Conformal holonomy of bi-invariant metrics

Leitner, Felipe

doi:10.1090/S1088-4173-08-00175-6

Conformal holonomy of bi-invariant metrics
HTML articles powered by AMS MathViewer

by Felipe Leitner

Conform. Geom. Dyn. 12 (2008), 18-31

DOI: https://doi.org/10.1090/S1088-4173-08-00175-6

Published electronically: March 5, 2008

PDF | Request permission

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

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, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. 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 10.1216/rmjm/1181072333
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 10.7146/math.scand.a-14413
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 10.1016/j.geomphys.2007.05.001
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. ac.at/\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.