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

SIGMA 7 (2011), 039, 16 pages      arXiv:1011.0288      https://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.

pdf (412 Kb)   tex (25 Kb)

References

  1. Alekseevski D.V., Groups of conformal transformations of Riemannian spaces, Sb. Math. 18 (1972), 285-301.
  2. Biquard O., Métriques d'Einstein asymptotiquement symétriques, Astérisque (2000), no. 265 (English transl.: Asymptotically symmetric Einstein metrics, SMF/AMS Texts and Monographs, Vol. 13, American Mathematical Society, Providence, 2006).
  3. Cap A., Infinitesimal automorphisms and deformations of parabolic geometries, J. Eur. Math. Soc. 10 (2008), 415-437, math.DG/0508535.
  4. Cap A., Slovák J., Weyl structures for parabolic geometries, Math. Scand. 93 (2003), 53-90, math.DG/0001166.
  5. Cap A., Slovák J., Parabolic geometries. I Background and general theory, Mathematical Surveys and Monographs, Vol. 154, American Mathematical Society, Providence, RI, 2009.
  6. Duistermaat J.J., Kolk J.A.C., Lie groups, Universitext, Springer-Verlag, Berlin, 2000.
  7. Ferrand J., The action of conformal transformations on a Riemannian manifold, Math. Ann. 304 (1996), 277-291.
  8. Frances C., Sur le groupe d'automorphismes des géométries paraboliques de rang un, Ann. Sci. École Norm. Sup. (4) 40 (2007), 741-764 (English version: A Ferrand-Obata theorem for rank one parabolic geometries, math.DG/0608537).
  9. Frances C., Essential conformal structures in Riemannian and Lorentzian geometry, in Recent Developments in Pseudo-Riemannian Geometry, ESI Lectures in Mathematics and Physics, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, 231-260.
  10. Frances C., Local dynamics of conformal vector fields, arXiv:0909.0044.
  11. Gover A.R., Laplacian operators and Q-curvature of conformally Einstein manifolds, Math. Ann. 336 (2006), 311-334, math.DG/0506037.
  12. Ivanov S., Vassilev D., Conformal quaternionic contact curvature and the local sphere theorem, J. Math. Pures Appl. (9) 93 (2010), 277-307, arXiv:0707.1289.
  13. Schoen R., On the conformal and CR automorphism groups, Geom. Funct. Anal. 5 (1995), 464-481.
  14. Webster S.M., On the transformation group of a real hypersurface, Trans. Amer. Math. Soc. 231 (1977), 179-190.

Previous article   Next article   Contents of Volume 7 (2011)