zbMATH — the first resource for mathematics

Twistor spinors and gravitational instantons. (English) Zbl 0860.53029
The authors construct a complete Riemannian metric on the four-dimensional real vector space $$R$$ which carries a two-dimensional space of twistor spinors having a common zero point. This metric is half conformally flat, but not conformally flat. Their construction uses a conformal completion at infinity of the Eguchi-Hanson metric on the exterior of a closed ball in $$R$$. Contents include: an introduction; the Eguchi-Hanson metric; half-conformally flat metrics admitting twistor spinors with zeros. The paper concludes with a list of references containing twenty items.

MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
Full Text:
References:
 [1] Atiyah, M. F. Hitchin, N. J. and Singer, I. M.: Self-duality in four-dimensional Riemannian geometry,Proc. Roy. Soc. London A 362 (1978), 425–461. · Zbl 0389.53011 [2] Bär, C., Real Killing spinors and holonomy,Comm. Math. Phys. 154 (1993), 509–521. · Zbl 0778.53037 [3] Bailey, T. N., Eastwood, M. G. and Gover, A. R.: Thomas’s structure bundle for conformal, projective and related structures,Rocky Mountains J. Math. 24 (1994), 1191–1217. · Zbl 0828.53012 [4] Baum, H., Friedrich, T., Grunewald, R. and Kath, I.:Twistors and Killing spinors on riemannian manifolds, Teubner Texte zur Math.. vol. 124, B. G. Teubner, Stuttgart, Leipzig, 1991. · Zbl 0734.53003 [5] Besse, A. L.:Einstein Manifolds, Ergebnisse der Math. 3. Folge, Band 10, Springer, Berlin, Heidelberg, New York, 1987. · Zbl 0613.53001 [6] Duff, M. J. and Nilsson, B. E. W.: Kaluza-Klein supergravity,Phys. Rep. 130 (1986), 1–142. [7] Eguchi, T. and Hanson, A. J.: Asymptotically flat self-dual solutions to euclidean gravity,Phys. Lett. B. 74 (1978), 249–251. [8] Eguchi, T., Gilkey, P. B. and Hanson, A. J.: Gravitation, gauge theories and differential geometry,Phys. Rep. 66 (1980), 213–393. [9] Friedrich, T.: On the conformal relation between twistors and Killing spinors. (Proc. Winterschool on geometry and physics, Srni 1989),Suppl. Rend. Circ. Mat. Palermo Serie II 22 (1989), 59–75. [10] Gibbons, G. W. and Hawking, S. W.: Classification of gravitational instanton symmetries,Comm. Math. Phys. 66 (1979), 291–310. [11] Gibbons, G. W. and Pope, C. N.: The positive action conjecture and asymptotically euclidean metrics in quantum gravity,Comm. Math. Phys. 66 (1979), 267–290. [12] Kronheimer, P. B.: ALE spaces as hyper-Kähler quotients,J. Differential Geom. 29 (1989), 665–683. · Zbl 0671.53045 [13] Kühnel, W. and Rademacher, H. B.: Twistor spinors with zeros,Int. J. Math. 5 (1994), 877–895. · Zbl 0818.53054 [14] Lichnerowicz, A.: Killing spinors, twistor-spinors and Hijazi inequality,J. Geom. Phys. 5 (1988), 2–18. · Zbl 0678.53018 [15] Lichnerowicz, A.: On the twistor spinors,Lett. Math. Phys. 18 (1989), 333–345. · Zbl 0685.53017 [16] Lichnerowicz, A.: Sur les zéros des spineurs-twisteurs,C.R. Acad. Sci. Paris, Série I,310 (1990), 19–22. · Zbl 0687.53043 [17] Nieuwenhuizen, P. van and Warner, N. P.: Integrability conditions for Killing spinors,Comm. Math. Phys. 93 (1984), 227–284. · Zbl 0549.53011 [18] Penrose, P. and Rindler, R.:Spinors and Space Time vol. 2, Cambridge Monogr. Math. Phys. 1986. · Zbl 0591.53002 [19] Schoen, R. S. and Yau, S. T.: Proof of the positive-action conjecture in quatum relativity,Phys. Rev. Lett. 42 (1979), 547–548. [20] Wang, M.: Parallel spinors and parallel forms,Ann. Global Anal. Geom. 7 (1989), 59–68. · Zbl 0688.53007
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.