اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe dancing metric, $mathrm G_2$-symmetry and projective rolling
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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.
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