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Register a new userThe 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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We consider topology-changing transitions between 7-manifolds of holonomy G_2 constructed as a quotient of CY x S^1 by an antiholomorphic involution. We classify involutions for Complete Intersection CY threefolds, focussing primarily on cases without fixed points. The ordinary conifold transition between CY threefolds descends to a transition between G_2 manifolds, corresponding in the N=1 effective theory incorporating the light black hole states either to a change of branch in the scalar potential or to a Higgs mechanism. A simple example of conifold transition with a fixed nodal point is also discussed. As a spin-off, we obtain examples of G_2 manifolds with the same value for the sum of Betti numbers b_2+b_3, and hence potential candidates for mirror manifolds.
We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold is a Calabi-Eckmann manifold. Moreover we show that a complete, simply connected, normal metric contact pair manifold such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, then the manifold is the product of globally $phi$-symmetric spaces and fibers over a locally symmetric space endowed with a symplectic pair.
Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. In recent work, it has become apparent that no-slip billiards resemble non-holonomic mechanical systems in a number of ways. Based on an idea by Borisov, Kilin and Mamaev, we show that no-slip billiards very generally arise as limits of non-holonomic (rolling) systems, in a way that is akin to how ordinary billiards arise as limits of geodesic flows through a flattening of the Riemannian manifold.
We obtain a compact Sobolev embedding for $H$-invariant functions in compact metric-measure spaces, where $H$ is a subgroup of the measure preserving bijections. In Riemannian manifolds, $H$ is a subgroup of the volume preserving diffeomorphisms: a compact embedding for the critical exponents follows. The results can be viewed as an extension of Sobolev embeddings of functions invariant under isometries in compact manifolds.
Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the body cone, along another, the space cone, with their common vertex at the fixed point. This description has been further refined by the second author in 1996, relating the geodesic curvatures of the spherical curves formed by intersecting the cones with the unit sphere in Euclidean $mathbb{R}^3$, thus enabling a reconstruction of the motion of the body from knowledge of the space cone together with the (time dependent) magnitude of the angular velocity vector. In this article we show that a similar description exists for a time dependent family of unimodular $ 2 times 2 $ matrices in terms of rolling cones in 3-dimensional Minkowski space $mathbb{R}^{2,1}$ and the associated pseudo spherical curves, in either the hyperbolic plane $H^2$ or its Lorentzian analog $H^{1,1}$. In particular, this yields an apparently new geometric interpretation of Schrodingers (or Hills) equation $ ddot x + q(t) x =0 $ in terms of rolling without slipping of curves in the hyperbolic plane.
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