## The Einstein-Dirac equation on Riemannian spin manifolds.(English)Zbl 0961.53023

For a Riemannian spin manifold $$(M,g)$$ we denote by $$D_g$$ the Dirae operator associated to $$g$$, by $$Ric_g$$ the Ricci-tensor and by $$S_g$$ the scalar curvature. The aim of the paper is to find exact solutions $$(g,\psi)$$ of the Riemannian Dirac-Einstein equation on a closed manifold: $$D_g\psi= \lambda \psi$$, $$Ric_g-{1\over 2}S_g \cdot g={\varepsilon \over 4}T_{(g,\psi)}$$, $$\varepsilon= \pm 1$$, $$\lambda\in \mathbb{R}$$. Special solutions of the Einstein-Dirac equation are weak Killing spinors, in dimension 3 they are the only ones. As an example special 1-connected Sasakian spin manifolds $$M^{2m+1}$$, $$m\geq 2$$, are considered. If the Ricci-tensor is related in a certain way to the contact form, then this Sasakian manifold admits weak Killing spinors. Even dimensional examples with Dirac-Einstein-spinors which are not weak Killing spinors are produced by considering products of 1-connected a nearly-Kähler manifold $$M^6$$ with manifold admitting usual geometric Killing spinors. The 3-dimensional case is examined more in detail.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C27 Spin and Spin$${}^c$$ geometry
Full Text:

### References:

 [1] Bär, C., Real Killing spinors and holonomy, Commun. math. phys., 154, 509-521, (1993) · Zbl 0778.53037 [2] Baum, H.; Friedrich, Th.; Grunewald, R.; Kath, I., Twistors and Killing spinors on Riemannian manifolds, (1991), Teubner Leipzig/Stuttgart · Zbl 0734.53003 [3] Bleecker, D., Gauge theory and variation principles, (1981), Addison-Wesley MA · Zbl 0481.58002 [4] Booss-Bavnbek, B.; Wojciechowski, K.P., Elliptic boundary problems for Dirac operators, (1993), Birkhäuser Basel · Zbl 0836.58041 [5] Bourguignon, J.P.; Gauduchon, P., Spineurs, opérateurs de Dirac et variations de Métriques, Commun. math. phys., 144, 581-599, (1992) · Zbl 0755.53009 [6] C.P. Boyer, K. Galicki, On Sasakian-Einstein geometry, math. DG/9811098. · Zbl 1022.53038 [7] Boyer, C.P.; Galicki, K., 3-Sasakian manifolds [8] Cahen, M.; Gutt, S.; Trautman, A., Pin structures and the modified Dirac operator, J. geom. phys., 17, 283-297, (1995) · Zbl 1010.58023 [9] Finster, F., Local U(2,2) symmetry in relativistic quantum mechanics, J. math. phys., 39, 12, 6276-6290, (1998) · Zbl 0935.81004 [10] F. Finster, J. Smoller, S.-T. Yau, Particle-like solutions of the Einstein-Dirac equations, gr-qc/9801079, Phys. Rev. D, to appear. [11] F. Finster, J. Smoller, S.-T. Yau, Particle-like solutions of the Einstein-Dirac-Maxwell equations, gr-qc/9802012. [12] F. Finster, J. Smoller, S.-T. Yau, Non-existence of black hole solutions for a spherically symmetric Einstein-Dirac-Maxwell system, gr-qc/9810048. · Zbl 0939.83037 [13] Finster, F.; Smoller, J.; Yau, S.-T., The coupling of gravity to spin and electromagnetism, (1999), preprint [14] Friedrich, Th., Der erste eigenwert des Dirac-operators einer kompakten riemannschen mannigfaltigkeit nichtnegativer skalarkrümmung, Math. nachr., 97, 117-146, (1980) · Zbl 0462.53027 [15] Friedrich, Th., Dirac-operatoren in der riemannschen geometrie, (1997), Vieweg Braunschweig Wiesbaden [16] Friedrich, Th.; Kath, I., Variétés riemanniennes compactes de dimension 7 admettant des spineurs de Killing, C.R. acad. sci. Paris serie I, 307, 967-969, (1988) · Zbl 0659.53017 [17] Friedrich, Th.; Kath, I., Einstein manifolds of dimension 5 with small first eigenvalue of the Dirac operator, J. diff. geom., 29, 2, 263-279, (1989) · Zbl 0633.53069 [18] Friedrich, Th.; Kath, I., Seven-dimensional compact Riemannian manifolds with Killing spinors, Commun. math. phys., 133, 543-561, (1990) · Zbl 0722.53038 [19] Friedrich, Th.; Kath, I.; Moroianu, A.; Semmelmann, U., On nearly parallel G2-structures, J. geom. phys., 23, 259-286, (1997) · Zbl 0898.53038 [20] Grunewald, R., Six-dimensional Riemannian manifolds with a real Killing spinor, Ann. glob. anal. geom., 8, 43-59, (1990) · Zbl 0704.53050 [21] Hsiung, C.C., Almost complex and complex structures, (1995), World Scientific Singapore · Zbl 0838.53050 [22] Kath, I., Variétés riemanniennes de dimension 7 admettant un spineur de Killing réel, C.R. acad. sci. Paris serie I, 311, 553-555, (1990) · Zbl 0715.53035 [23] Kim, E.C., Die Einstein-Dirac-gleichung über riemannschen spin-mannigfaltigkeiten, () · Zbl 1009.58503 [24] Kim, E.C., Lower eigenvalue estimates for non-vanishing eigenspinors of the Dirac operator, (1999), preprint [25] Kim, E.C., The Einstein-Dirac equation on pseudo-Riemmanian spin manifolds, (1999), preprint [26] Kosmann, Y., Derivees de Lie des spineurs, Ann. mat. pura ed appl., 91, 317-395, (1972) · Zbl 0231.53065 [27] Lichnerowicz, A., Spin manifolds, Killing spinors and the universality of hijazi inequality, Lett. math. phys., 13, 331-344, (1987) · Zbl 0624.53034 [28] Lichnerowicz, A., Sur LES resultates de H. baum et th. friedrich concernant LES spineurs de Killing a valeur propre imaginaire, C.R. acad. sci. Paris serie I, 306, 41-45, (1989) · Zbl 0676.53052 [29] Schrödinger, E., Diracsches elektron im schwerefeld I, (), 436-460 · JFM 58.0937.01 [30] Tanno, S., The topology of contact Riemannian manifolds, Illinois J. math., 12, 700-717, (1968) · Zbl 0165.24703 [31] van Nieuwenhuizen, P.; Warner, N.P., Integrability conditions for Killing spinors, Commun. math. phys., 93, 277-284, (1984) · Zbl 0549.53011 [32] Wang, M., Preserving parallel spinors under metric deformations, Indiana univ. math. J., 40, 815-844, (1991) · Zbl 0724.53031 [33] Yano, K.; Kon, M., Structures on manifolds, (1984), World Scientific Singapore · Zbl 0557.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.