Abstract
In [17] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler manifolds. In the present article we study the limiting case, i.e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kähler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevskii, D. V.: Riemannian spaces with exceptional holonomy groups, Funktional Anal. Appl. 2 (1968), 97–105.
Alekseevskii, D. V.: Compact quaternion spaces, Funktional Anal. Appl. 2 (1968), 106–114.
Alekseevskii, D. V. and Marchiafava, S.: Transformations of a quaternionic Kähler manifold C.R. Acad. Sci. Paris 320 (1995), 703–708.
Bär, Ch.: Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993), 509–521.
Barker, R. and Salamon, S.: Analysis on a generalized Heisenberg group, J. London Math. Soc. 28 (1983), 184–192.
Bonan, E.: Sur les G-structures de type quaternionien, Cahiers Top. et Geom. Diff. 9 (1967), 389–461.
Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 ( 1980), 117–146.
Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac Operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151–162.
Hijazi, O.: Eigenvalues of the Dirac operator on Compact Kähler Manifolds, Comm. Math. Phys. 160 (1994), 563–579.
Hijazi, O. and Milhorat, J.-L.: Décomposition du fibré des spineurs d’une variété spin Kähler-quaternionienne sous l’action de la 4-forme fondametale, J. Geom. Phys. 15 (1995), 320–332.
Hijazi, O. and Milhorat, J.-L.: Minoration des valeurs propres de l’opérateur de Dirac sur les variétés spin Kähler-quaternioniennes, J. Math. Pures Appl. 74 (1995), 387–414.
Hijazi, O. and Milhorat, J.-L.: Twistor operators and eigenvalues of the Dirac operator on compact quaternionic-Kähler spin manifolds, Ann. Global Anal. Geom. 15 (1997), 117–131.
Ishihara, S.: Quaternionic Kählerian manifolds, J. Diff. Geom. 9 (1974), 483–500.
Kirchberg, K.-D.: An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Global Anal. Geom. 4 (1986), 291–325.
Kirchberg, K.-D.: The first eigenvalue of the Dirac operator on Kähler manifolds, J. Geom. Phys. 7 (1990), 449–468.
Kraines, V. Y.: Topology of quaternionic manifolds, Trans. Amer. Math. Soc. 122 (1966), 357–367.
Kramer, W., Semmelmann, U., and Weingart, G.: Eigenvalue estimates for the Dirac operator on quaternionic Kähler manifolds, Preprint 507, SFB 256, Bonn, 1997.
Kramer, W., Semmelmann, U., and Weingart, G.: The first eigenvalue of the Dirac operator on quaternionic Kähler manifolds, Preprint (90), Max-Planck-Institut für Mathematik, Bonn, 1997.
LeBrun, C. and Salamon, S.: Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118(1) (1994), 109–132.
LeBrun, C.: Fano Manifolds, contact structures and quaternionic geometry, Int. J. of Math. 6(3) (1995), 419–437.
Lichnerowicz, A.: Spin manifolds, Killing spinors and the universality of the Hijazi inequality, Lett. Math. Phys. 13 (1987), 331–344.
Lichnerowicz, A.: La première valeur propre de l’opérateur de Dirac pour une variètès Kählèrienne et son cas limite, C.R. Acad. Sci. Paris 311 (1990), 717–722.
Milhorat, J.-L.: Spectre de l’opérateur de Dirac sur les espaces projectifs quaternioniens, C.R. Acad. Sci. Paris 314 (1992), 69–72.
Moroianu, A.: La première valeur propre de l’opérateur de Dirac sur les variétés kählériennes compactes, Comm. Math. Phys. 169 (1995), 373–384.
Nagatomo, Y. and Nitta, T.: Vanishing theorems for quaternionic complexes, Preprint, 1996.
Poon, Y. S. and Salamon, S. M.: Eight-dimensional quaternionic Kähler manifolds with positive scalar curvature, J. Differential Geom. 33 (1991), 363–378.
Salamon, S. M.: Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143–171.
Wang, M. Y. and Ziller, W.: Symmetric spaces and strongly irreducible spaces, Math. Ann. 296 (1993), 285–326.
Wang, M. Y.: Parallel spinors and parallel forms, Anal. Global Anal. Geom. 7 (1989), 59–68.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kramer, W., Semmelmann, U. & Weingart, G. Quaternionic Killing Spinors. Annals of Global Analysis and Geometry 16, 63–87 (1998). https://doi.org/10.1023/A:1006545828324
Issue Date:
DOI: https://doi.org/10.1023/A:1006545828324