Abstract
We construct a complete Riemannian metric on the four-dimensional vector space ℝ4 which carries a two-dimensional space of twistor spinor with common zero point. This metric is half-conformally flat but not conformally flat. The construction uses a conformal completion at infinity of theEguchi-Hanson metric on the exterior of a closed ball in ℝ4.
Similar content being viewed by others
References
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.
Bär, C., Real Killing spinors and holonomy,Comm. Math. Phys. 154 (1993), 509–521.
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.
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.
Besse, A. L.:Einstein Manifolds, Ergebnisse der Math. 3. Folge, Band 10, Springer, Berlin, Heidelberg, New York, 1987.
Duff, M. J. and Nilsson, B. E. W.: Kaluza-Klein supergravity,Phys. Rep. 130 (1986), 1–142.
Eguchi, T. and Hanson, A. J.: Asymptotically flat self-dual solutions to euclidean gravity,Phys. Lett. B. 74 (1978), 249–251.
Eguchi, T., Gilkey, P. B. and Hanson, A. J.: Gravitation, gauge theories and differential geometry,Phys. Rep. 66 (1980), 213–393.
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.
Gibbons, G. W. and Hawking, S. W.: Classification of gravitational instanton symmetries,Comm. Math. Phys. 66 (1979), 291–310.
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.
Kronheimer, P. B.: ALE spaces as hyper-Kähler quotients,J. Differential Geom. 29 (1989), 665–683.
Kühnel, W. and Rademacher, H. B.: Twistor spinors with zeros,Int. J. Math. 5 (1994), 877–895.
Lichnerowicz, A.: Killing spinors, twistor-spinors and Hijazi inequality,J. Geom. Phys. 5 (1988), 2–18.
Lichnerowicz, A.: On the twistor spinors,Lett. Math. Phys. 18 (1989), 333–345.
Lichnerowicz, A.: Sur les zéros des spineurs-twisteurs,C.R. Acad. Sci. Paris, Série I,310 (1990), 19–22.
Nieuwenhuizen, P. van and Warner, N. P.: Integrability conditions for Killing spinors,Comm. Math. Phys. 93 (1984), 227–284.
Penrose, P. and Rindler, R.:Spinors and Space Time vol. 2, Cambridge Monogr. Math. Phys. 1986.
Schoen, R. S. and Yau, S. T.: Proof of the positive-action conjecture in quatum relativity,Phys. Rev. Lett. 42 (1979), 547–548.
Wang, M.: Parallel spinors and parallel forms,Ann. Global Anal. Geom. 7 (1989), 59–68.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kühnel, W., Rademacher, HB. Twistor spinors and gravitational instantons. Lett Math Phys 38, 411–419 (1996). https://doi.org/10.1007/BF01815523
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01815523