Generalized Killing spinors in dimension 5

Conti, Diego; Salamon, Simon

doi:10.1090/S0002-9947-07-04307-3

Generalized Killing spinors in dimension 5

Authors: Diego Conti and Simon Salamon
Journal: Trans. Amer. Math. Soc. 359 (2007), 5319-5343
MSC (2000): Primary 53C25; Secondary 14J32, 53C29, 53C42, 58A15
DOI: https://doi.org/10.1090/S0002-9947-07-04307-3
Published electronically: May 1, 2007
MathSciNet review: 2327032
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, $\mathrm{SU}(2)$ -structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy $\mathrm{SU}(3)$ .

1. B.S. Acharya, F. Denef, C. Hofman, and N. Lambert, Freund-Rubin revisited, hep-th/0308046.
2. Michael Atiyah and Edward Witten, 𝑀-theory dynamics on a manifold of 𝐺₂ holonomy, Adv. Theor. Math. Phys. 6 (2002), no. 1, 1–106. MR 1992874, https://doi.org/10.4310/ATMP.2002.v6.n1.a1
3. Christian Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), no. 3, 509–521. MR 1224089
4. Christian Bär, Paul Gauduchon, and Andrei Moroianu, Generalized cylinders in semi-Riemannian and Spin geometry, Math. Z. 249 (2005), no. 3, 545–580. MR 2121740, https://doi.org/10.1007/s00209-004-0718-0
5. Helga Baum, Thomas Friedrich, Ralf Grunewald, and Ines Kath, Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 124, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991. With German, French and Russian summaries. MR 1164864
6. David E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, vol. 203, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1874240
7. C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surv. Differ. Geom. 7 (1999), 123-184.
8. Robert L. Bryant, Calibrated embeddings in the special Lagrangian and coassociative cases, Ann. Global Anal. Geom. 18 (2000), no. 3-4, 405–435. Special issue in memory of Alfred Gray (1939–1998). MR 1795105, https://doi.org/10.1023/A:1006780703789
9. R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. MR 1083148
10. R.L. Bryant, private communication.
11. Simon G. Chiossi and Anna Fino, Conformally parallel 𝐺₂ structures on a class of solvmanifolds, Math. Z. 252 (2006), no. 4, 825–848. MR 2206629, https://doi.org/10.1007/s00209-005-0885-7
12. Simon G. Chiossi and Andrew Swann, 𝐺₂-structures with torsion from half-integrable nilmanifolds, J. Geom. Phys. 54 (2005), no. 3, 262–285. MR 2139083, https://doi.org/10.1016/j.geomphys.2004.09.009
13. D. Conti, Special holonomy and hypersurfaces, Ph.D. thesis, Scuola Normale Superiore, Pisa, 2005.
14. Thomas Friedrich, On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phys. 28 (1998), no. 1-2, 143–157. MR 1653146, https://doi.org/10.1016/S0393-0440(98)00018-7
15. Th. Friedrich and I. Kath, Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator, J. Differential Geom. 29 (1989), no. 2, 263–279. MR 982174
16. Eui Chul Kim and Thomas Friedrich, The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys. 33 (2000), no. 1-2, 128–172. MR 1738150, https://doi.org/10.1016/S0393-0440(99)00043-1
17. D. Giovannini, Special structures and symplectic geometry, Ph.D. thesis, Università degli Studi di Torino, 2003.
18. Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, https://doi.org/10.1007/BF02392726
19. Nigel Hitchin, Stable forms and special metrics, Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000) Contemp. Math., vol. 288, Amer. Math. Soc., Providence, RI, 2001, pp. 70–89. MR 1871001, https://doi.org/10.1090/conm/288/04818
20. Dominic Joyce, Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications, J. Differential Geom. 63 (2003), no. 2, 279–347. MR 2015549
21. B. Morel, The energy-momentum tensor as a second fundamental form, DG/0302205, 2003.
22. Simon Salamon, Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics Series, vol. 201, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1004008
23. S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 311–333. MR 1812058, https://doi.org/10.1016/S0022-4049(00)00033-5