The dancing metric, $\mathrm G_2$-symmetry and projective rolling

Abstract

The "dancing metric" is a pseudo-riemannian metric $\pmb{g}$ 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 $(M^4,\pmb{g})$ are given by the "dancing condition": the point is moving towards a point on the line, about which the line is turning. We establish a dictionary between classical projective geometry (incidence, cross ratio, projective duality, projective invariants of plane curves...) and pseudo-riemannian 4-dimensional conformal geometry (null-curves and geodesics, parallel transport, self-dual null 2-planes, the Weyl curvature,...). There is also an unexpected bonus: by applying a twistor construction to $(M^4,\pmb{g})$, a $\mathrm G_2$-symmetry emerges, hidden deep in classical projective geometry. To uncover this symmetry, one needs to refine the "dancing condition" by a higher-order condition, expressed in terms of the osculating conic along a plane curve. The outcome is a correspondence between curves in the projective plane and its dual, a projective geometry analog of the more familiar "rolling without slipping and twisting" for a pair of riemannian surfaces.