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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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:
  • Luis Hernández Lamoneda
  • Affiliation: CIMAT, A.P. 402, Guanajuato, Gto. 36000, Mexico
  • Email:
  • Pawel Nurowski
  • Affiliation: Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/46, 02-668 Warszawa, Poland
  • MR Author ID: 250343
  • Email:
  • 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:
  • MathSciNet review: 3811534