Its geometric operations of joining two distinct points by a line and of intersecting two lines in a point are not only continuous but even smooth (infinitely differentiable
[1] Smooth planes exist only with point spaces of dimension 2m where
, because this is true for compact connected projective topological planes.
The point manifold of a smooth projective plane is homeomorphic to its classical counterpart, and so is the line manifold.
[4] Automorphisms play a crucial role in the study of smooth planes.
A bijection of the point set of a projective plane is called a collineation, if it maps lines onto lines.
The continuous collineations of a compact projective plane
is a smooth plane, then each continuous collineation of
All other smooth planes have much smaller groups.
A projective plane is called a translation plane if its automorphism group has a subgroup that fixes each point on some line
and acts sharply transitively on the set of points not on
Every smooth projective translation plane
[6] This shows that there are many compact connected topological projective planes that are not smooth.
On the other hand, the following construction yields real analytic non-Desarguesian planes of dimension 2, 4, and 8, with a compact group of automorphisms of dimension 1, 4, and 13, respectively:[7] represent points and lines in the usual way by homogeneous coordinates over the real or complex numbers or the quaternions, say, by vectors of length
(a closed subset homeomorphic to the circle), and any two distinct points are joined by a unique line.
is homeomorphic to the point space of the real plane
, any two distinct lines intersect in a unique point, and the geometric operations are continuous (apply Salzmann et al. 1995, §31 to the complement of a line).
A familiar family of examples was given by Moulton in 1902.
[8][9] These planes are characterized by the fact that they have a 4-dimensional automorphism group.
[10] More generally, all non-classical compact 2-dimensional planes
Hence, the result on translation planes implies: Theorem.
the connected component of its full automorphism group.
[18][19] This yields a result similar to the case of 4-dimensional planes: Theorem.
denote the automorphism group of a compact 16-dimensional topological projective plane
is the smooth classical octonion plane or
also fixes an incident point-line pair, but neither
The last four results combine to give the following theorem: If
is a non-classical compact 2m-dimensional topological projective plane, then
The condition, that the geometric operations of a projective plane are complex analytic, is very restrictive.
In fact, it is satisfied only in the classical complex plane.