Abstract
In this paper we prove the Spinc analog of the Hijazi inequality on the first eigenvalue of the Dirac operator on compact Riemannian manifolds and study its equality case. During this study, we are naturally led to consider generalized Killing spinors on Spinc manifolds and we prove that such objects can only exist on low-dimensional manifolds (up to dimension three). This allows us to give a nice geometrical description of the manifolds satisfying the equality case of the above-mentioned inequality and to classify them in dimension three and four.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atiyah, M., Hitchin, N. and Singer, I.: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London A 362 (1978), 425–461.
Bando, S. and Mabuchi, T.: Uniqueness of Kähler-Einstein metrics modulo connected group actions, in Algebraic Geometry, Sendai, Adv. Stud. Pure Math. 10, Academic Press, Boston, 1987, pp. 11–40.
Bär, C.: Lower eigenvalues estimates for Dirac operators, Math. Ann. 293 (1992), 39–46.
Bär, C.:, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509–521.
Barth, W., Peters, C. and Van de Ven, A.: Compact Complex Surfaces, Ergeb. Math. Grenzgeb. 4, Springer-Verlag, Berlin, 1984.
Baum, H.: Eigenvalue estimates for Dirac operators coupled to instantons, Humboldt University, Berlin, SFB 288, Preprint 88, 1993.
Baum, H., Friedrich, T., Grunewald, R. and Kath, I.: Twistors and Killing Spinors in Riemannian Geometry, Teubner, Berlin, 1990.
Besse, A. L.: Einstein Manifolds, Ergeb. Math. Grenzgeb. 10, Springer-Verlag, Berlin, 1987.
Blair, D. E.: Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer-Verlag, Berlin, 1976.
Friedrich, Th.: Der erste Eigenwert des Dirac Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 (1980), 117–146.
Friedrich, Th., Kath, I., Moroianu, A. and Semmelmann, U.: On nearly-parallel G 2-structures, J. Geom. Phys. 23 (1997), 259–286.
Gauduchon, P.: Hermitian connections and dirac operators, Bolletino UMI (7) 11-B (1997), Suppl. fasc. 2, 257–288.
Gursky, M. and LeBrun, C.: Yamabe invariants and Spinc structures, Preprint, dg-ga/9708002, 1997.
Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151–162.
Hitchin, N. L.: On compact four-dimensional Einstein manifolds, J. Differential Geom. 9 (1974), 435–442.
Kronheimer, P. and Mrowka, T.: The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797–808.
Lawson, H. B. and Michelsohn, M. L.: Spin Geometry, Princeton Math. Ser. 38, Princeton University Press, Princeton, NJ, 1989.
Lott, J.: Eigenvalue bounds for the Dirac Operator, Pacific J. Math. 125 (1986), 117–128.
Maier, S.: Generic metrics and connections on Spin-and Spinc-manifolds, Comm. Math. Phys. 188 (1997), 407–437.
Moroianu, A.: On the infinitesimal isometries of manifolds with Killing spinors, Humboldt University, Berlin, SFB 288, Preprint 240, 1996.
Moroianu, A.: Parallel and Killing spinors on Spinc manifolds, Comm. Math. Phys. 187 (1997), 417–428.
Obata, M.: The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), 247–258.
Schrödinger, E.: Diracsches Elektron im Schwerfeld I, Sitzungsbericht der Preusischen Akademie der Wissenschaften Phys.-Math. Klasse 1932, Verlag der Akademie der Wissenschaften Berlin, 1932, pp. 436–460.
Tian, G.: On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), 101–172.
Wolf, J.: Spaces of Constant Curvature, McGraw-Hill, New York, 1967.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Herzlich, M., Moroianu, A. Generalized Killing Spinors and Conformal Eigenvalue Estimates for Spinc Manifolds. Annals of Global Analysis and Geometry 17, 341–370 (1999). https://doi.org/10.1023/A:1006546915261
Issue Date:
DOI: https://doi.org/10.1023/A:1006546915261