11,834 research outputs found

    Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy

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    Nurowski showed that any generic 2-plane field DD on a 5-manifold MM determines a natural conformal structure cDc_D on MM; these conformal structures are exactly those (on oriented MM) whose normal conformal holonomy is contained in the (split, real) simple Lie group G2G_2. Graham and Willse showed that for real-analytic DD the same holds for the holonomy of the real-analytic Fefferman-Graham ambient metric of cDc_D, and that both holonomy groups are equal to G2G_2 for almost all DD. We investigate here independently interesting plane fields for which the associated holonomy groups are a proper subset of G2G_2. Cartan solved the local equivalence problem for 22-plane fields DD and constructed the fundamental curvature tensor AA for these objects. He furthermore claimed to describe locally all DD whose infinitesimal symmetry algebra has rank at least 66 and gave a local quasi-normal form, depending on a single function of one variable, for those that furthermore satisfy a natural degeneracy condition on AA, but Doubrov and Govorov recently rediscovered a counterexample to Cartan's claim. We show that for all DD given by Cartan's alleged quasi-normal form, the conformal structures cDc_D induced via Nurowski's construction are almost Einstein, that we can write their ambient metrics explicitly, and that the holonomy groups associated to cDc_D are always the 55-dimensional Heisenberg group, which here acts indecomposably but not irreducibly. (Not all of these properties hold, however, for Doubrov and Govorov's counterexample.) We also show that the similar results hold for the related class of 22-plane fields defined on suitable jet spaces by ordinary differential equations z(x)=F(y(x))z'(x) = F(y''(x)) satisfying a simple genericity condition.Comment: 34 pages. Revised to accommodate a counterexample to a cited classification of Cartan found by Doubrov and Govorov; fixed some minor error

    On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces

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    We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in GG-spaces, whether homogeneous or not, provided that a certain kthk^{\rm th} order jet bundle over the GG-space admits a GG-invariant local coframe field of constant structure. As a corollary, we note that the differential order of a minimal complete set of congruence invariants is bounded by k+1k+1. We demonstrate the method by rediscovering the speed and curvature invariants of Euclidean planar curves, the Schwarzian derivative of holomorphic immersions in the complex projective line, and equivalents of the first and second fundamental forms of surfaces in R3{\mathbb R}^3 subject to rotation
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