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A non-homogeneous, symmetric contact projective structure. (English) Zbl 1302.53058
The author adapts the construction of a projective symmetric space which is not projectively homogeneous presented by F. Podestà in [Bull. Soc. Math. Fr. 117, No. 3, 343–360 (1989; Zbl 0697.53047)], for the context of contact projective symmetries. They obtain a contact projective manifold which is non-homogeneous but carries a contact projective symmetry at each of its points. From the viewpoint of symmetric parabolic geometries, both constructions are relevant since they give as result the only known examples of symmetric parabolic geometries which are not homogeneous.

MSC:
53C30 Differential geometry of homogeneous manifolds
53A20 Projective differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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[1] Čap A., Slovák J., Parabolic Geometries I, Math. Surveys Monogr., 154, American Mathematical Society, Providence, 2009
[2] Fox D.J.F., Contact projective structures, Indiana Univ. Math. J., 2005, 54(6), 1547-1598 http://dx.doi.org/10.1512/iumj.2005.54.2603 · Zbl 1093.53083
[3] Gregorovič J., General construction of symmetric parabolic structures, Differential Geom. Appl., 2012, 30(5), 450-476 http://dx.doi.org/10.1016/j.difgeo.2012.06.006
[4] Gregorovič J., Zalabová L., Symmetric parabolic contact geometries and symmetric spaces, Transform. Groups, 2013, 18(3), 711-737 http://dx.doi.org/10.1007/s00031-013-9231-z
[5] Podesta F., A class of symmetric spaces, Bull. Soc. Math. France, 1989, 117(3), 343-360 · Zbl 0697.53047
[6] Zalabová L., Parabolic symmetric spaces, Ann. Global Anal. Geom., 2010, 37(2), 125-141 http://dx.doi.org/10.1007/s10455-009-9177-5
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