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The geometry of almost Einstein \((2,3,5)\) distributions. (English) Zbl 1372.32033
Authors’ abstract: We analyze the classical problem of the existence of Einstein metrics in a given conformal structure for the class of conformal structures induced, following Nurowski’s construction, by (oriented) \((2,3,5)\) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures \(\mathbf{c}\) that are induced by at least two distinct oriented \((2,3,5)\) distributions; in this case there is a \(1\)-parameter family of such distributions that induce \(\mathbf{c}\). Second, they are characterized by the existence of a holonomy reduction to \(\mathrm{SU}(1,2)\), \(\mathrm{SL}(3,{\mathbb R})\), or a particular semidirect product \(\mathrm{SL}(2,{\mathbb R}) \ltimes Q_+\), according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each one with a geometric structure. This establishes novel links between \((2,3,5)\) distributions and many other geometries – several classical geometries among them – including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension \(5\); Kähler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension \(4\); CR geometry and the point geometry of second-order ordinary differential equations in dimension \(3\); and projective geometry in dimension \(2\). We describe a generalized Fefferman construction that builds from a \(4\)-dimensional Kähler-Einstein or para-Kähler-Einstein structure a family of \((2,3,5)\) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein \((2,3,5)\) conformal structures for which the Einstein constant is positive and negative.

MSC:
32Q20 Kähler-Einstein manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32V05 CR structures, CR operators, and generalizations
53A30 Conformal differential geometry (MSC2010)
53B35 Local differential geometry of Hermitian and Kählerian structures
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[1] Agrachev, A. A. and Sachkov, Yu. L., An intrinsic approach to the control of rolling bodies, Proceedings of the 38th IEEE Conference on Decision and Control, Vol. 5 (Phoenix, Arizona, USA, December 7-10, 1999), 431-435, (1999), IEEE Control Systems Society, Piscataway, NJ
[2] An, Daniel and Nurowski, Pawe{\l}, Twistor space for rolling bodies, Communications in Mathematical Physics, 326, 2, 393-414, (2014) · Zbl 1296.53100
[3] Armstrong, Stuart, Projective holonomy. {II}. {C}ones and complete classifications, Annals of Global Analysis and Geometry, 33, 2, 137-160, (2008) · Zbl 1147.53038
[4] Bailey, T. N. and Eastwood, M. G. and Gover, A. R., Thomas’s structure bundle for conformal, projective and related structures, The Rocky Mountain Journal of Mathematics, 24, 4, 1191-1217, (1994) · Zbl 0828.53012
[5] Blair, David E., Contact manifolds in {R}iemannian geometry, Lecture Notes in Math., 509, vi+146, (1976), Springer-Verlag, Berlin – New York · Zbl 0319.53026
[6] Bor, Gil and Lamoneda, Luis Hern\'andez and Nurowski, Pawel, The dancing metric, {\({\rm G}_2\)}-symmetry and projective rolling · Zbl 1387.53021
[7] Bor, Gil and Montgomery, Richard, {\({\rm G}_2\)} and the rolling distribution, L’Enseignement Math\'ematique. Revue Internationale. 2e S\'erie, 55, 1-2, 157-196, (2009) · Zbl 1251.70008
[8] Bor, Gil and Nurowski, Pawel
[9] Brinkmann, H. W., Riemann spaces conformal to {E}instein spaces, Mathematische Annalen, 91, 3-4, 269-278, (1924) · JFM 50.0504.01
[10] Brinkmann, H. W., Einstein spaces which are mapped conformally on each other, Mathematische Annalen, 94, 1, 119-145, (1925) · JFM 51.0568.03
[11] Brown, Robert B. and Gray, Alfred, Vector cross products, Commentarii Mathematici Helvetici, 42, 222-236, (1967) · Zbl 0155.35702
[12] Bryant, Robert L., Some remarks on {\({\rm G}_2\)}-structures, Proceedings of {G}\"okova {G}eometry-{T}opology {C}onference 2005, 75-109, (2006), G\"okova Geometry/Topology Conference (GGT), G\"okova · Zbl 1115.53018
[13] Bryant, Robert L. and Hsu, Lucas, Rigidity of integral curves of rank {\(2\)} distributions, Inventiones Mathematicae, 114, 2, 435-461, (1993) · Zbl 0807.58007
[14] Calderbank, David M. J. and Diemer, Tammo, Differential invariants and curved {B}ernstein–{G}elfand–{G}elfand sequences, Journal f\"ur die Reine und Angewandte Mathematik. [Crelle’s Journal], 537, 67-103, (2001) · Zbl 0985.58002
[15] \v{C}ap, Andreas, Infinitesimal automorphisms and deformations of parabolic geometries, Journal of the European Mathematical Society (JEMS), 10, 2, 415-437, (2008) · Zbl 1161.32020
[16] \v{C}ap, Andreas and Gover, A. Rod, A holonomy characterisation of {F}efferman spaces, Annals of Global Analysis and Geometry, 38, 4, 399-412, (2010) · Zbl 1298.53041
[17] \v{C}ap, A. and Gover, A. R. and Hammerl, M., Holonomy reductions of {C}artan geometries and curved orbit decompositions, Duke Mathematical Journal, 163, 5, 1035-1070, (2014) · Zbl 1298.53042
[18] \v{C}ap, Andreas and Slov\'ak, Jan, Parabolic geometries. {I}. Background and general theory, Mathematical Surveys and Monographs, 154, x+628, (2009), Amer. Math. Soc., Providence, RI · Zbl 1183.53002
[19] \v{C}ap, Andreas and Slov\'ak, Jan and Sou\v{c}ek, Vladim\'{\i}r, Bernstein–{G}elfand–{G}elfand sequences, Annals of Mathematics. Second Series, 154, 1, 97-113, (2001) · Zbl 1159.58309
[20] Cartan, \'{E}lie, Les syst\`emes de {P}faff, \`a cinq variables et les \'equations aux d\'eriv\'ees partielles du second ordre, Annales Scientifiques de l’\'Ecole Normale Sup\'erieure. Troisi\`eme S\'erie, 27, 109-192, (1910) · JFM 41.0417.01
[21] Cartan, \'{E}lie, Sur la structure des groupes simples finis et continus, C. R. Acad. Sci. Paris, 113, 784-786, (1893) · JFM 25.0637.01
[22] Chudecki, Adam, On some examples of para-{H}ermite and para-{K}\"{a}hler {E}instein spaces with {\( \Lambda \neq 0\)}, Journal of Geometry and Physics, 112, 2, 175-196, (2017) · Zbl 1355.53024
[23] Cort\'es, V. and Leistner, T. and Sch\"afer, L. and Schulte-Hengesbach, F., Half-flat structures and special holonomy, Proceedings of the London Mathematical Society. Third Series, 102, 1, 113-158, (2011) · Zbl 1225.53024
[24] Curry, Sean and Gover, A. Rod, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity · Zbl 1416.83014
[25] Doubrov, Boris and Komrakov, Boris, The geometry of second-order ordinary differential equations · Zbl 0934.53007
[26] Doubrov, Boris and Kruglikov, Boris, On the models of submaximal symmetric rank 2 distributions in 5{D}, Differential Geometry and its Applications, 35, suppl., suppl., 314-322, (2014) · Zbl 1319.58003
[27] Dunajski, Maciej and Przanowski, Maciej, Null {K}\"ahler structures, symmetries and integrability, Topics in Mathematical Physics, General Relativity and Cosmology in Honor of {J}erzy {P}leba\'nski, 147-155, (2006), World Sci. Publ., Hackensack, NJ · Zbl 1135.53050
[28] Engel, F., Sur un groupe simple \`{a} quatorze param\`{e}tres, C. R. Acad. Sci. Paris, 116, 786-788, (1893) · JFM 25.0631.01
[29] Fefferman, Charles L., Monge–{A}mp\`ere equations, the {B}ergman kernel, and geometry of pseudoconvex domains, Annals of Mathematics. Second Series, 103, 3, 395-416, (1976) · Zbl 0322.32012
[30] Fefferman, Charles and Graham, C. Robin, The ambient metric, Annals of Mathematics Studies, 178, x+113, (2012), Princeton University Press, Princeton, NJ · Zbl 1243.53004
[31] Fox, Daniel J. F., Contact projective structures, Indiana University Mathematics Journal, 54, 6, 1547-1598, (2005) · Zbl 1093.53083
[32] Gover, A. Rod, Almost conformally {E}instein manifolds and obstructions, Differential Geometry and its Applications, 247-260, (2005), Matfyzpress, Prague · Zbl 1121.53032
[33] Gover, A. Rod, Almost {E}instein and {P}oincar\'e–{E}instein manifolds in {R}iemannian signature, Journal of Geometry and Physics, 60, 2, 182-204, (2010) · Zbl 1194.53038
[34] Gover, A. Rod and Macbeth, Heather R., Detecting {E}instein geodesics: {E}instein metrics in projective and conformal geometry, Differential Geometry and its Applications, 33, suppl., suppl., 44-69, (2014) · Zbl 1372.53017
[35] Gover, A. Rod and Neusser, K. and Willse, Travis, Sasaki and {K}\"{a}hler structures and their compactifications via projective geometry · Zbl 1462.53009
[36] Gover, A. Rod and Nurowski, Pawe{\l}, Obstructions to conformally {E}instein metrics in {\(n\)} dimensions, Journal of Geometry and Physics, 56, 3, 450-484, (2006) · Zbl 1098.53014
[37] Gover, A. Rod and Panai, Roberto and Willse, Travis, Nearly {K}\"ahler geometry and \((2,3,5)\)-distributions via projective holonomy, Indiana University Mathematics Journal · Zbl 1390.53013
[38] Graham, C. Robin and Willse, Travis, Parallel tractor extension and ambient metrics of holonomy split {\({\rm G}_2\)}, Journal of Differential Geometry, 92, 3, 463-505, (2012) · Zbl 1268.53075
[39] Hammerl, Matthias, Natural prolongations of {BGG}-operators, (2009), Universit\"at Wien · Zbl 1212.53014
[40] Hammerl, Matthias and Sagerschnig, Katja, Conformal structures associated to generic rank 2 distributions on 5-manifolds – characterization and {K}illing-field decomposition, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 5, 081, 29 pages, (2009) · Zbl 1191.53016
[41] Hammerl, Matthias and Sagerschnig, Katja, The twistor spinors of generic 2- and 3-distributions, Annals of Global Analysis and Geometry, 39, 4, 403-425, (2011) · Zbl 1229.53058
[42] Hammerl, Matthias and Sagerschnig, Katja and \v{S}ilhan, Josef and Taghavi-Chabert, Arman and \v{Z}\'{a}dn\'{i}k, Vojte\v{e}ch, A projective-to-conformal {F}efferman-type construction · Zbl 1378.53019
[43] Kath, I., {\(G_{2(2)}^*\)}-structures on pseudo-{R}iemannian manifolds, Journal of Geometry and Physics, 27, 3-4, 155-177, (1998) · Zbl 0957.53018
[44] Haantjes, J. and Wrona, W., \"Uber konformeuklidische und {E}insteinsche {R}\"aume gerader {D}imension, 42, 626-636, (1939) · JFM 65.0798.04
[45] Kath, I., Killing spinors on pseudo-{R}iemannian manifolds · Zbl 0957.53018
[46] Kol\'a\v{r}, Ivan and Michor, Peter W. and Slov\'ak, Jan, Natural operations in differential geometry, vi+434, (1993), Springer-Verlag, Berlin · Zbl 0782.53013
[47] Kozameh, Carlos N. and Newman, Ezra T. and Tod, K. P., Conformal {E}instein spaces, General Relativity and Gravitation, 17, 4, 343-352, (1985) · Zbl 0564.53011
[48] Kruglikov, Boris and The, D., The gap phenomenon in parabolic geometries, Journal f\"ur die Reine und Angewandte Mathematik · Zbl 1359.58019
[49] Leistner, Thomas and Nurowski, Pawe{\l}, Ambient metrics with exceptional holonomy, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 11, 2, 407-436, (2012) · Zbl 1255.53018
[50] Leitner, Felipe, A remark on unitary conformal holonomy, Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., 144, 445-460, (2008), Springer, New York · Zbl 1142.53040
[51] Leitner, Felipe, Conformal {K}illing forms with normalisation condition, Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento, 75, 279-292, (2005) · Zbl 1101.53040
[52] Nurowski, Pawe{\l}, Differential equations and conformal structures, Journal of Geometry and Physics, 55, 1, 19-49, (2005) · Zbl 1082.53024
[53] Nurowski, Pawe{\l} and Sparling, George A., Three-dimensional {C}auchy–{R}iemann structures and second-order ordinary differential equations, Classical and Quantum Gravity, 20, 23, 4995-5016, (2003) · Zbl 1051.32019
[54] Sagerschnig, Katja, Split octonions and generic rank two distributions in dimension five, Universitatis Masarykianae Brunensis. Facultas Scientiarum Naturalium. Archivum Mathematicum, 42, suppl., suppl., 329-339, (2006) · Zbl 1164.53362
[55] Sagerschnig, Katja and Willse, Travis, The almost {E}instein operator for \((2, 3, 5)\) distributions · Zbl 1372.32033
[56] Sagerschnig, Katja and Willse, Travis, Fefferman constructions for K\"ahler surfaces · Zbl 1449.53033
[57] Sasaki, S., On a relation between a {R}iemannian space which is conformal with {E}instein spaces and normal conformally connected spaces whose groups of holonomy fix a point or a hypersphere, 5, 66-72, (1942)
[58] Schouten, J. A., Ricci-calculus. {A}n introduction to tensor analysis and its geometrical applications, Die Grundlehren der Mathematischen Wissenschaften, 10, xx+516, (1954), Springer-Verlag, Berlin – G\"ottingen – Heidelberg · Zbl 0057.37803
[59] Schulte-Hengesbach, F., Half-flat structure on {L}ie groups, (2010), Universit\"at Hamburg · Zbl 1246.53073
[60] Semmelmann, Uwe, Conformal {K}illing forms on {R}iemannian manifolds, Mathematische Zeitschrift, 245, 3, 503-527, (2003) · Zbl 1061.53033
[61] Sharpe, R. W., Differential geometry. {C}artan’s generalization of {K}lein’s {E}rlangen program, Graduate Texts in Mathematics, 166, xx+421, (1997), Springer-Verlag, New York · Zbl 0876.53001
[62] Susskind, Leonard, The world as a hologram, Journal of Mathematical Physics, 36, 11, 6377-6396, (1995) · Zbl 0850.00013
[63] Szekeres, P., Spaces conformal to a class of spaces in general relativity, 274, 206-212, (1963) · Zbl 0113.44805
[64] Willse, Travis, Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional {H}eisenberg group holonomy, Differential Geometry and its Applications, 33, suppl., suppl., 81-111, (2014) · Zbl 1293.53040
[65] Wolf, Joseph A., Isotropic manifolds of indefinite metric, Commentarii Mathematici Helvetici, 39, 21-64, (1964) · Zbl 0125.39203
[66] Wong, Yung-Chow, Some {E}instein spaces with conformally separable fundamental tensors, Transactions of the American Mathematical Society, 53, 157-194, (1943) · Zbl 0060.38606
[67] Yano, Kentaro, Conformal and concircular geometries in {E}instein spaces, 19, 444-453, (1943) · Zbl 0060.38906
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