Abstract
We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those flows carrying non-trivial solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alexandrov, B., Grantcharov, G., Ivanov, S.: An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting a parallel one-form. J. Geom. Phys. 28(3–4), 263–270 (1998)
Ammann, B.: A variational problem in conformal spin geometry. Habilitation Thesis, Universität Hamburg (2003)
Ammann, B., Bär, C.: The Dirac operator on Nilmanifolds and collapsing circle bundles. Ann. Glob. Anal. Geom. 16, 221–253 (1998)
Bär, C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)
Bär, C.: Extrinsic bounds for eigenvalues of the Dirac operator. Ann. Glob. Anal. Geom. 16, 573–596 (1998)
Baum, H.: Complete Riemannian manifolds with imaginary Killing spinors. Ann. Glob. Anal. Geom. 7, 205–226 (1989)
Baum, H., Friedrich, Th., Grunewald, R., Kath, I.: Twistor and Killing spinors on Riemannian manifolds. Teubner-Texte zur Mathematik, vol. 124. Teubner, Stuttgart (1991)
Belgun, F.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317(1), 1–40 (2000)
Blumenthal, R.: Foliated manifolds with flat basic connection. J. Differ. Geom. 16, 401–406 (1981)
Boyer, C., Galicki, K.: 3-Sasakian manifolds. In: Surveys in Differential Geometry: Essays on Einstein Manifolds. Surv. Differ. Geom., vol. VI, pp. 123–184. International Press, Boston (1999)
Carrière, Y.: Flots Riemanniens. Structure transverse des feuilletages, Toulouse. Astérique 116, 31–52 (1984)
Friedrich, Th.: Zur Abhängigkeit des Dirac-Operators von der Spin-Struktur. Coll. Math. J. 48, 57–62 (1984)
Friedrich, Th., Kim, E.C.: The Einstein Dirac equations on Riemannian spin manifolds. J. Geom. Phys. 33, 128–172 (2000)
Ginoux, N., Habib, G.: A spectral estimate for the Dirac operator on Riemannian flows. Preprint (2007)
Habib, G.: Tenseur d’impulsion-énergie et feuilletages. Ph.D. Thesis, Institut Élie Cartan—Université Henri Poincaré, Nancy (2006)
Hijazi, O.: Spectral properties of the Dirac operator and geometrical structures. In: Proceedings of the Summer School on Geometric Methods in Quantum Field Theory, Villa de Leyva, Colombia, July 12–30, 1999. World Scientific, Singapore (2001)
Kirchberg, K.-D.: An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature. Ann. Glob. Anal. Geom. 4, 291–325 (1986)
Moroianu, A.: Opérateur de Dirac et submersions riemanniennes. Ph.D. Thesis, École Polytechnique (1996)
Okumura, M.: Some remarks on space with a certain contact structure. Tôhoku Math. J. (2) 14, 135–145 (1962)
O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)
Pfäffle, F.: The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35(4), 367–385 (2000)
Reinhart, B.: Foliated manifolds with bundle-like metrics. Ann. Math. 69, 119–132 (1959)
Tanno, S.: The topology of contact Riemannian manifolds. Ill. J. Math. 12, 700–717 (1968)
Tondeur, Ph.: Foliations on Riemannian Manifolds. Springer, New York (1959)
Wang, Mc.K.: Parallel spinors and parallel forms. Ann. Glob. Anal. Geom. 7, 59–68 (1989)
Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill, New York (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Cortés.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Ginoux, N., Habib, G. Geometric aspects of transversal Killing spinors on Riemannian flows. Abh. Math. Semin. Univ. Hambg. 78, 69–90 (2008). https://doi.org/10.1007/s12188-008-0006-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-008-0006-8