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Asymptotically Euclidean manifolds and twistor spinors. (English) Zbl 0929.53023
Commun. Math. Phys. 196, No. 1, 67-76 (1998); erratum ibid. 207, 735 (1999).
The paper under review continues the considerations of the authors on twistor-spinors initiated mainly by investigations of A. Lichnerowicz [Lett. Math. Phys. 18, 333-345 (1989; Zbl 0685.53017); J. Geom. Phys. 5, 1-18 (1988; Zbl 0678.53018)] and Th. Friedrich [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 22, 59-75 (1990; Zbl 0703.53012); with O. Pokorna, ibid. 26, 149-154 (1991; Zbl 0760.53033)]. In [W. Kühnel and H.-B. Rademacher, Int. J. Math. 5, 877-895 (1994; Zbl 0818.53054)], the authors proved that a Riemannian spin manifold admitting a nontrivial twistor-spinor with zeros and with nontrivial associated conformal field is conformally flat.
In the present paper, all Riemannian spin manifolds carrying a twistor-spinor with at least one zero are classified. In particular, outside the zero set the metric is conformal to either a flat metric or a Ricci flat and locally irreducible metric. Furthermore, using the Berger-Simons holonomy classification, one obtains that the dimension of the manifold is either even or 7.

53C27 Spin and Spin\({}^c\) geometry
53C28 Twistor methods in differential geometry
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