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Transactions of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/tran/7277
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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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
Email: nurowski@cft.edu.pl

DOI: https://doi.org/10.1090/tran/7277
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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