Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-24T09:15:44.339Z Has data issue: false hasContentIssue false
  • English
  • Français

The WeitzenbÕck formula for the Bach operator

Published online by Cambridge University Press:  22 January 2016

Mitsuhiro Itoh
Mitsuhiro Itoh*
Affiliation:
Institute of Mathematics, University of Tsukuba 305, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

(Anti-)self-dual metrics are 4-dimensional Riemannian metrics whose Weyl conformai tensor W half vanishes. The Weyl conformai tensor W of an arbitrary metric on an oriented 4-manifold has in general the self-dual part W+ and the anti-self-dual part W− with respect to the Hodge star operator * and one says that a metric is self-dual or anti-self-dual if W− = 0 or W+ = 0, respectively.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 137 , March 1995 , pp. 149 - 181
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1995

References

[ 1 ] Baily, W. L., The decomposition theorem for V-manifolds, Amer. J. Math., 78 (1956), 862–888.Google Scholar
[ 2 ] Bourguignon, J. P., Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein, Invent. Math., 63 (1981), 263 – 286.Google Scholar
[ 3 ] Derdzinski, A., Self-dual Kahler manifolds and Einstein manifolds of dimension four, Comp. Math., 49 (1983), 405–433.Google Scholar
[ 4 ] Donaldson, S. K. and Friedman, R., Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity, 2 (1989), 197–239.Google Scholar
[ 5 ] Floer, A., Self-dual conformal structures on ICP, J. Differential Geom., 33 (1991), 551–573.Google Scholar
[ 6 ] Friendrich, T. and Kurke, H., Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math. Nachr., 106 (1982), 271–299.Google Scholar
[ 7 ] Furuta, M., On self-dual pseudo-connections on some orbifolds, Math. Zeit., 209 (1992), 319–337.Google Scholar
[ 8 ] Galicki, K. and Lawson, H. B., Quaternionic reduction and quaternionic orbifolds, Math. Ann., 282 (1988), 1–21.Google Scholar
[ 9 ] Galicki, K. and Nitta, T., Non-zero scalar curvature generalizations of the ALE hyperkähler metrics (1991), preprint.Google Scholar
[10] Hitchin, N., On compact four-dimensional Einstein manifolds, J. Differential Geom., 9 (1974), 435–442.Google Scholar
[11] Hitchin, N., Kählerian twistor spaces, Proc. London Math. Soc, 43 (1981), 133–150.Google Scholar
[12] Itoh, M., Moduli of half conformally flat structures, Math. Ann., 296 (1993), 687–708.Google Scholar
[13] Itoh, M., Half conformally flat structures and the deformation obstruction space, Tsukuba J. Math., 17 (1993), 143–158.Google Scholar
[14] Itoh, M., Half conformally flat metrics and connected sums of orbifolds, preparation.Google Scholar
[15] King, A. D. and Kotshick, D., The deformation theory of anti-self-dual conformal structures, Math. Ann., 294 (1992), 591–609.Google Scholar
[16] Kronheimer, P. B., A Torelli-type theorem for gravitational instantons, J. Differential Geom., 29 (1989), 685–697.Google Scholar
[17] Poon, Y. S., Compact self-dual manifolds of positive scalar curvature, J. Differential Geom., 24 (1986), 97–132.Google Scholar
[18] Satake, I., On a generalization of the notion of manifolds, Proc. Natl. Acad. Sci. Amer., 42 (1956), 359–363.Google Scholar
[19] Taubes, C. H., The existence of anti-dual conformal structures, J. Differential Geom., 36 (1992), 163–253.Google Scholar