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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The dancing metric, $\textrm {G}_2$-symmetry and projective rolling
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by Gil Bor, Luis Hernández Lamoneda and Pawel Nurowski PDF
Trans. Amer. Math. Soc. 370 (2018), 4433-4481 Request permission

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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Additional Information
  • Gil Bor
  • Affiliation: CIMAT, A.P. 402, Guanajuato, Gto. 36000, Mexico
  • Email: gil@cimat.mx
  • Luis Hernández Lamoneda
  • Affiliation: CIMAT, A.P. 402, Guanajuato, Gto. 36000, Mexico
  • Email: lamoneda@cimat.mx
  • Pawel Nurowski
  • Affiliation: Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/46, 02-668 Warszawa, Poland
  • MR Author ID: 250343
  • Email: nurowski@cft.edu.pl
  • Received by editor(s): May 31, 2016
  • Received by editor(s) in revised form: March 2, 2017
  • Published electronically: February 14, 2018
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 4433-4481
  • MSC (2010): Primary 53A20, 53A30, 53A40, 53A55, 53C26
  • DOI: https://doi.org/10.1090/tran/7277
  • MathSciNet review: 3811534