On 5-manifolds admitting rank two distributions of Cartan type

Dave, Shantanu; Haller, Stefan

doi:10.1090/tran/7495

On 5-manifolds admitting rank two distributions of Cartan type
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Trans. Amer. Math. Soc. 371 (2019), 4911-4929 Request permission

Abstract:

We consider the question whether an orientable $5$-manifold can be equipped with a rank two distribution of Cartan type and what $2$-plane bundles can be realized. We obtain a complete answer for open manifolds. In the closed case, we settle the topological part of this problem and present partial results concerning its geometric aspects and new examples.

References

D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, Spin cobordism, Bull. Amer. Math. Soc. 72 (1966), 256–260. MR 190939, DOI 10.1090/S0002-9904-1966-11486-6
D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR 219077, DOI 10.2307/1970690
Michael F. Atiyah, Vector fields on manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, Westdeutscher Verlag, Cologne, 1970 (English, with German and French summaries). MR 0263102, DOI 10.1007/978-3-322-98503-3
M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl, suppl. 1, 3–38. MR 167985, DOI 10.1016/0040-9383(64)90003-5
M. F. Atiyah and J. L. Dupont, Vector fields with finite singularities, Acta Math. 128 (1972), 1–40. MR 451256, DOI 10.1007/BF02392157
M. F. Atiyah and I. M. Singer, The index of elliptic operators. V, Ann. of Math. (2) 93 (1971), 139–149. MR 279834, DOI 10.2307/1970757
D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR 184241, DOI 10.2307/1970702
Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015), no. 2, 281–361. MR 3455235, DOI 10.1007/s11511-016-0134-4
Steven R. Bell and Steven G. Krantz, Smoothness to the boundary of conformal maps, Rocky Mountain J. Math. 17 (1987), no. 1, 23–40. MR 882882, DOI 10.1216/RMJ-1987-17-1-23
Gil Bor and Richard Montgomery, $G_2$ and the rolling distribution, Enseign. Math. (2) 55 (2009), no. 1-2, 157–196. MR 2541507, DOI 10.4171/LEM/55-1-8
Robert L. Bryant and Lucas Hsu, Rigidity of integral curves of rank $2$ distributions, Invent. Math. 114 (1993), no. 2, 435–461. MR 1240644, DOI 10.1007/BF01232676
Andreas Čap and Katharina Neusser, On automorphism groups of some types of generic distributions, Differential Geom. Appl. 27 (2009), no. 6, 769–779. MR 2552685, DOI 10.1016/j.difgeo.2009.05.003
Andreas Čap and Katja Sagerschnig, On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five, J. Geom. Phys. 59 (2009), no. 7, 901–912. MR 2536853, DOI 10.1016/j.geomphys.2009.04.001
Andreas Čap and Jan Slovák, Parabolic geometries. I, Mathematical Surveys and Monographs, vol. 154, American Mathematical Society, Providence, RI, 2009. Background and general theory. MR 2532439, DOI 10.1090/surv/154
Andreas Čap, Jan Slovák, and Vladimír Souček, Bernstein-Gelfand-Gelfand sequences, Ann. of Math. (2) 154 (2001), no. 1, 97–113. MR 1847589, DOI 10.2307/3062111
Andreas Čap and Vladimír Souček, Subcomplexes in curved BGG-sequences, Math. Ann. 354 (2012), no. 1, 111–136. MR 2957620, DOI 10.1007/s00208-011-0726-4
Elie 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 (French). MR 1509120, DOI 10.24033/asens.618
D. Crowley, $5$-manifolds: $1$-connected, Bulletin of the Manifold Atlas 2011, 49–55. http://www.map.mpim-bonn.mpg.de/5-manifolds:_1-connected
S. Dave and S. Haller, Graded hypoellipticity of BGG sequences, arXiv:1705.01659 (2017).
A. Dold and H. Whitney, Classification of oriented sphere bundles over a $4$-complex, Ann. of Math. (2) 69 (1959), 667–677. MR 123331, DOI 10.2307/1970030
Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, DOI 10.1007/BF01393840
Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245, DOI 10.1090/gsm/048
Hansjörg Geiges, Contact structures on $1$-connected $5$-manifolds, Mathematika 38 (1991), no. 2, 303–311 (1992). MR 1147828, DOI 10.1112/S002557930000663X
Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
Matthias Hammerl and Katja Sagerschnig, The twistor spinors of generic 2- and 3-distributions, Ann. Global Anal. Geom. 39 (2011), no. 4, 403–425. MR 2776770, DOI 10.1007/s10455-010-9240-2
Michel Kervaire, Courbure intégrale généralisée et homotopie, Math. Ann. 131 (1956), 219–252 (French). MR 86302, DOI 10.1007/BF01342961
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
G. Lusztig, J. Milnor, and F. P. Peterson, Semi-characteristics and cobordism, Topology 8 (1969), 357–359. MR 246308, DOI 10.1016/0040-9383(69)90021-4
J. Milnor, Spin structures on manifolds, Enseign. Math. (2) 9 (1963), 198–203. MR 157388
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
PawełNurowski, Differential equations and conformal structures, J. Geom. Phys. 55 (2005), no. 1, 19–49. MR 2157414, DOI 10.1016/j.geomphys.2004.11.006
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
Katja Sagerschnig, Split octonions and generic rank two distributions in dimension five, Arch. Math. (Brno) 42 (2006), no. suppl., 329–339. MR 2322419
Katja Sagerschnig, Weyl structures for generic rank two distributions in dimension five. Dissertation, University of Vienna, 2008. http://othes.univie.ac.at/2186/.
Stephen Smale, On the structure of $5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR 141133, DOI 10.2307/1970417
Zizhou Tang and Weiping Zhang, A generalization of the Atiyah-Dupont vector fields theory, Commun. Contemp. Math. 4 (2002), no. 4, 777–796. MR 1938494, DOI 10.1142/S0219199702000841
René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
Emery Thomas, The index of a tangent $2$-field, Comment. Math. Helv. 42 (1967), 86–110. MR 215317, DOI 10.1007/BF02564413
Emery Thomas, Vector fields on low dimensional manifolds, Math. Z. 103 (1968), 85–93. MR 224109, DOI 10.1007/BF01110620
Emery Thomas, Vector fields on manifolds, Bull. Amer. Math. Soc. 75 (1969), 643–683. MR 242189, DOI 10.1090/S0002-9904-1969-12240-8
Thomas Vogel, Existence of Engel structures, Ann. of Math. (2) 169 (2009), no. 1, 79–137. MR 2480602, DOI 10.4007/annals.2009.169.79