Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 039, 16 pages      arXiv:1011.0288      http://dx.doi.org/10.3842/SIGMA.2011.039

Essential Parabolic Structures and Their Infinitesimal Automorphisms

Jesse Alt
School of Mathematics, University of the Witwatersrand, P O Wits 2050, Johannesburg, South Africa

Received November 02, 2010, in final form April 11, 2011; Published online April 14, 2011

Abstract
Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As a corollary of the generalized Ferrand-Obata theorem proved by C. Frances, this proves a generalization of the ''Lichnérowicz conjecture'' for conformal Riemannian, strictly pseudo-convex CR, and quaternionic/octonionic contact manifolds in positive-definite signature. For an infinitesimal automorphism with a singularity, we give a generalization of the dictionary introduced by Frances for conformal Killing fields, which characterizes (local) essentiality via the so-called holonomy associated to a singularity of an infinitesimal automorphism.

Key words: essential structures; infinitesimal automorphisms; parabolic geometry; Lichnérowicz conjecture.

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