No Access

New relations between $G_{2}$ $G_{2}$ geometries in dimensions 5 and 7

Thomas Leistner

School of Mathematical Sciences, University of Adelaide, SA 5005, Australia

E-mail Address: thomas.leistner@adelaide.edu.au

Search for more papers by this author

Paweł Nurowski

Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/46, 02-668 Warszawa, Poland

E-mail Address: nurowski@cft.edu.pl

Search for more papers by this author

, and

Katja Sagerschnig

http://orcid.org/0000-0003-2497-9455

Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca Degli Abruzzi 24, 10129 Torino, Italy

E-mail Address: katja.sagerschnig@univie.ac.at

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0129167X1750094XCited by:3 (Source: Crossref)

Abstract

There are two well-known parabolic split $G_{2}$ $G_{2}$ geometries in dimension 5, $(2, 3, 5)$ $(2, 3, 5)$ distributions and $G_{2}$ $G_{2}$ contact structures. Here we link these two geometries with yet another $G_{2}$ $G_{2}$ related contact structure, which lives on a $7$ $7$ -manifold. More precisely, we present a natural geometric construction that associates to a $(2, 3, 5)$ $(2, 3, 5)$ distribution a $7$ $7$ -dimensional bundle endowed with a canonical Lie contact structure. We further study the relation between the canonical normal Cartan connections associated with the two structures and we show that the Cartan holonomy of the induced Lie contact structure reduces to $G_{2}$ $G_{2}$ . This motivates the study of the curved orbit decomposition associated with a $G_{2}$ $G_{2}$ reduced Lie contact structure on a $7$ $7$ -manifold. It is shown that, provided an additional curvature condition is satisfied, in a neighborhood of each point in the open curved orbit the structure descends to a $(2, 3, 5)$ $(2, 3, 5)$ distribution on a local leaf space. The closed orbit carries an induced $G_{2}$ $G_{2}$ contact structure.

Keywords:

AMSC: 53B15, 53C15, 53C29, 53D99, 58A15, 58A30

References

1. J. Baez and J. Huerta, $G_{2}$ $G_{2}$ and the rolling ball, Trans. Amer. Math. Soc. 366(10) (2014) 5257–5293. Crossref, Web of Science, Google Scholar
2. R. L. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2) 126(3) (1987) 525–576. Crossref, Web of Science, Google Scholar
3. R. L. Bryant, Élie Cartan and geometric duality, Journées É. Cartan 1998 et 1999 (Institut Élie Cartan, 2000), pp. 5–20. Google Scholar
4. A. Čap, Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005) 143–172. Crossref, Web of Science, Google Scholar
5. A. Čap, Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo (2) Suppl. 79 (2006) 11–37. Google Scholar
6. A. Čap, A. R. Gover and M. Hammerl, Holonomy reductions of Cartan geometries and curved orbit decompositions, Duke Math. J. 163(5) (2014) 1035–1070. Crossref, Web of Science, Google Scholar
7. A. Čap and T. Salač, Parabolic conformally symplectic structures I; definition and distinguished connections, preprint (2016), arXiv:1605.01161. Google Scholar
8. A. Čap and J. Slovák, Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI, 2009). Crossref, Google Scholar
9. A. Čap, J. Slovák and V. Souček, Bernstein–Gelfand–Gelfand sequences, Ann. of Math. 154(1) (2001) 97–113. Crossref, Web of Science, Google Scholar
10. É. Cartan, Sur la structure des groupes simples finis et continus, C. R. Acad. Sci. Paris 116 (1893) 784–786. Google Scholar
11. É. Cartan, Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. (3) 27 (1910) 109–192. Crossref, Google Scholar
12. F. Engel, Sur un groupe simple à quatorze paramètres, C. R. Acad. Sci. Paris 116 (1893) 786–788. Google Scholar
13. É. Goursat, Lecons sur le Problème de Pfaff (Librairie Scientifique J. Hermann, Paris, 1922). Google Scholar
14. C. R. Graham and T. Willse, Parallel tractor extension and ambient metrics of holonomy split $G_{2}$ $G_{2}$ , J. Differential Geom. 92(3) (2012) 463–505. Crossref, Web of Science, Google Scholar
15. M. Hammerl and K. Sagerschnig, Conformal structures associated to generic rank 2 distributions on 5-manifolds — characterization and Killing-field decomposition, Symmetry Integrability Geom. Methods Appl. 5 (2009) 29 pp. Web of Science, Google Scholar
16. B. Kostant, Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. 74(2) (1961) 329–387. Crossref, Web of Science, Google Scholar
17. J. M. Landsberg and L. Manivel, On the projective geometry of rational homogeneous varieties, Comment. Math. Helv. 78(1) (2003) 65–100. Crossref, Web of Science, Google Scholar
18. J. M. Landsberg and L. Manivel, The sextonions and $E_{7 \frac{1}{2}}$ $E_{7 \frac{1}{2}}$ , Adv. Math. 201(1) (2006) 143–179. Crossref, Web of Science, Google Scholar
19. T. Leistner and P. Nurowski, Ambient metrics with exceptional holonomy, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(2) (2012) 407–436. Google Scholar
20. R. Miyaoka, Lie contact structures and conformal structures, Kodai Math. J. 14(1) (1991) 42–71. Crossref, Google Scholar
21. R. Miyaoka, Lie contact structures and normal Cartan connections, Kodai Math. J. 14(1) (1991) 13–41. Crossref, Google Scholar
22. T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993) 263–347. Crossref, Google Scholar
23. P. Nurowski, Differential equations and conformal structures, J. Geom. Phys. 55(1) (2005) 19–49. Crossref, Web of Science, Google Scholar
24. P. Nurowski, Conformal structures with explicit ambient metrics and conformal $G_{2}$ $G_{2}$ holonomy, in Symmetries and Overdetermined System of Partial Differential Equations, IMA Volumes in Mathematics and its Applications, Vol. 144 (Springer, New York, 2008), pp. 515–526. Crossref, Google Scholar
25. B. Pasquier, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann. 344(4) (2009) 963–987. Crossref, Web of Science, Google Scholar
26. K. Sagerschnig, Split octonions and generic rank two distributions in dimension five, Arch. Math. (Brno) 42(Suppl.) (2006) 329–339. Google Scholar
27. H. Sato and K. Yamaguchi, Lie contact manifolds, in Geometry of Manifolds, Perspectives in Mathematics, Vol. 8 (Academic Press, Boston, MA, 1989), pp. 191–238. Google Scholar
28. H. Sato and K. Yamaguchi, Lie contact manifolds II, Math. Ann. 297(1) (1993) 33–57. Crossref, Web of Science, Google Scholar
29. N. Tanaka, On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8(1) (1979) 23–84. Crossref, Google Scholar
30. T. Willse, Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy, Differential Geom. Appl. 33(Suppl.) (2014) 81–111. Crossref, Web of Science, Google Scholar
31. K. Yamaguchi, Differential systems associated with simple graded Lie algebras, in Progress in Differential Geometry, Advanced Studies in Pure Mathematics, Vol. 22 (Mathematical Society Japan, Tokyo, 1993), pp. 413–494. Crossref, Google Scholar
32. V. Žádník, Lie contact structures and chains, preprint (2009), arXiv:0901.4433. Google Scholar