Abstract
We give a description of all complete simply connected Riemannian manifolds carrying real Killing spinors. Furthermore, we present a construction method for manifolds with the exceptional holonomy groupsG 2 and Spin(7).
Similar content being viewed by others
References
Baum, H.: Variétés riemanniennes admettant des spineurs de Killing imaginaries. C.R. Acad. Sci. Paris309, 47–49 (1989)
Baum, H.: Odd-dimensional Riemannian manifolds with imaginary Killing spinors. Ann. Glob. Anal. Geom.7, 141–153 (1989)
Baum, H.: Complete Riemannian manifolds with imaginary Killing spinors. Ann. Glob. Anal. Geom.7, 205–226 (1989)
Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistor and Killing spinors on Riemannian manifolds. Seminarbericht108, Humboldt-Universität Berlin, 1990
Besse, A.L.: Einstein manifolds. Berlin, Heidelberg, New York: Springer 1987
Blair, D.E.: Contact manifolds in Riemannian geometry. Berlin, Heidelberg, New York: Springer 1976
Bonan, E.: Sur les variétés riemanniennes à groupe d'holonomieG 2 ou Spin(7). C.R. Acad. Sci. Paris262, 127–129 (1966)
Bryant, R.: Metrics with exceptional holonomy. Ann. Math.126, 525–576 (1987)
Bryant, R., Salamon, S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J.58, 829–850 (1989)
Cahen, M., Gutt, S., Lemaire, L., Spindel, P.: Killing spinors. Bull. Soc. Math. Belg.38, 75–102 (1986)
Duff, M.J., Nilsson, B.E.W., Pope, C.N.: Kaluza-Klein supergravity. Phys. Rep.130, 1–142 (1986)
Franc, A.: Spin structures and Killing spinors on lens spaces. J. Geom. Phys.4, 277–287 (1987).
Friedrich, T.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nach.97, 117–146 (1980)
Friedrich, T.: A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds. Math. Nach.102, 53–56 (1981)
Friedrich, T.: Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten. Colloq. Math.44, 277–290 (1981)
Friedrich, T., Grunewald, R.: On the first eigenvalue of the Dirac operator on 6-dimensional manifolds. Ann. Glob. Anal. Geom.3, 265–273 (1985)
Friedrich, T., Kath, I.: Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator. J. Diff. Geom.29, 263–279 (1989)
Friedrich, T., Kath, I.: Variétés riemanniennes compactes de dimension 7 admettant des spineurs de Killing. C.R. Acad. Sci. Paris307, 967–969 (1988)
Friedrich, T., Kath, I.: 7-dimensional compact Riemannian manifolds with Killing spinors. Commun. Math. Phys.133, 543–561 (1990)
Gallot, S.: Equations différentielles caractéristiques de la sphère. Ann. Sc. Ec. Norm. Sup.12, 235–267 (1979)
Gibbons, G.W., Page, D.N., Pope, C.N.: Einstein metrics onS 3, ℝ3 and ℝ4 bundles. Commun. Math. Phys.127, 529–553 (1990)
Gray, A.: Vector cross products on manifolds. Trans. Am. Math. Soc.141, 465–504 (1969)
Gray, A.: Nearly Kähler manifolds. J. Diff. Geom.4, 283–309 (1970)
Gray, A.: Riemannian manifolds with geodesic symmetries of order 3. J. Diff. Geom.7, 343–369 (1972)
Gray, A.: The structure of nearly Kähler manifolds. Math. Ann.223, 233–248 (1976)
Grunewald, R.: Six-dimensional Riemannian manifolds with a real Killing spinor. Ann. Glob. Anal. Geom.8, 43–59 (1990)
Hijazi, O.: Caractérisation de la sphère par les premières valeurs propres de l'opérateur de Dirac en dimension 3, 4, 7 et 8. C.R. Acad. Sci. Paris303, 417–419 (1986)
Hitchin, N.: Harmonic spinors. Adv. Math.14, 1–55 (1974)
Kreck, M., Stolz, S.: Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature. J. Diff. Geom.33, 465–486 (1991)
van Nieuwenhuizen, P., Warner, N.P.: Integrability conditions for Killing spinors. Commun. Math. Phys.93, 277–284 (1984)
O'Neill, B.: Semi-Riemannian geometry. New York: Acad. Press 1983
Rademacher, H.-B.: Generalized Killing spinors with imaginary Killing function and conformal fields. To appear in Proc. Conf. Anal. Glob. Diff. Geom., Berlin 1990, Springer Lecture Notes
Salamon, S.: Riemannian geometry and holonomy groups. London: Longman Scientific & Technical 1989
Wang, M.: Some examples of homogeneous Einstein manifolds in dimension 7. Duke Math. J.49, 23–28 (1982)
Wang, M.: Parallel spinors and parallel forms. Ann. Glob. Anal. Geom.7, 59–68 (1989)
Ziller, W.: Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann.259, 351–358 (1982)
Author information
Authors and Affiliations
Additional information
Communicated by A. Connes
Rights and permissions
About this article
Cite this article
Bär, C. Real Killing spinors and holonomy. Commun.Math. Phys. 154, 509–521 (1993). https://doi.org/10.1007/BF02102106
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02102106