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

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.