Topological geometry deals with incidence structures consisting of a point set
As in the case of topological groups, many deeper results require the point space to be (locally) compact and connected.
Linear geometries are incidence structures in which any two distinct points
The dual of a linear geometry is obtained by interchanging the roles of points and lines.
[2] Earlier, the topological properties of the real plane had been introduced via ordering relations on the affine lines, see, e.g., Hilbert,[3] Coxeter,[4] and O.
[5] The completeness of the ordering is equivalent to local compactness and implies that the affine lines are homeomorphic to
[6] 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 manifolds of dimension
Early examples not isomorphic to the classical real plane
[12] 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
In general, it holds in a projective plane if, and only if, the plane can be coordinatized by a (not necessarily commutative) field,[3][15][13] hence it implies that the group of automorphisms is transitive on the set of quadrangles (
In the present setting, a much weaker homogeneity condition characterizes
, taken with the topology of uniform convergence on the point space, is a locally compact group of dimension at most
This is possible due to the following basic theorem: Topology of compact planes.
[18] Special aspects of 4-dimensional planes are treated in,[19] more recent results can be found in.
are exactly the projective closures of the affine planes coordinatized by a so-called mutation
[20][29][30][31][32] 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.
Subplanes of projective spaces of geometrical dimension at least 3 are necessarily Desarguesian, see [35] §1 or [4] §16 or.
[36] Therefore, all compact connected projective spaces can be coordinatized by the real or complex numbers or the quaternion field.
[40] As in the case of projective planes, line pencils are compact and homotopy equivalent to a sphere of dimension
denote a maximal compact subgroup of the automorphism group of the classical
is isomorphic to the interior of the absolute sphere of the hyperbolic polarity of a classical plane; see.
is an elliptic cone without its vertex, the geometry is called a Laguerre plane.
[56] A large class of examples is given by the plane sections of an egg-like surface in real
[60][61] The classical model of a Laguerre plane consists of a circular cylindrical surface
Pairs of points which are not joined by a circle are called parallel.
-dimensional Laguerre plane is related to the geometry of complex quadratic polynomials.
Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder
However, transitivity of the automorphism group on the set of circles does not suffice to characterize the classical model among the
as point space, circles are the graphs of real fractional linear maps on