On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five. (English) Zbl 1172.53014

Čap, Andreas; Sagerschnig, Katja

J. Geom. Phys. 59, No. 7, 901-912 (2009).

The authors introduce the notion of a generalized contact form for a generic distribution of rank two on a manifold \(M\) of dimension five. For this form they associate a generalized Reeb field and a partial connection. In this setting they give an explicit construction of a pseudo-Riemannian manifold \(M\) having a split signature, and prove that a change of the generalized contact form yields only a conformal rescaling of the pseudo-Riemannian metric. Consequently the corresponding conformal class is intrinsic to the generalized contact form. This conformal class is then related to the canonical Cartan connection associated to the original distribution, and this allows them to prove that it coincides with conformal class of P. Nurowski [J. Geom. Phys. 55, No. 1, 19–49 (1986)]. Contents include: Introduction (containing a nice overview of the topic); A canonical conformal structure; The relationship to Nurowski’s construction; and References (thirteen items).

Reviewer: Joseph D. Zund (Las Cruces)

MSC:

53A30	Conformal differential geometry (MSC2010)
53A40	Other special differential geometries
53B15	Other connections
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

generalized contact form; generic rank two distribution in dimension five; Cartan connection; conformal structure; parabolic geometry; Weyl structure

References:

[1]	Cartan, E., Les systèmes de Pfaff a cinq variables et les équations aux derivées partielles du second ordre, Ann. Ec. Normale, 27, 109-192 (1910) · JFM 41.0417.01
[2]	K. Sagerschnig, Split octonions and generic rank two distributions in dimension five, Arch. Math. (Brno) 42 (Suppl.) 329-339; K. Sagerschnig, Split octonions and generic rank two distributions in dimension five, Arch. Math. (Brno) 42 (Suppl.) 329-339 · Zbl 1164.53362
[3]	Tanaka, N., On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J., 8, 23-84 (1979) · Zbl 0409.17013
[4]	Čap, A., Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo Suppl. Ser. II, 79, 11-37 (2006) · Zbl 1120.53013
[5]	Sagerschnig, K., Parabolic geometries determined by filtrations of the tangent bundle, Rend. Circ. Mat. Palermo Suppl. Ser. II, 79, 175-181 (2006) · Zbl 1114.53029
[6]	Nurowski, P., Differential equations and conformal structures, J. Geom. Phys., 55, 1, 19-49 (2005) · Zbl 1082.53024
[7]	Burns, D.; Diederich, K.; Shnider, S., Distinguished curves in pseudoconvex boundaries, Duke Math. J., 44, 2, 407-431 (1977) · Zbl 0382.32011
[8]	Lee, J. M., The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc., 296, 1, 411-429 (1986) · Zbl 0595.32026
[9]	Lee, J. M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math., 110, 1, 157-178 (1988) · Zbl 0638.32019
[10]	K. Sagerschnig, Weyl structures for generic rank two distributions in dimension five, Doctoral thesis, University of Vienna, 2008. http://othes.univie.ac.at/2186/; K. Sagerschnig, Weyl structures for generic rank two distributions in dimension five, Doctoral thesis, University of Vienna, 2008. http://othes.univie.ac.at/2186/ · Zbl 1164.53362
[11]	Springer, T. A.; Feldenkamp, F. D., Octonions, Jordan algebras and exceptional groups (2000), Springer: Springer Berlin · Zbl 1087.17001
[12]	Čap, A.; Slovák, J., Weyl structures for parabolic geometries, Math. Scand., 93, 1, 53-90 (2003) · Zbl 1076.53029
[13]	Calderbank, D. M.J.; Diemer, T.; Souček, V., Ricci-corrected derivatives and invariant differential operators, Differential Geom. Appl., 23, 2, 149-175 (2005) · Zbl 1082.58037