Smooth projective plane

In geometry, smooth projective planes are special projective planes. The most prominent example of a smooth projective plane is the real projective plane . 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 ). Similarly, the classical planes over the complex numbers, the quaternions, and the octonions are smooth planes. However, these are not the only such planes.

Definition and basic properties

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A smooth projective plane   consists of a point space   and a line space   that are smooth manifolds 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.[1] Smooth planes exist only with point spaces of dimension 2m where  , because this is true for compact connected projective topological planes.[2][3] 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.[4]

Automorphisms

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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   form the group  . This group is taken with the topology of uniform convergence. We have:[5]

Theorem. If   is a smooth plane, then each continuous collineation of   is smooth; in other words, the group of automorphisms of a smooth plane   coincides with  . Moreover,   is a smooth Lie transformation group of   and of  .

The automorphism groups of the four classical planes are simple Lie groups of dimension 8, 16, 35, or 78, respectively. All other smooth planes have much smaller groups. See below.

Translation planes

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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  .

Theorem. Every smooth projective translation plane   is isomorphic to one of the four classical planes.[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  . Then the incidence of the point   and the line   is defined by  , where   is a fixed real parameter such that  . These planes are self-dual.

2-dimensional planes

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Compact 2-dimensional projective planes can be described in the following way: the point space is a compact surface  , each line is a Jordan curve in   (a closed subset homeomorphic to the circle), and any two distinct points are joined by a unique line. Then   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. They are not isomorphic to a smooth plane.[10] More generally, all non-classical compact 2-dimensional planes   such that   are known explicitly; none of these is smooth:

Theorem. If   is a smooth 2-dimensional plane and if  , then   is the classical real plane  .[11]

4-dimensional planes

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All compact planes   with a 4-dimensional point space and   have been classified.[12] Up to duality, they are either translation planes or they are isomorphic to a unique so-called shift plane.[13] According to Bödi (1996, 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  .[14]

8-dimensional planes

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Compact 8-dimensional topological planes   have been discussed in Salzmann et al. (1995, Chapter 8) and, more recently, in Salzmann (2014). Put  . Either   is the classical quaternion plane or  . If  , then   is a translation plane, or a dual translation plane, or a Hughes plane.[15] The latter can be characterized as follows:   leaves some classical complex subplane   invariant and induces on   the connected component of its full automorphism group.[16][17] The Hughes planes are not smooth.[18][19] This yields a result similar to the case of 4-dimensional planes:

Theorem. If   is a smooth 8-dimensional plane, then   is the classical quaternion plane or  .

16-dimensional planes

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Let   denote the automorphism group of a compact 16-dimensional topological projective plane  . Either   is the smooth classical octonion plane or  . If  , then   fixes a line   and a point  , and the affine plane   and its dual are translation planes.[20] If  , then   also fixes an incident point-line pair, but neither   nor   are known explicitly. Nevertheless, none of these planes can be smooth:[21][22][23]

Theorem. If   is a 16-dimensional smooth projective plane, then   is the classical octonion plane or  .

Main theorem

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The last four results combine to give the following theorem:

If   is the largest value of  , where   is a non-classical compact 2m-dimensional topological projective plane, then   whenever   is even smooth.

Complex analytic planes

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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.[24][25]

Theorem. Every complex analytic projective plane is isomorphic as an analytic plane to the complex plane with its standard analytic structure.

Notes

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  1. ^ Salzmann et al. 1995, 42.4
  2. ^ Löwen, R. (1983), "Topology and dimension of stable planes: On a conjecture of H. Freudenthal", J. Reine Angew. Math., 343: 108–122
  3. ^ Salzmann et al. 1995, 54.11
  4. ^ Kramer, L. (1994), "The topology of smooth projective planes", Arch. Math., 63: 85–91, doi:10.1007/bf01196303, S2CID 15480568
  5. ^ Bödi, R. (1998), "Collineations of smooth stable planes", Forum Math., 10 (6): 751–773, doi:10.1515/form.10.6.751, hdl:11475/3260, S2CID 54504153
  6. ^ Otte, J. (1995), "Smooth Projective Translation Planes", Geom. Dedicata, 58 (2): 203–212, doi:10.1007/bf01265639, S2CID 120238728
  7. ^ Immervoll, S. (2003), "Real analytic projective planes with large automorphism groups", Adv. Geom., 3 (2): 163–176, doi:10.1515/advg.2003.011
  8. ^ Moulton, F. R. (1902), "A simple non-desarguesian plane geometry", Trans. Amer. Math. Soc., 3 (2): 192–195, doi:10.1090/s0002-9947-1902-1500595-3
  9. ^ Salzmann et al. 1995, §34
  10. ^ Betten, D. (1971), "2-dimensionale differenzierbare projektive Ebenen", Arch. Math., 22: 304–309, doi:10.1007/bf01222580, S2CID 119885473
  11. ^ Bödi 1996, (9.1)
  12. ^ Salzmann et al. 1995, 74.27
  13. ^ Salzmann et al. 1995, §74
  14. ^ Bödi 1996, (10.11)
  15. ^ Salzmann 2014, 1.10
  16. ^ Salzmann et al. 1995, §86
  17. ^ Salzmann, H. (2003), "Baer subplanes", Illinois J. Math., 47 (1–2): 485–513, doi:10.1215/ijm/1258488168 3.19
  18. ^ Bödi, R. (1999), "Smooth Hughes planes are classical", Arch. Math., 73: 73–80, doi:10.1007/s000130050022, hdl:11475/3229, S2CID 120222293
  19. ^ Salzmann 2014, 9.17
  20. ^ Salzmann et al. 1995, 87.7
  21. ^ Bödi 1996, Chap. 12
  22. ^ Bödi, R. (1998), "16-dimensional smooth projective planes with large collineation groups", Geom. Dedicata, 72 (3): 283–298, doi:10.1023/A:1005020223604, hdl:11475/3238, S2CID 56094550
  23. ^ Salzmann 2014, 9.18 for a sketch of the proof
  24. ^ Breitsprecher, S. (1967), "Einzigkeit der reellen und der komplexen projektiven Ebene", Math. Z., 99 (5): 429–432, doi:10.1007/bf01111021, S2CID 120984088
  25. ^ Salzmann et al. 1995, 75.1

References

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