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Conformal completion of \(\mathbb{U}(n)\)-invariant Ricci-flat Kähler metrics at infinity. (English) Zbl 0870.53040
In a recent paper [Lett. Math. Phys. 38, 411-419 (1996; Zbl 0860.53029)], the authors exhibited the first example of a Riemannian spin manifold of \(m=4\) dimensions which is not conformally flat and which admits twistor spinors with zeros. It was then shown that under a conformal change, a pair of linearly independent parallel spinors of the Eguchi-Hanson metric become twistor spinors with zeros at infinity.
The present investigation is intended to extend this result to \(m=2n\) dimensions, \(n\geq 2\), for a \(U(n)\)-invariant cohomogenity one metric which is not conformally flat, and admits twistor spinors with zeros. Their construction makes use of previously known results of E. Calabi, D. Z. Freedman and G. W. Gibbons on the conformal completion of \(U(n)\)-invariant metrics as described in the title of the paper. Presumably this work has implications for the theory of gravitational instantons; however, these are not discussed in the present paper.

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
Full Text: DOI
[1] Baum, II., Friedrich, T., Grunewald, R. and I. Math: Twistors and Killing Spinors on Rie- mannian Manifolds (Teubner-Texte zur Mathematik: Vol. 124). Leipzig und Stuttgart: B.G. Teubner 1991. · Zbl 0734.53003
[2] Calabi, E.: Métriques kiihlerzennes et fibres holomorphes. Ann. Ecol. Norm. Sup. 12 (1979), 269 - 294. · Zbl 0431.53056
[3] Eguchi, ’1’. and A. J. Hanson: Asymptotically flat self-dual solutions to Euclidean gravity. Phys. Lett. 74B (1978), 249- 251.
[4] Freedman, D. Z. and C. W. Gibbons: Remarks on supersymmetry and Kàhler geometry. In: Superspace and Supergravity. Proc. Nuffield workshop, Cambridge 1980 (eds.: S. W. Hawking and M. Roek). Cambridge: Univ. Press 1981.
[5] KühneI, W. and H. B. Rademacher: Twistor spinors with zeros. Int. J. Math. 5 (1994), 877 - 895. · Zbl 0818.53054
[6] Kühnel, W. and H. B. Rademacher: Twistor spinors and gravitational instantons. Lett. Math. Phys. (to appear). · Zbl 0860.53029
[7] Lichncrowicz, A.: Killing spinors, twistor spinors and Hijazi inequality. J. Geom. Physics 5 (1988), 2 - 18. · Zbl 0678.53018
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