View source for Smooth projective plane

In [[geometry]], '''smooth projective planes''' are special [[projective plane]]s. The most prominent example of a smooth projective plane is the [[real projective plane]] <math>{\mathcal E}</math>. 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 <math>= C^\infty</math>). Similarly, the classical planes over the [[complex number]]s, the [[quaternion]]s, and the [[octonion]]s are smooth planes. However, these are not the only such planes.
==Definition and basic properties==
A smooth projective plane <math>{\mathcal P} = (P,\mathfrak{L})</math> consists of a point space <math>P</math> and a line space <math>\mathfrak{L}</math> that are smooth [[manifold]]s and where both geometric operations of joining and intersecting are smooth.
The geometric operations of smooth planes are continuous; hence, each smooth plane is a ''compact'' topological plane.<ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc= 42.4}}</ref> Smooth planes exist only with point spaces of dimension 2<sup>''m''</sup> where <math>1\le m\le 4</math>, because this is true for compact connected projective topological planes.<ref>{{citation|first= R.|last=Löwen|year=1983|title=Topology and dimension of stable planes: On a conjecture of H. Freudenthal|journal= J. Reine Angew. Math.|volume=343|pages= 108–122}}</ref><ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc= 54.11}}</ref> These four cases will be treated separately below.
Theorem. ''The point manifold of a smooth projective plane is homeomorphic to its classical counterpart, and so is the line manifold''.<ref>{{citation|first= L.|last= Kramer|year= 1994|title=The topology of smooth projective planes|journal= Arch. Math.|volume=63|pages= 85–91|doi=10.1007/bf01196303|s2cid= 15480568}}</ref>
==Automorphisms==
[[Automorphism]]s 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 <math>{\mathcal P}</math> form the group <math>\operatorname{Aut}{\mathcal P}</math>. This group is taken with the topology of [[uniform convergence]]. We have:<ref>{{citation|first= R.|last= Bödi|year= 1998|title=Collineations of smooth stable planes|journal= Forum Math.|volume=10|issue= 6|pages= 751–773|doi=10.1515/form.10.6.751|s2cid= 54504153|url= https://digitalcollection.zhaw.ch/handle/11475/3260|hdl= 11475/3260|hdl-access= free}}</ref>
Theorem. ''If <math>{\mathcal P} = (P,\mathfrak{L})</math> is a smooth plane, then each continuous collineation of <math>{\mathcal P}</math> is smooth''; ''in other words, the group of automorphisms of a smooth plane <math>{\mathcal P}</math> coincides with <math>\operatorname{Aut}{\mathcal P}</math>. Moreover, <math>\operatorname{Aut}{\mathcal P}</math> is a smooth Lie transformation group of <math>P</math> and of <math>\mathfrak{L}</math>''.
The automorphism groups of the four classical planes are [[simple Lie group]]s of dimension 8, 16, 35, or 78, respectively. All other smooth planes have much smaller groups. See below.
==Translation planes==
A projective plane is called a
[[translation plane]] if its automorphism group has a subgroup that fixes each point on some line <math>W</math> and [[Group action (mathematics)|acts]] [[Sharply multiply transitive|sharply transitively]] on the set of points not on <math>W</math>.
Theorem. ''Every smooth projective translation plane <math>{\mathcal P}</math> is isomorphic to one of the four classical planes''.<ref>{{citation|first= J.|last= Otte|year= 1995|title=Smooth Projective Translation Planes|journal= Geom. Dedicata|volume=58|issue= 2|pages= 203–212|doi=10.1007/bf01265639|s2cid= 120238728}}</ref>
This shows that there are many compact connected topological projective planes that are not smooth. On the other hand, the following construction yields [[Analytic function|real analytic]] [[non-Desarguesian plane]]s of dimension 2, 4, and 8, with a compact group of automorphisms of dimension 1, 4, and 13, respectively:<ref>{{citation|first= S.|last= Immervoll|year= 2003|title=Real analytic projective planes with large automorphism groups|journal= Adv. Geom.|volume=3|issue= 2|pages= 163–176|doi=10.1515/advg.2003.011}}</ref> represent points and lines in the usual way by [[homogeneous coordinates]] over the real or complex numbers or the [[quaternion]]s, say, by vectors of length <math>1</math>. Then the incidence of the point <math>(x,y,z)</math> and the line <math>(a,b,c)</math> is defined by <math>ax+by+cz=t |c|^2|z|^2cz</math>, where <math>t</math> is a fixed real parameter such that <math>|t|<1/9</math>. These planes are self-dual.
==2-dimensional planes==
Compact 2-dimensional projective planes can be described in the following way: the point space is a compact [[Surface (topology)|surface]] <math>S</math>, each line is a [[Jordan curve theorem|Jordan curve]] in <math>S</math> (a closed subset homeomorphic to the circle), and any two distinct points are joined by a unique line. Then <math>S</math> is homeomorphic to the point space of the real plane <math>{\mathcal E}</math>, any two distinct lines intersect in a unique point, and the geometric operations are continuous (apply {{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc=§31}} to the complement of a line). A familiar family of examples was given by
[[Forest Ray Moulton|Moulton]] in 1902.<ref>{{citation|first= F. R.|last= Moulton|year= 1902|title=A simple non-desarguesian plane geometry|journal= Trans. Amer. Math. Soc.|volume=3|issue= 2|pages= 192–195|doi=10.1090/s0002-9947-1902-1500595-3|doi-access= free}}</ref><ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc= §34}}</ref>  These planes are characterized by the fact that they have a 4-dimensional automorphism group. They are not isomorphic to a smooth plane.<ref>{{citation|first= D.|last= Betten|year= 1971|title=2-dimensionale differenzierbare projektive Ebenen|journal= Arch. Math.|volume=22|pages= 304–309|doi=10.1007/bf01222580|s2cid= 119885473}}</ref> More generally, all non-classical compact 2-dimensional planes <math>{\mathcal P}</math> such that <math>\dim\operatorname{Aut}{\mathcal P} \ge 3</math> are known explicitly; none of these is smooth:
Theorem. ''If <math>{\mathcal P}</math> is a smooth 2-dimensional plane and if <math>\dim\operatorname{Aut}{\mathcal P} \ge 3</math>, then <math>{\mathcal P}</math> is the classical real plane <math>{\mathcal E}</math>''.<ref>{{harvnb|Bödi|1996|loc= (9.1)}}</ref>
==4-dimensional planes==
All compact planes <math>{\mathcal P}</math> with a 4-dimensional point space and <math>\operatorname{Aut}{\mathcal P} \ge 7</math> have been classified.<ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc=74.27}}</ref> Up to duality, they are either translation planes or they are isomorphic to a unique so-called shift plane.<ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc=§74}}</ref> According to {{harvtxt|Bödi|1996|loc= Chap. 10}}, this shift plane is not smooth. Hence, the result on translation planes implies:
Theorem. ''A smooth 4-dimensional plane is isomorphic to the classical complex plane, or <math>\dim\operatorname{Aut}{\mathcal P}\le 6</math>''.<ref>{{harvnb|Bödi|1996|loc= (10.11)}}</ref>
==8-dimensional planes==
Compact 8-dimensional ''topological'' planes <math>{\mathcal P}</math> have been discussed in {{harvtxt|Salzmann|Betten|Grundhöfer|Hähl|1995|loc=Chapter 8}} and, more recently, in {{harvtxt|Salzmann|2014}}. Put <math>\Sigma = \operatorname{Aut}{\mathcal P}</math>. Either <math>{\mathcal P}</math> is the classical quaternion plane or <math>\dim\Sigma \le 18</math>. If <math>\dim\Sigma \ge 17</math>, then <math>{\mathcal P}</math> is a translation plane, or a dual translation plane, or a Hughes plane.<ref>{{harvnb|Salzmann|2014|loc=1.10}}</ref> The latter can be characterized as follows: <math>\Sigma</math> leaves some classical complex subplane <math>{\mathcal C}</math> invariant and induces on <math>{\mathcal C}</math> the connected component of its full automorphism group.<ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc=§86}}</ref><ref>{{citation|first= H.|last= Salzmann|year= 2003|title=Baer subplanes|journal= Illinois J. Math.|volume=47|issue= 1–2|pages= 485–513|doi=10.1215/ijm/1258488168|doi-access= free}} 3.19</ref> The Hughes planes are not smooth.<ref>{{citation|first= R.|last= Bödi|year= 1999|title=Smooth Hughes planes are classical|journal= Arch. Math.|volume=73|pages= 73–80|doi=10.1007/s000130050022|hdl= 11475/3229|s2cid= 120222293|hdl-access= free}}</ref><ref>{{harvnb|Salzmann|2014|loc=9.17}}</ref> This yields a result similar to the case of 4-dimensional planes:
Theorem. ''If <math>{\mathcal P}</math> is a smooth 8-dimensional plane, then <math>{\mathcal P}</math> is the classical quaternion plane or <math>\dim\Sigma \le 16</math>''.
==16-dimensional planes==
Let <math>\Sigma</math> denote the automorphism group of a compact 16-dimensional topological projective plane <math>{\mathcal P}</math>. Either <math>{\mathcal P}</math> is the smooth classical octonion plane or <math>\dim\Sigma\le 40</math>. If <math>\dim\Sigma = 40</math>, then <math>\Sigma</math> fixes a line <math>W</math> and a point <math>v\in W</math>, and the affine plane <math>{\mathcal P}\smallsetminus W</math> and its dual are translation planes.<ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc= 87.7}}</ref> If <math>\dim\Sigma = 39</math>, then <math>\Sigma</math> also fixes an incident point-line pair, but neither <math>{\mathcal P}</math> nor <math>\Sigma</math> are known explicitly. Nevertheless, none of these planes can be smooth:<ref>{{harvnb|Bödi|1996|loc= Chap. 12}}</ref><ref>{{citation|first= R.|last= Bödi|year=1998|title=16-dimensional smooth projective planes with large collineation groups|journal= Geom. Dedicata|volume=72|issue= 3|pages= 283–298|doi= 10.1023/A:1005020223604|s2cid= 56094550|url= https://digitalcollection.zhaw.ch/handle/11475/3238|hdl= 11475/3238|hdl-access= free}}</ref><ref>{{harvnb|Salzmann|2014|loc=9.18 for a sketch of the proof}}</ref>
Theorem. ''If <math>{\mathcal P}</math> is a 16-dimensional smooth projective plane, then <math>{\mathcal P}</math> is the classical octonion plane or <math>\dim\Sigma \le 38</math>''.
==Main theorem==
The last four results combine to give the following theorem:
If <math>c_m</math> is the largest value of <math>\dim\operatorname{Aut}{\mathcal P}</math>, where <math>{\mathcal P}</math> is a non-classical compact 2<sup>''m''</sup>-dimensional ''topological'' projective plane, then <math>\dim\operatorname{Aut}{\mathcal P} \le c_m-2</math> whenever <math>{\mathcal P}</math> is even smooth.
==Complex analytic planes==
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.<ref>{{citation|first=S.|last=Breitsprecher|year=1967|title=Einzigkeit der reellen und der komplexen projektiven Ebene|journal= Math. Z.|volume=99|issue=5|pages= 429–432|doi=10.1007/bf01111021|s2cid=120984088}}</ref><ref>{{harvnb|Salzmann|Betten|Grundhöfer|Hähl|1995|loc= 75.1}}</ref>
Theorem. ''Every complex analytic projective plane is isomorphic as an analytic plane to the complex plane with its standard analytic structure''.
==Notes==
{{reflist|30em}}
==References==
* {{citation|first=  R.|last= Bödi|title=Smooth stable and projective planes|journal=Thesis, Tübingen|year= 1996|url=http://www.mathematik.uni-tuebingen.de/ab/Geometrie.alt/Smoothstableplanes.ps}}
* {{citation|first1=H.|last1=Salzmann|first2= D.|last2= Betten|first3=T.|last3= Grundhöfer|first4=H.|last4= Hähl|first5= R.|last5= Löwen|first6= M.|last6=Stroppel|year= 1995|title=Compact Projective Planes|publisher= W. de Gruyter}}
* {{citation|first=H.|last= Salzmann|year= 2014|title=Compact planes, mostly 8-dimensional. A retrospect|arxiv=1402.0304|bibcode=2014arXiv1402.0304S}}
[[Category:Projective geometry]]
[[Category:Surfaces]]