Group IV. Axiom of continuity.

Let the interior points of a segment $ [ AB] _ {M} $ be divided into two classes such that: 1) each point of the segment falls into one of these classes; 2) each class is not empty; and 3) if a point $ X $ belongs to the first class and a point $ Y $ to the second, then $ X $ is always an interior point of the segment $ [ AY] _ {M} $. Thus, in the segment $ [ AB] _ {M} $ there is a point $ C $ such that each interior point of $ [ AC] _ {M} $ belongs to the first class and each interior point of $ [ CB] _ {M} $ belongs to the second.

There are other systems of axioms for elliptic geometry, at the basis of which there are other basic concepts and relations (see, for example, [3], [5]).

The metrical properties of "local" elliptic geometry coincide with those of a certain hypersphere in the corresponding Euclidean space. For example, for any point in an elliptic plane there is a part of the plane containing this point which is isometric to a part of the sphere in three-dimensional Euclidean space; the radius $ r $ of this sphere is the same for all planes of a given elliptic space, and is called the radius of curvature of this space. The metrical properties of a three-dimensional elliptic space "locally" coincide with those of a hypersphere in four-dimensional Euclidean space, etc. The number $ k = 1/r ^ {2} $ is called the curvature of the elliptic space. For most purposes it is natural to assume $ r= 1 $.

The basic facts of elliptic geometry for a (straight) line, a plane and a three-dimensional space are given below.

The elliptic line is a closed finite line $ El ^ {1} $. A circle of unit radius with diametrically-opposite points identified in the Euclidean plane $ E ^ {1} $ can serve as a model of this line. Two distinct points of the line divide it into two parts. The mutual arrangement of points on the line is determined using the concept of "separation of two pairs of points" . The distance between two points of the line is defined in two ways: the shorter does not exceed $ \pi /2 $, the longer exceeds $ \pi /2 $. The length of the entire line is $ \pi $. Two points which are $ \pi /2 $ apart are called orthogonal; each point of the line has a corresponding unique point orthogonal to it.

The elliptic plane is a closed finite one-sided surface $ El ^ {2} $ homeomorphic to a Möbius strip whose boundary is glued to a circle (i.e. the elliptic plane is homeomorphic to the real projective plane). A sphere of unit radius with antipoles identified in three-dimensional Euclidean space can serve as a model for the plane of elliptic geometry with curvature $ 1 $. A line does not divide the plane into two regions.

Any two lines in the plane have a common perpendicular of length $ \alpha $, where $ \alpha $ is the angle between the lines. Two distinct lines divide the plane into two regions, known as angles. A trilateral arrangement (i.e. three lines having three intersections) divides the entire plane into four regions, known as triangles.

The metrical relationships in a triangle in the plane $ El ^ {2} $ are expressed by the corresponding relationships of spherical trigonometry on a sphere of unit radius in the Euclidean space $ E ^ {3} $. Generally speaking, the trigonometric formulas in $ El ^ {2} $ of elliptic geometry are the same as the formulas of spherical trigonometry.

The set of points of a plane located at a distance $ \pi /2 $ from a given point (the pole) is a line, the polar of the pole. Any line is uniquely determined by its pole, and conversely determines its pole. Poles of lines which pass through a given point are situated on the polar of this point, and polars of points lying on a line intersect at the pole of this line. Mutually polar triangles have as vertices the poles of corresponding sides. For two mutually polar triangles Chasles' theorem holds: The three straight lines joining the corresponding vertices of these triangles intersect at one point. If the vertices of the triangle are the poles of its sides, then the triangle is called a self-polar triangle. The sum of the angles of a triangle is greater than $ \pi $; its area $ \Delta $ is equal to the angular excess $ \widehat{A} + \widehat{B} + \widehat{C} - \pi $.

A circle in the elliptic plane is the set of points located at the same distance from a certain point (its centre). The radius $ R $ of a circle may be taken to be $ \leq \pi / 2 $. A circle is equidistant from a certain line (the axis of the circle). When $ R = \pi /2 $, the circle is a line — the polar of its centre. The length of a circle of radius $ R $ is $ 2 \pi \sin R $ and the area of the disc is $ 4 \pi \sin ^ {2} ( R /2) $.

There are four and only four circles which pass through three given points that do not lie on one line. Two distinct circles can intersect at not more than four distinct points. The total area of the elliptic plane is $ 2 \pi $.

The duality principle is applicable in the elliptic geometry of a plane: In every true statement the terms "point" and "line" can be interchanged, and the result is a true statement.

Three-dimensional elliptic space is a closed finite two-sided (orientable) space $ El ^ {3} $. A hypersphere of unit radius with diametrically-opposite points identified in the Euclidean space $ E ^ {4} $ can serve as a model for the space $ El ^ {3} $. The volume of the elliptic space is $ \pi ^ {2} $.

A plane does not divide the space into two regions. Two distinct planes in $ El ^ {3} $ intersect along a line. A line which does not lie in a plane intersects it at one point.

The set of points of the space at distance $ \pi /2 $ from a given point (the pole) is a plane — the polar of the pole. Any plane is uniquely determined by its pole, and conversely determines its pole. If three planes pass through one line, then their poles lie on one line, and, conversely, if the poles of three planes lie on one line, these planes intersect along one line. For each line there is a another line such that the poles of the planes which pass through the first line lie on the second, while the polar planes of the points which lie on the first line pass through the second line. These skew lines are called mutually polar. Two skew lines are called oblique if they are not mutually polar or if each of them does not intersect the polar of the other. Two oblique lines have two common perpendiculars which are mutually polar. If the two common perpendiculars of oblique lines are of different lengths, then the lengths of the common perpendiculars give the least and greatest distances of one line from the other. Two oblique lines with an infinite set of common perpendiculars of the same length are called Clifford parallels (equidistant or paratactic lines). Through each point of the space not belonging to either of two given polar lines, two Clifford parallels to the given lines can be drawn. The set of points at the same distance less than $ \pi /2 $ from a given line is called a Clifford surface. The given line is called the axis and the distance $ \rho $ of the points from the axis is the radius of the Clifford surface. This surface has two mutually polar axes and two corresponding radii which complement each other up to $ \pi /2 $. A Clifford surface is a surface of revolution in two distinct ways. Through each point of a Clifford surface two lines equidistant from its axes and entirely belonging to the surface can be drawn. These lines are known as rectilinear generators of the Clifford surface. Any three lines equidistant from each other determine a Clifford surface for which they are generators. Each pair of generators of different families intersect at a constant angle. A Clifford surface is isometric to a Euclidean rhombus with an acute angle equal to the angle between the generators of the different families, with side-length $ \pi $ and with opposite sides identified. In other words, a Clifford surface is locally Euclidean. The area of the Clifford surface with radius $ \rho $ is $ \pi ^ {2} \sin 2 \rho $.

A sphere in $ El ^ {3} $ is the set of points located at the same distance from a given point (its centre). A sphere is equidistant from a plane (the axial plane of the sphere). The area of a sphere of radius $ R $ is $ 4 \pi \sin ^ {2} R $.

The duality principle is applicable in the elliptic geometry of space: The terms "point" and "plane" can be interchanged in every true statement, and a true statement is obtained.

The concept of elliptic geometry was apparently introduced by B. Riemann in his lecture [1] (1854, published in 1867). In it, elliptic geometry was examined as a special case of Riemannian geometry.


[1] B. Riemann, "Über die Hypothesen, welche der Geometrie zugrunde liegen" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)
[2] N.V. Efimov, "Higher geometry" , MIR (1980) (Translated from Russian)
[3] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[4] V.F. Kagan, "Foundations of geometry" , 2 , Moscow-Leningrad (1956) (In Russian)
[5] S.A. Bogomolov, "An introduction to Riemann's non-Euclidean geometry" , Leningrad-Moscow (1934) (In Russian)


At the end of his celebrated habilitation address (1854, published in 1867), Riemann turned to spaces with the same curvature at each point and in each two-dimensional direction. For figures in such spaces he asserted the same free mobility as governs Euclidean space, although he did not mention explicitly their relation to non-Euclidean geometry. This was done in 1868 by H. Helmholtz, who, starting from the postulate of free mobility of figures in space, also arrived at the spaces of constant curvature — vanishing curvature for Euclidean space, positive for spheres, and negative for absolute geometry. This led to an extension of the term non-Euclidean geometry to spaces of positive constant curvature (Riemannian geometries), where great circles should play the role of straight lines, and great two-dimensional spheres that of planes. The disadvantage that two distinct lines in the same plane then intersect in two points (rather than in one) can be overcome by identifying antipodal points ( "elliptic geometry" ) with each other. The topology of such a space is the same as that of projective space of the same dimension. The metric relations are those of spherical geometry; spherical trigonometry applies in a natural way, but since straight lines are topological circles, one has to be as careful about distances as one is in Euclidean geometry about angles. Elliptic $ 3 $- space is most easily studied by means of quaternions.

It was F. Klein who interpreted Cayley's metric within a conic in the projective plane as a model of absolute geometry, which he generalized in order to include non-Euclidean geometries of arbitrary dimension. Depending on the type of the basic hyperquadric in projective $ n $- space,

$$ x _ {1} ^ {2} + x _ {2} ^ {2} + \dots + x _ {n+} 1 ^ {2} = 0 $$


$$ x _ {1} ^ {2} + x _ {2} ^ {2} + \dots - x _ {n+} 1 ^ {2} = 0, $$

he spoke of elliptic and hyperbolic geometries, which eventually became the usual terminology.

Many axiomatic approaches to elliptic geometry have been proposed. If one prefers to keep close to Hilbert's axiomatics of Euclidean geometry, one has to replace Hilbert's axioms on linear order by axioms on cyclic order: 1) On each line there are two (mutually opposite) cyclic orders distinguished; and 2) projections within a plane map distinguished orders on each other.

(Cyclic order is defined as follows. For a finite set a cyclic order is an equivalence class of linear orders, defined by the relations (for all $ i $)

$$ a _ {1} < \dots < a _ {i} < a _ {i+} 1 < \dots < a _ {n\ } \approx $$

$$ \approx \ a _ {i+} 1 < \dots < a _ {n} < a _ {1} < \dots < a _ {i} . $$

An arbitrary set is cyclically ordered by providing every finite subset with a cyclic order, such that for $ V \subset W $ the cyclic order on $ W $ extends that on $ V $. Replacing $ < $ by $ > $ changes an order into its opposite.)

For a modern treatment of elliptic geometry see [a3].


[a1] H. Helmholtz, "Über die Tatsachen, die der Geometrie zum Grunde liegen" , Wissenschaftliche Abhandlungen , II (1883) pp. 618–639
[a2] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965)
[a3] F. Bachmann, "Aufbau der Geometrie aus dem Spiegelungbegriff" , Springer (1959)
[a4] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1989)
[a5] J. Gray, "Ideas of space" , Oxford (1989) pp. Chapt. 14
[a6] H.P. Manning, "Introductory non-Euclidean geometry" , New York (1963) pp. Chapt. III
[a7] O. Veblen, J.W. Young, "Projective geometry" , II , Blaisdell (1946) pp. Chapt. VII
[a8] M. Berger, "Geometry" , II , Springer (1987) pp. Chapt. 19
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Riemann geometry. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article