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The dancing metric, \(G_2\)-symmetry and projective rolling. (English) Zbl 1387.53021
Authors’ abstract: The “dancing metric” is a pseudo-Riemannian metric \(g\) of signature \((2,2)\) on the space \(M^4\) of non-incident point-line pairs in the real projective plane \(\mathbb{R}\mathbb{P}^2\). The null curves of \((M^4,g)\) 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 \(\text{SL}_3(\mathbb{R})/\text{GL}_2(\mathbb{R})\), also known as the “para-Kähler Fubini-Study metric”, introduced by P. Libermann [Ann. Mat. Pura Appl. (4) 36, 27–120 (1954; Zbl 0056.15401)]. 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 \((M^4,g)\), 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.

MSC:
53A20 Projective differential geometry
53A30 Conformal differential geometry (MSC2010)
53A40 Other special differential geometries
53A55 Differential invariants (local theory), geometric objects
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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