×

Conformal Killing forms with normalisation condition. (English) Zbl 1101.53040

Slovák, Jan (ed.) et al., The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 279-292 (2005).
The author studies a special class of conformal Killing forms. These solve certain normalized conformally covariant equations. The existence of non-trivial solutions is related to the conformal symmetry of the conformal class of a pseudo-Riemannian metric. The basic theory is developed, integrability conditions are derived, solutions for normal conformal Killing forms are studied on Einstein spaces, and a class of examples for non-Einstein spaces (Fefferman spaces) is presented which contain solutions of these equations. The equations arise naturally in the context of conformal geometry from the canonical Cartan connection and the existence of solutions induces a reduction of the holonomy of the canonical connection.
For the entire collection see [Zbl 1074.53001].

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53B15 Other connections
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

References:

[1] Ch. Bär. Real Killing spinors and holonomy, Comm. Math. Phys. 154(1993), p. 509-521. · Zbl 0778.53037
[2] H. Baum. Complete Riemannian manifolds with imaginary Killing spinors, Ann. Glob. Anal. Geom. 7(1989), p. 205-226. · Zbl 0694.53043
[3] H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistor and Killing spinors on Riemannian manifolds, Teubner-Text Nr. 124, Teubner-Verlag Stuttgart-Leipzig, 1991. · Zbl 0734.53003
[4] W. Kühnel & H.-B. Rademacher. Conformal vector fields on pseudo-Riemannian spaces, J. Diff. Geom. and its Appl. 7(1997), 237-250. · Zbl 0901.53048
[5] U. Semmelmann. Conformal Killing forms on Riemannian manifolds, Habilitations-schrift, LMU München, 2001.
[6] Tachibana, S., On conformal Killing tensors in a Riemannian space, Tohoku Math. J. (2) 21, 1969, p. 56-64. · Zbl 0182.55301
[7] Kashiwada, T., On conformal Killing tensor, Natur. Sci. Rep. Ochanomizu Univ. 19, 1968, p. 67-74. · Zbl 0179.26902
[8] C.R. Graham. On {S}parling’s characterisation of {F}efferman metrics. Amer. J. Math., 109:853–874, 1987. · Zbl 0663.53050
[9] A. Lichnerowicz. Killing spinors, twistor spinors and Hijazi inequality, J. Geom. Phys. 5(1988), p. 2-18. · Zbl 0678.53018
[10] S. Kobayashi. Transformation Groups in Differential Geometry. Springer-Verlag Berlin Heidelberg, 1972. · Zbl 0246.53031
[11] A. Čap, J. Slovak, V. Souček. 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). · Zbl 1024.53007
[12] H.W. Brinkmann, Einstein spaces which are mapped conformally on each other, Math. Ann. 94 (1925), p. 119-145. [Küh88]{Kuh88} W. Kühnel, Conformal transformations between Einstein spaces, in Conformal geometry, eds., R.S. Kulkarni and U. Pinkall, Aspects of Math. E12 (Vieweg-Verlag, Braunschweig-Wiesbaden, 1988), p. 105-146. · JFM 51.0568.03
[13] N. Hitchin, The geometry of three-forms in six dimensions, J. Differential Geometry 55 (2000), p. 547-576. · Zbl 1036.53042
[14] C. Fefferman. Monge-Ampère equations, the Bergman kernel and geometry of pseudoconvex domains, Ann. Math. 103(1976), p. 395-416. · Zbl 0322.32012
[15] W. Kühnel & H.-B. Rademacher. Twistor spinors and gravitational instantons, Lett. Math. Phys. 38(1996) 411-419. · Zbl 0860.53029
[16] W. Reichel, Über die Trilinearen Alternierenden Formen in 6 und 7 Veränderlichen, Dissertation, Greifswald, 1907. · JFM 38.0674.03
[17] R. Penrose. Twistor algebra, J. Math. Phys. 8(1967), p. 345-366. · Zbl 0163.22602
[18] R. Penrose & W. Rindler. Spinors and Space-time II, Cambr. Univ. Press, 1986. · Zbl 0591.53002
[19] H. Baum, F. Leitner. The twistor equation in Lorentzian spin geometry. to appear Math. Zeitschrift (2004). · Zbl 1068.53031
[20] H. Baum. Twistor and Killing spinors in Lorentzian geometry, in Global analysis and harmonic analysis, ed. J.P. Bourguignon, T. Branson and O. Hijazi, Seminaires & Congres 4, SMF 2000, p. 35-52. · Zbl 1012.53044
[21] H. Baum. Spin-Strukturen und Dirac-Operatoren über pseudo-Riemannschen Mannigfaltigkeiten. Nummer 41 in Teubner-Texte zur Mathematik. Teubner, 1981. · Zbl 0519.53054
[22] H. Baum. Lorentzian twistor spinors and CR-geometry, J. Diff. Geom. and its Appl. 11(1999), no. 1, p. 69-96. · Zbl 0930.53033
[23] Ch. Bouble. Sur l’holonomy des varietes pseudo-riemanniennes, PhD thesis, Institut Elie Cartan (Nancy), 2000.
[24] F. Leitner. Imaginary Killing spinors in Lorentzian geometry. J. Math. Phys. 44 (2003), no. 10, 4795–4806 · Zbl 1062.53039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.