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Four–dimensional metrics conformal to Kähler

Published online by Cambridge University Press:  05 January 2010

MACIEJ DUNAJSKI  and
PAUL TOD
MACIEJ DUNAJSKI
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA. e-mail: [email protected]
PAUL TOD
Affiliation:
The Mathematical Institute, Oxford University, 24–29 St Giles, Oxford OX1 3LB. e-mail: [email protected]
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Abstract

We derive some necessary conditions on a Riemannian metric (M, g) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti–self–dual, the self–dual Weyl spinor must be of algebraic type D and satisfy a simple first order conformally invariant condition which is necessary and sufficient for the existence of a Kähler metric in the conformal class. In the anti–self–dual case we establish a one to one correspondence between Kähler metrics in the conformal class and non–zero parallel sections of a certain connection on a natural rank ten vector bundle over M. We use this characterisation to provide examples of ASD metrics which are not conformal to Kähler.

We establish a link between the ‘conformal to Kähler condition’ in dimension four and the metrisability of projective structures in dimension two. A projective structure on a surface U is metrisable if and only if the induced (2, 2) conformal structure on M = TU admits a Kähler metric or a para–Kähler metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 3 , May 2010 , pp. 485 - 503
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[1]Atiyah, M. F., Hitchin, N. J. and Singer, I. M.Self-duality in four-dimensional Riemannian geometry. Proc. Lon. Math. Soc A 362 (1978), 425–461.Google Scholar
[2]Apostolov, V. and Gauduchon, P.The Riemannian Goldberg-Sachs theorem. Internat. J. Math. 8 (1997), 421–439.CrossRefGoogle Scholar
[3]Bailey, T. N., Eastwood, M. G. and Gover, A. R.Thomas's structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. 24 (1994), 1191–1217.CrossRefGoogle Scholar
[4]Bryant, R. L.Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor. Global analysis and harmonic analysis. Semin. Congr. 4 (2000), 63–94.Google Scholar
[5]Bryant, R. L., Dunajski, M. and Eastwood, M. (2008) Metrisability of two-dimensional projective structures arXiv:0801.0300v1, to appear in J. Diff. Geom.CrossRefGoogle Scholar
[6]Chave, T., Valent, G. and Tod, K. P.(4, 0) and (4, 4) sigma models with a tri-holomorphic Killing vector. Phys. Lett. B 383 (1996), 262–270.CrossRefGoogle Scholar
[7]Derdziński, A.Self-dual Khler manifolds and Einstein manifolds of dimension four. Compositio Math. 49 (1983), 405–433.Google Scholar
[8]Dunajski, M.Anti-self-dual four-manifolds with a parallel real spinor, Proc. Royal Soc. Lond. A 458 (2002), 1205–1222.CrossRefGoogle Scholar
[9]Dunajski, M. and West, S.Anti-self-dual conformal structures from projective structures. Comm. Math. Phys. 272 (2007), 85–118.CrossRefGoogle Scholar
[10]Kim, J., LeBrun, C. and Pontecorvo, M.Scalar-flat Kähler surfaces of all genera. J. Reine Angew. Math. 486 (1997), 69–95.Google Scholar
[11]LeBrun, C. R.-space with a cosmological constant. Proc. Royal. Soc. London Ser. A 380, no. 1778 (1982), 171–185.Google Scholar
[12]LeBrun, C.On the topology of self-dual 4-manifolds. Proc. Amer. Math. Soc. 98 (1986), 637–640.CrossRefGoogle Scholar
[13]LeBrun, C.R.Explicit self-dual metrics on ℂℙ2# ··· #ℂℙ2. J. Diff. Geom. 34 (1991), 233–253.Google Scholar
[14]LeBrun, C. and Maskit, B.On optimal 4-dimensional metrics. J. Geom. Anal. 18 (2008), 537–564.CrossRefGoogle Scholar
[15]Nurowski, P., Sparling, G. A. J.Three-dimensional Cauchy–Riemann structures and second-order ordinary differential equations, Class. Quant. Grav. 20 (2003), 4995–5016.CrossRefGoogle Scholar
[16]Papadopoulos, G.Elliptic monopoles and (4, 0)-supersymmetric sigma models with torsion. Phys. Lett. B 356 (1995), 249–255.CrossRefGoogle Scholar
[17]Penrose, R.Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7 (1976), 31–52.CrossRefGoogle Scholar
[18]Penrose, R. and Rindler, W.Spinors and space-time. Two-spinor calculus and relativistic fields. Cambridge Monogr. Math. Phys. (Cambridge University Press, 1987, 1988).Google Scholar
[19]Pontecorvo, M.On twistor spaces of anti-self-dual hermitian surfaces. Trans. Amer. Math. Soc. 331 (1992), 653–661.CrossRefGoogle Scholar
[20]Przanowski, M.Killing vector fields in self-dual, Euclidean Einstein spaces with Λ ≠ 0. J. Math. Phys. 32 (1991), 1004–1010.CrossRefGoogle Scholar
[21]Rollin, Y., Singer, M.Non-minimal scalar-flat Kähler surfaces and parabolic stability. Invent. Math. 162 (2005), 235–270.CrossRefGoogle Scholar
[22]Semmelmann, U.Conformal Killing forms on Riemannian manifolds. Math. Z. 245 (2003), 503–527.CrossRefGoogle Scholar
[23]Tod, K. P. (1995) The SU(∞)-Toda field equation and special four-dimensional metrics.Geometry and physics (Aarhus, 1995), 307–312. Lecture Notes in Pure and Appl. Math., 184. (Dekker, 1997).CrossRefGoogle Scholar
[24]Walker, A. G. Riemann extensions of non-Riemannian spaces. In Convegno di Geometria Differenziale. (Venice), (1953).Google Scholar
[25]Ward, R.S.On self-dual gauge fields. Phys. Lett. 61A (1977), 81–2.CrossRefGoogle Scholar
[26]Yano, K. and Ishihara, S.Tangent and cotangent bundles: differential geometry. Pure and Appl. Math. 16 (Marcel Dekker, 1973).Google Scholar