Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Conformal holonomy of bi-invariant metrics

Author: Felipe Leitner
Journal: Conform. Geom. Dyn. 12 (2008), 18-31
MSC (2000): Primary 53A30, 53C29; Secondary 53B15
Published electronically: March 5, 2008
MathSciNet review: 2385406
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss in this paper the conformal geometry of bi-invariant metrics on compact semisimple Lie groups. For this purpose, we develop an invariant Cartan calculus. Our main goal is to derive an iterative formula for the holonomy algebra of the normal conformal Cartan connection of a bi-invariant metric. As an example, we demonstrate the application of our invariant calculus to the computation of the conformal holonomy of $\mathrm {SO}(4)$. Its conformal holonomy algebra is $\mathfrak {so}(7)$.

References [Enhancements On Off] (What's this?)

  • S. Sasaki. On the spaces with normal conformal connexions whose groups of holonomy fix a point or a hypersphere. Jap. J. Math. 18, (1943).
  • Marcel Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279–330 (French). MR 79806
  • S. Kobayashi, K. Nomizu. Foundations of differential geometry I & II, John Wiley & Sons, New York, 1963/69.
  • Shoshichi Kobayashi, Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. MR 0355886
  • T. N. Bailey, M. G. Eastwood, and A. R. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), no. 4, 1191–1217. MR 1322223, DOI
  • R. W. Sharpe, Differential geometry, Graduate Texts in Mathematics, vol. 166, Springer-Verlag, New York, 1997. Cartan’s generalization of Klein’s Erlangen program; With a foreword by S. S. Chern. MR 1453120
  • A. $\check \textrm {{C}}$ap, J. Slovák, V. Sou$\check \textrm {{c}}$ek. Invariant Operators on Manifolds with Almost Hermitian Symmetric Structures I & II. Acta. Math. Univ. Comen., New Ser. 66, No. 1, p. 33-69 & No. 2, p. 203-220(1997).
  • Andreas Čap and Jan Slovák, Weyl structures for parabolic geometries, Math. Scand. 93 (2003), no. 1, 53–90. MR 1997873, DOI
  • F. Leitner. Normal conformal Killing forms. e-print: arXiv:math.DG/0406316 (2004).
  • Stuart Armstrong, Definite signature conformal holonomy: a complete classification, J. Geom. Phys. 57 (2007), no. 10, 2024–2048. MR 2348277, DOI
  • Felipe Leitner, Conformal Killing forms with normalisation condition, Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 279–292. MR 2152367
  • A. $\check \textrm {{C}}$ap, A.R. Gover. A holonomy characterisation of Fefferman spaces. e-print: arXiv:math/ 0611939 (2006).
  • A.R. Gover, F. Leitner. A sub-product construction of Poincaré-Einstein metrics. arXiv: math/0608044 (2006).
  • M. Hammerl. Homogeneous Cartan Geometries (master thesis). http://www.mat.univie.\symbol{126}cap/files/Hammerl.pdf, 2006.
  • M. Hammerl. Homogeneous Cartan Geometries. e-print: arXiv:math/0703627 (2007).
  • Felipe Leitner, A remark on unitary conformal holonomy, IMA Volumes in Mathematics and its Applications: Symmetries and Overdetermined Systems of Partial Differential Equations, Editors: Michael Eastwood and Willard Miller, Jr., Springer New York, Volume 144 (2007), p. 445-461.

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 53A30, 53C29, 53B15

Retrieve articles in all journals with MSC (2000): 53A30, 53C29, 53B15

Additional Information

Felipe Leitner
Affiliation: Institut für Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart-Vaihingen, D-70569, Germany

Keywords: Conformal geometry, holonomy theory
Received by editor(s): April 19, 2007
Published electronically: March 5, 2008
Article copyright: © Copyright 2008 American Mathematical Society