Spin-harmonic structures and nilmanifolds

Thumbnail Image
Official URL
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim = 4, 5), SU(3) (dim = 6) and G2 (dim = 7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.
UCM subjects
Unesco subjects
[1] Ilka Agricola, Simon G. Chiossi, Thomas Friedrich, and Jos Höll. Spinorial description of SU(3)- and G2-manifolds. J. Geom. Phys., 98:535–555, 2015. (cited on p. 1, 2, 8, 10, 11, 12) [2] Bernd Ammann and Christian Bär. The Dirac operator on nilmanifolds and collapsing circle bundles. Ann. Global Anal. Geom., 16(3):221–253, 1998. (cited on p. 20) [3] Christian Bär. Real Killing spinors and holonomy. Comm. Math. Phys., 154(3):509–521, 1993. (cited on p. 1) [4] Christian Bär. On nodal sets for Dirac and Laplace operators. Comm. Math. Phys., 188(3):709–721, 1997. (cited on p. 2) [5] Helga Baum, Thomas Friedrich, Ralf Grunewald, and Ines Kath. Twistors and Killing spinors on Riemannian manifolds, volume 124 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991. With German, French and Russian summaries. (cited on p. 1) [6] Giovanni Bazzoni and Vicente Muñoz. Classification of minimal algebras over any field up to dimension 6. Trans. Amer. Math. Soc., 364(2):1007–1028, 2012. (cited on p. 3, 24, 29, 31) [7] Lucio Bedulli and Luigi Vezzoni. The Ricci tensor of SU(3)-manifolds. J. Geom. Phys., 57(4):1125–1146, 2007. (cited on p. 11) [8] Lucio Bedulli and Luigi Vezzoni. Torsion of SU(2)-structures and Ricci curvature in dimension 5. Differential Geom. Appl., 27(1):85–99, 2009. (cited on p. 1, 13) [9] Robert L. Bryant. Some remarks on G2-structures. In Proceedings of Gökova GeometryTopology Conference 2005, pages 75–109. Gökova Geometry/Topology Conference (GGT), Gökova, 2006. (cited on p. 10) [10] Francisco M. Cabrera. On Riemannian manifolds with Spin(7)-structure. Publ. Math. Debrecen, 46(3-4):271–283, 1995. (cited on p. 2) [11] Simon Chiossi and Simon Salamon. The intrinsic torsion of SU(3) and G2 structures. In Differential geometry, Valencia, 2001, pages 115–133. World Sci. Publ., River Edge, NJ, 2002. (cited on p. 10) [12] Diego Conti and Simon Salamon. Generalized Killing spinors in dimension 5. Trans. Amer. Math. Soc.,35(11):5319–5343, 2007. (cited on p. 1, 12, 13, 19) [13] Marisa Fernández. A classification of Riemannian manifolds with structure group Spin(7). Ann. Mat. Pura Appl. (4), 143:101–122, 1986. (cited on p. 2, 9) [14] Marisa Fernández. A new example of compact Riemannian manifold with structure group Spin(7). Portugal. Math., 44(2):161–165, 1987. (cited on p. 2) [15] Thomas Friedrich. Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr., 97:117–146, 1980. (cited on p. 1) [16] Thomas Friedrich. Dirac operators in Riemannian geometry, volume 25 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000. Translated from the 1997 German original by Andreas Nestke. (cited on p. 3, 4, 5, 20) [17] Nigel Hitchin. Harmonic spinors. Advances in Math., 14:1–55, 1974. (cited on p. 2) [18] Nigel Hitchin. The geometry of three-forms in six dimensions. J. Differential Geom., 55(3):547–576, 2000. (cited on p. 10) [19] Stefan Ivanov. Connections with torsion, parallel spinors and geometry of Spin(7) manifolds. Math. Res. Lett., 11(2-3):171–186, 2004. (cited on p. 2) [20] H. Blaine Lawson, Jr. and Marie-Louise Michelsohn. Spin geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1989. (cited on p. 3, 4) [21] Lucía Martín-Merchán. Spinorial classification of Spin(7) structures. 2018. (cited on p. 2, 8, 9) [22] Andrei Moroianu and Uwe Semmelmann. Parallel spinors and holonomy groups. J. Math. Phys., 41(4):2395–2402, 2000. (cited on p. 1) [23] Dietmar A. Salamon and Thomas Walpuski. Notes on the octonions. In Proceedings of the Gökova Geometry-Topology Conference 2016, pages 1–85. Gökova Geometry/Topology Conference (GGT), Gökova, 2017. (cited on p. 9, 10) [24] Simon Salamon. Riemannian geometry and holonomy groups, volume 201 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. (cited on p. 9, 10)