7-dimensional compact Riemannian manifolds with Killing spinors. (English) Zbl 0722.53038

Let M be a compact Riemannian spin manifold. A section \(\psi\) of the spin bundle is called a Killing spinor if \(\nabla_ X\psi =\lambda X\cdot \psi,\) where \(\lambda \neq 0\) is a (real) constant and X any tangent vector. The authors show that M has at least two independent Killing spinors if dim(M) is odd and M is simply connected and Einstein-Sasaki. In dimension 7 they classify (in a certain sense) the cases with two or three independent Killing or parallel spinors. Generalizations of these results have been obtained recently by Ch. Bär (Preprint, Bonn 1991).
Reviewer: W.Ballmann (Bonn)


53C20 Global Riemannian geometry, including pinching
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[1] Atiyah, M.F., Hitchin, N., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A326, 425–461 (1978) · Zbl 0389.53011
[2] Baum, H.: Odd-dimensional Riemannian manifolds with imaginary Killing spinors. Ann. Global Anal. Geom.7, 2 (1989) · Zbl 0708.53039
[3] Baum, H.: Complete Riemannian manifolds with imaginary Killing spinors. Ann. Global Anal. Geom. (to appear) · Zbl 0694.53043
[4] Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. Lecture Notes in Mathematics, Vol. 194. Berlin, Heidelberg, New York: Springer 1971 · Zbl 0223.53034
[5] Blair, D.: Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics, Vol. 509. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0319.53026
[6] Duff, M.J., Nilsson, B., Pope, C.N.: Kaluza-Klein supergravity. Phys. Rep.130, 1–142 (1986)
[7] Fischer, A.E., Wolf, J.A.: The structure of compact Ricci-flat Riemannian manifolds. J. Diff. Geom.10, 277–288 (1975) · Zbl 0313.53020
[8] Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr.97, 117–146 (1980) · Zbl 0462.53027
[9] Friedrich, Th.: A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds. Math. Nachr.102, 53–56 (1981) · Zbl 0481.53039
[10] Friedrich, Th.: Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten. Colloq. Math.44, 277–290 (1981) · Zbl 0487.53016
[11] Friedrich, Th., Grunewald, R.: On the first eigenvalue of the Dirac operator on 6-dimensional manifolds. Ann. Global Anal. Geom.3, 265–273 (1985) · Zbl 0577.58034
[12] Friedrich, Th., Kath, I.: Einstein manifolds of dimension five with small eigenvalue of the Dirac operator, J. Diff. Geom.29, 263–279 (1989) · Zbl 0633.53069
[13] Friedrich, Th., Kath, I.: Compact 5-dimensional Riemannian manifolds with parallel spinors, Math. Nachr. (to appear) · Zbl 0709.53022
[14] Friedrich, Th., Kath, I.: Variétés Riemanniennes compactes de dimension 7 admettant des spineurs de Killing. C.R. Acad. Sci. Paris Ser. I. t.307, 967–969 (1988) · Zbl 0659.53017
[15] Friedrich, Th., Kurke, H.: Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature. Math. Nachr.106, 271–299 (1982) · Zbl 0503.53035
[16] Friedrich, Th. (Ed.): Self-dual Riemannian geometry and instantons. Leipzig: Teubner 1981 · Zbl 0471.00021
[17] Futaki, A.: Kähler-Einstein metrics and integral invariants. Lecture Notes in Mathematics, Vol. 1314. Berlin, Heidelberg, New York: Springer 1988 · Zbl 0646.53045
[18] 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. Paris, Ser. I303, 417–419 (1986) · Zbl 0606.53024
[19] Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys.104, 151–162 (1986) · Zbl 0593.58040
[20] Hitchin, N.: Compact four-dimensional Einstein manifolds. J. Diff. Geom.9, 435–441 (1974) · Zbl 0281.53039
[21] Hitchin, N.: Kählerian twistor spaces. Proc. Lond. Math. Soc.43, 133–150 (1981) · Zbl 0474.14024
[22] Husemoller, D.: Fibre bundles. New York: 1966 · Zbl 0144.44804
[23] Ishihara, S., Konishi, M.: Differential geometry of fibred spaces. Kyoto: 1973 · Zbl 0337.53001
[24] Kirchberg, K.D.: Compact six-dimensional Kähler spin manifolds of positive scalar curvature with the smallest positive first eigenvalue of the Dirac operator. Math. Ann.282, 157–176 (1988) · Zbl 0628.53061
[25] Kobayashi, S.: On compact Kähler manifolds with positive definite Ricci tensor. Ann. Math.74, 570–574 (1961) · Zbl 0107.16002
[26] Kobayashi, S.: Topology of positively pinched Kähler geometry. Tohoku Math. J.15, 121–139 (1963) · Zbl 0114.37601
[27] Koiso, N., Sakane, Y.: Non-homogeneous Kähler-Einstein metrics on compact complex manifolds. Preprint · Zbl 0591.53056
[28] Kostant, B.: Quantization and unitary representations. Lecture Notes in Mathematics, Vol. 170, pp. 87–207. Berlin, Heidelberg, New York: Springer 1970 · Zbl 0223.53028
[29] Kreck, M., Stolz, St.: A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds withSU(3){\(\times\)}SU(2){\(\times\)}U(1)-symmetry. Ann. Math.127, 373–388 (1988) · Zbl 0649.53029
[30] Lichnerowicz, A.: Spin manifolds, Killing spinors, and universality of the Hijazi inequality. Lett. Math. Phys.13, 331–344 (1987) · Zbl 0624.53034
[31] Lichnerowicz, A.: Les spineurs-twisteurs sur une varieté spinorielle compacte. C.R. Acad. Sci. Paris, Ser. I306, 381–385 (1988) · Zbl 0641.53014
[32] Nieuwenhuizen, P. van, Warner, N.P.: Integrability conditions for Killing spinors. Commun. Math. Phys.93, 277–284 (1984) · Zbl 0549.53011
[33] Postnikov, M.M.: Lectures on Geometry V, Lie groups and algebras. Moscow: 1982 (Russ.) · Zbl 0597.22001
[34] Salamon, S.: Topics in four-dimensional Riemannian geometry. In: Geometry Seminar ”Luigi Bianchi”. Lecture Notes in Mathematics, Vol. 1022. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0532.53035
[35] Tian, G.: On Kähler-Einstein metrics on certain Kähler manifolds withc 1(M)>0. Invent. Math.89, 225–246 (1987) · Zbl 0599.53046
[36] Tian, G., Yau, S.-T.: Kähler-Einstein metrics on complex surfaces withc 1>0. Commun. Math. Phys.112, 175–203 (1987) · Zbl 0631.53052
[37] Séminaire Palaiseau, Géométrie des surfacesK3: modules et périodes. Asterisque126 (1985)
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