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The dancing metric, $ {G}_2$-symmetry and projective rolling

Authors: Gil Bor, Luis Hernández Lamoneda and Pawel Nurowski
Journal: Trans. Amer. Math. Soc. 370 (2018), 4433-4481
MSC (2010): Primary 53A20, 53A30, 53A40, 53A55, 53C26
Published electronically: February 14, 2018
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Abstract: The ``dancing metric'' is a pseudo-Riemannian metric $ \bf {g}$ of signature (2,2) on the space $ M^4$ of non-incident point-line pairs in the real projective plane $ \mathbb{RP}^2$. The null curves of $ (M^4,\bf {g})$ are given by the ``dancing condition'': at each moment, the point is moving towards or away from the point on the line about which the line is turning. This is the standard homogeneous metric on the pseudo-Riemannian symmetric space $ \mathrm {SL}_3(\mathbb{R})/\mathrm {GL}_2(\mathbb{R})$, also known as the ``para-Kähler Fubini-Study metric'', introduced by P. Libermann. We establish a dictionary between classical projective geometry (incidence, cross ratio, projective duality, projective invariants of plane curves, etc.) and pseudo-Riemannian 4-dimensional conformal geometry (null curves and geodesics, parallel transport, self-dual null 2-planes, the Weyl curvature, etc.). Then, applying a twistor construction to $ (M^4,\bf {g})$, a $ G_2$-symmetry is revealed, hidden deep in classical projective geometry. To uncover this symmetry, one needs to refine the ``dancing condition'' to a higher-order condition. The outcome is a correspondence between curves in the real projective plane and its dual, a projective geometric analog of the more familiar ``rolling without slipping and twisting'' for a pair of Riemannian surfaces.

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  • [1] A. A. Agrachev, Rolling balls and octonions, Tr. Mat. Inst. Steklova 258 (2007), no. Anal. i Osob. Ch. 1, 17-27; English transl., Proc. Steklov Inst. Math. 258 (2007), no. 1, 13-22. MR 2400520,
  • [2] D. V. Alekseevsky, K. Medori, and A. Tomassini, Homogeneous para-Kählerian Einstein manifolds, Uspekhi Mat. Nauk 64 (2009), no. 1(385), 3-50; English transl., Russian Math. Surveys 64 (2009), no. 1, 1-43. MR 2503094,
  • [3] Daniel An and Paweł Nurowski, Twistor space for rolling bodies, Comm. Math. Phys. 326 (2014), no. 2, 393-414. MR 3165459,
  • [4] Gil Bor and Richard Montgomery, $ G_2$ and the rolling distribution, Enseign. Math. (2) 55 (2009), no. 1-2, 157-196. MR 2541507,
  • [5] Robert L. Bryant and Lucas Hsu, Rigidity of integral curves of rank $ 2$ distributions, Invent. Math. 114 (1993), no. 2, 435-461. MR 1240644,
  • [6] Élie Cartan, Leçons sur la géométrie projective complexe. La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile. Leçons sur la théorie des espaces à connexion projective, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1992. Reprint of the editions of 1931, 1937 and 1937. MR 1190006
  • [7] É. Cartan, Sur la structure des groupes de transformations finis et continus, Thèse, Paris, Nony 1894; 2e édition, Vuibert, 1933. (Reprinted in Œuvres Complètes, Partie I, Vol. 1, 137-288.)
  • [8] Élie Cartan, Œuvres complètes. Partie I. Groupes de Lie, Gauthier-Villars, Paris, 1952. MR 0050516
  • [9] Élie 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. MR 1509120
  • [10] É. Cartan, Sur la structure des groupes finis et continus, C. R. Acad. Sci. Paris 116 (1893), 784-786.
  • [11] É. Cartan, Nombres complexes. Exposé d'après l'article allemand de E. Study, Encyclopedia Math. Sci, Tome I, vol. 1, I-5 (1908), 331-468. (Reprinted in Œuvres Complètes 2, 107-246, MR0058523.)
  • [12] V. Cruceanu, P. Fortuny, and P. M. Gadea, A survey on paracomplex geometry, Rocky Mountain J. Math. 26 (1996), no. 1, 83-115. MR 1386154,
  • [13] B. Doubrov and I. Zelenko, Geometry of curves in parabolic homogeneous spaces, preprint, arXiv:1110.0226 (2011).
  • [14] F. Engel, Sur un groupe simple à quatorze paramètres, C. R. Acad. Sci. Paris 116 (1893), 786-788.
  • [15] Étienne Ghys, Sergei Tabachnikov, and Vladlen Timorin, Osculating curves: around the Tait-Kneser theorem, Math. Intelligencer 35 (2013), no. 1, 61-66. MR 3041992,
  • [16] A. Rod Gover, C. Denson Hill, and Paweł Nurowski, Sharp version of the Goldberg-Sachs theorem, Ann. Mat. Pura Appl. (4) 190 (2011), no. 2, 295-340. MR 2786175,
  • [17] Irving Kaplansky, Linear algebra and geometry, Reprint of the 1974 revised edition, Dover Publications, Inc., Mineola, NY, 2003. A second course. MR 2001037
  • [18] Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389
  • [19] N. G. Konovenko and V. V. Lychagin, On projective classification of plane curves, Global and Stoch. Anal. 1 (2011), no. 2, 241-264.
  • [20] Felix Klein and Sophus Lie, Ueber diejenigen ebenen Curven, welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sich übergehen, Math. Ann. 4 (1871), no. 1, 50-84. MR 1509730,
  • [21] Paulette Libermann, Sur le problème d'équivalence de certaines structures infinitésimales, Ann. Mat. Pura Appl. (4) 36 (1954), 27-120. MR 0066020,
  • [22] Paweł Nurowski, Differential equations and conformal structures, J. Geom. Phys. 55 (2005), no. 1, 19-49. MR 2157414,
  • [23] P. J. Olver, Moving frames and differential invariants in centro-affine geometry, Lobachevskii J. Math. 31 (2010), no. 2, 77-89. MR 2661301,
  • [24] V. Ovsienko and S. Tabachnikov, Projective differential geometry old and new: From the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge Tracts in Mathematics, vol. 165, Cambridge University Press, Cambridge, 2005. MR 2177471
  • [25] Roger Penrose, Nonlinear gravitons and curved twistor theory: The riddle of gravitation-on the occasion of the 60th birthday of Peter G. Bergmann (Proc. Conf., Syracuse Univ., Syracuse, N. Y., 1975), General Relativity and Gravitation 7 (1976), no. 1, 31-52. MR 0439004
  • [26] Katja Sagerschnig, Split octonions and generic rank two distributions in dimension five, Arch. Math. (Brno) 42 (2006), no. suppl., 329-339. MR 2322419
  • [27] I. M. Singer and J. A. Thorpe, The curvature of $ 4$-dimensional Einstein spaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 355-365. MR 0256303
  • [28] E. J. Wilczynski, Projective differential geometry of curves and ruled surfaces, Chelsea Publishing Co., New York, 1962. MR 0131232
  • [29] Igor Zelenko, On variational approach to differential invariants of rank two distributions, Differential Geom. Appl. 24 (2006), no. 3, 235-259. MR 2216939,
  • [30] Max Zorn, Theorie der alternativen ringe, Abh. Math. Sem. Univ. Hamburg 8 (1931), no. 1, 123-147. MR 3069547,

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Additional Information

Gil Bor
Affiliation: CIMAT, A.P. 402, Guanajuato, Gto. 36000, Mexico

Luis Hernández Lamoneda
Affiliation: CIMAT, A.P. 402, Guanajuato, Gto. 36000, Mexico

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

Received by editor(s): May 31, 2016
Received by editor(s) in revised form: March 2, 2017
Published electronically: February 14, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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