Topological geometry deals with incidence structures consisting of a point set
and a family
of subsets of
called lines or circles etc. such that both
and
carry a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s, many deeper results require the point space to be (locally) compact and connected. This generalizes the observation that the line joining two distinct points in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
depends continuously on the pair of points and the intersection point of two lines is a continuous function of these lines.
Linear geometries
Linear geometries are
incidence structures in which any two distinct points
and
are joined by a unique line
. Such geometries are called ''topological'' if
depends continuously on the pair
with respect to given topologies on the point set and the line set. The ''dual'' of a linear geometry is obtained by interchanging the roles of points and lines. A survey of linear topological geometries is given in Chapter 23 of the ''Handbook of incidence geometry''. The most extensively investigated topological linear geometries are those which are also dual topological linear geometries. Such geometries are known as topological
projective planes.
History
A systematic study of these planes began in 1954 with a paper by Skornyakov. Earlier, the topological properties of the
real plane
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
had been introduced via
ordering relations on the affine lines, see, e.g.,
Hilbert,
Coxeter,
and O. Wyler. The completeness of the ordering is equivalent to
local compactness and implies that the affine lines are
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to
and that the point space is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. Note that the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s do not suffice to describe our intuitive notions of plane geometry and that some extension of the rational field is necessary. In fact, the equation
for a circle has no rational solution.
Topological projective planes
The approach to the topological properties of projective planes via ordering relations is not possible, however, for the planes coordinatized by the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s or the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
algebra. The point spaces as well as the line spaces of these ''classical'' planes (over the real numbers, the complex numbers, the quaternions, and the octonions) are compact
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s of dimension
.
Topological dimension
The notion of the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of a topological space plays a prominent rôle in the study of topological, in particular of compact connected planes. For a
normal space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
, the dimension
can be characterized as follows:
If
denotes the
-sphere, then ''
if, and only if, for every closed subspace
each continuous map
has a continuous extension
''.
For details and other definitions of a dimension see and the references given there, in particular Engelking or Fedorchuk.
2-dimensional planes
The lines of a compact topological plane with a 2-dimensional point space form a family of curves homeomorphic to a circle, and this fact characterizes these planes among the topological projective planes.
Equivalently, the point space is a
surface. Early examples not isomorphic to the classical real plane
have been given by Hilbert
and
Moulton. The continuity properties of these examples have not been considered explicitly at that time, they may have been taken for granted. Hilbert’s construction can be modified to obtain uncountably many pairwise non-isomorphic
-dimensional compact planes. The traditional way to distinguish
from the other
-dimensional planes is by the validity of
Desargues’s theorem or the
theorem of Pappos (see, e.g., Pickert
for a discussion of these two configuration theorems). The latter is known to imply the former (
Hessenberg). The theorem of Desargues expresses a kind of homogeneity of the plane. In general, it holds in a projective plane if, and only if, the plane can be coordinatized by a (not necessarily commutative) field,
hence it implies that the group of
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s is
transitive on the set of quadrangles (
points no
of which are collinear). In the present setting, a much weaker homogeneity condition characterizes
:
Theorem. ''If the automorphism group
of a
-dimensional compact plane
is transitive on the point set
(or the line set), then has a compact subgroup which is even transitive on the set of flags'' (=incident point-line pairs), ''and is classical''.
The automorphism group of a -dimensional compact plane , taken with the topology of uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
on the point space, is a locally compact group of dimension at most , in fact even a Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
. All -dimensional planes such that can be described explicitly; those with are exactly the Moulton planes, the classical plane is the only -dimensional plane with ; see also.
Compact connected planes
The results on -dimensional planes have been extended to compact planes of dimension . This is possible due to the following basic theorem:
Topology of compact planes. ''If the dimension of the point space of a compact connected projective plane is finite, then with . Moreover, each line is a homotopy sphere of dimension '', see or.
Special aspects of 4-dimensional planes are treated in, more recent results can be found in. The lines of a -dimensional compact plane are homeomorphic to the -sphere; in the cases the lines are not known to be manifolds, but in all examples which have been found so far the lines are spheres. A subplane of a projective plane is said to be a Baer subplane, if each point of is incident with a line of and each line of contains a point of . A closed subplane is a Baer subplane of a compact connected plane if, and only if, the point space of and a line of have the same dimension. Hence the lines of an 8-dimensional plane are homeomorphic to a sphere if has a closed Baer subplane.
Homogeneous planes. ''If is a compact connected projective plane and if is transitive on the point set of , then has a flag-transitive compact subgroup and is classical'', see or. In fact, is an elliptic motion group.
Let be a compact plane of dimension , and write . If , then is classical, and is a simple Lie group of dimension respectively. All planes with are known explicitly. The planes with are exactly the projective closures of the affine planes coordinatized by a so-called ''mutation'' of the octonion algebra , where the new multiplication is defined as follows: choose a real number with and put . Vast families of planes with a group of large dimension have been discovered systematically starting from assumptions about their automorphism groups, see, e.g.,. Many of them are projective closures of translation planes (affine planes admitting a sharply transitive group of automorphisms mapping each line to a parallel), cf.; see also for more recent results in the case and for .
Compact projective spaces
Subplanes of projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s of ''geometrical'' dimension at least 3 are necessarily Desarguesian, see §1 or §16 or. Therefore, all compact connected projective spaces can be coordinatized by the real or complex numbers or the quaternion field.
Stable planes
The classical non-euclidean hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
can be represented by the intersections of the straight lines in the real plane with an open circular disk. More generally, open (convex) parts of the classical affine planes are typical stable planes. A survey of these geometries can be found in, for the -dimensional case see also.
Precisely, a ''stable plane'' is a topological linear geometry such that
# is a locally compact space of positive finite dimension,
# each line is a closed subset of , and is a Hausdorff space,
# the set is an open subspace ( ''stability''),
# the map is continuous.
Note that stability excludes geometries like the -dimensional affine space over or .
A stable plane is a projective plane if, and only if, is compact.
As in the case of projective planes, line pencils are compact and homotopy equivalent to a sphere of dimension , and with , see or. Moreover, the point space is locally contractible.
Compact groups of (proper) stable planes
are rather small. Let denote a maximal compact subgroup of the automorphism group of the classical -dimensional projective plane . Then the following theorem holds:
''If a -dimensional stable plane admits a compact group of automorphisms such that , then '', see.
Flag-homogeneous stable planes. ''Let be a stable plane. If the automorphism group is flag-transitive, then is a classical projective or affine plane, or is isomorphic to the interior of the absolute sphere of the hyperbolic polarity of a classical plane''; see.
In contrast to the projective case, there is an abundance of point-homogeneous stable planes, among them vast classes of translation planes, see and.
Symmetric planes
Affine translation planes have the following property:
* There exists a point transitive closed subgroup of the automorphism group which contains a unique reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
at some and hence at each point.
More generally, a ''symmetric plane'' is a stable plane satisfying the aforementioned condition; see, cf. for a survey of these geometries. By Corollary 5.5, the group is a Lie group and the point space is a manifold. It follows that is a symmetric space. By means of the Lie theory of symmetric spaces, all symmetric planes with a point set of dimension or have been classified. ''They are either translation planes or they are determined by a Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
''. An easy example is the real hyperbolic plane.
Circle geometries
Classical models are given by the plane sections of a quadratic surface in real projective -space; if is a sphere, the geometry is called a Möbius plane In mathematics, a Möbius plane (named after August Ferdinand Möbius) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real Affine plane (incide ...
. The plane sections of a ruled surface (one-sheeted hyperboloid) yield the classical Minkowski plane, cf. for generalizations. If is an elliptic cone without its vertex, the geometry is called a Laguerre plane In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre.
The classical Laguerre ...
. Collectively these planes are sometimes referred to as Benz plane
In mathematics, a Benz plane is a type of 2-dimensional geometrical structure, named after the German mathematician Walter Benz. The term was applied to a group of objects that arise from a common axiomatization of certain structures and split int ...
s. ''A topological Benz plane is classical, if each point has a neighbourhood which is isomorphic to some open piece of the corresponding classical Benz plane''.
Möbius planes
Möbius planes consist of a family of circles, which are topological 1-spheres, on the -sphere such that for each point the ''derived'' structure is a topological affine plane. In particular, any distinct points are joined by a unique circle. The circle space is then homeomorphic to real projective -space with one point deleted. A large class of examples is given by the plane sections of an egg-like surface in real -space.
Homogeneous Möbius planes
''If the automorphism group of a Möbius plane is transitive on the point set or on the set of circles, or if , then is classical and '', see.
In contrast to compact projective planes there are no topological Möbius planes with circles of dimension , in particular no compact Möbius planes with a -dimensional point space. All 2-dimensional Möbius planes such that can be described explicitly.
Laguerre planes
The classical model of a Laguerre plane consists of a circular cylindrical surface in real -space as point set and the compact plane sections of as circles. Pairs of points which are not joined by a circle are called ''parallel''. Let denote a class of parallel points. Then is a plane , the circles can be represented in this plane by parabolas of the form .
In an analogous way, the classical -dimensional Laguerre plane is related to the geometry of complex quadratic polynomials. In general, the axioms of a locally compact connected Laguerre plane require that the derived planes embed into compact projective planes of finite dimension. A circle not passing through the point of derivation induces an oval
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one ...
in the derived projective plane. By or, circles are homeomorphic to spheres of dimension or . Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder or it is a -dimensional manifold, cf. A large class of -dimensional examples, called ovoidal Laguerre planes, is given by the plane sections of a cylinder in real 3-space whose base is an oval in .
The automorphism group of a -dimensional Laguerre plane () is a Lie group with respect to the topology of uniform convergence on compact subsets of the point space; furthermore, this group has dimension at most . All automorphisms of a Laguerre plane which fix each parallel class form a normal subgroup, the ''kernel'' of the full automorphism group. The -dimensional Laguerre planes with are exactly the ovoidal planes over proper skew parabolae. The classical -dimensional Laguerre planes are the only ones such that , see, cf. also.
Homogeneous Laguerre planes
''If the automorphism group of a -dimensional Laguerre plane is transitive on the set of parallel classes, and if the kernel is transitive on the set of circles, then is classical'', see 2.1,2.
However, transitivity of the automorphism group on the set of circles does not suffice to characterize the classical model among the -dimensional Laguerre planes.
Minkowski planes
The classical model of a Minkowski plane has the torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
as point space, circles are the graphs of real fractional linear maps on . As with Laguerre planes, the point space of a locally compact connected Minkowski plane is - or -dimensional; the point space is then homeomorphic to a torus or to , see.
Homogeneous Minkowski planes
''If the automorphism group of a Minkowski plane of dimension is flag-transitive, then is classical''.
The automorphism group of a -dimensional Minkowski plane is a Lie group of dimension at most . All -dimensional Minkowski planes such that can be described explicitly. The classical -dimensional Minkowski plane is the only one with , see.
Notes
References
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* {{citation, first=G., last=Steinke, year=1995, title=Topological circle geometries, journal=Handbook of Incidence Geometry, pages=1325–1354, location=Amsterdam, publisher=North-Holland, doi=10.1016/B978-044488355-1/50026-8, isbn=9780444883551
Topology
Incidence geometry