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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
Published electronically:
May 1, 2007
MathSciNet review:
2327032
Full-text PDF Free Access
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Abstract: We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally,  -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  .
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& Sons, Inc., New York, 1989. MR 1004008
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- 1.
- B.S. Acharya, F. Denef, C. Hofman, and N. Lambert, Freund-Rubin revisited, hep-th/0308046.
- 2.
- M. Atiyah and E. Witten, M-theory dynamics on a manifold of
 holonomy, Adv. Theor. Math. Phys. 6 (2003), 1-106. MR 1992874 (2004f:53046)
- 3.
- C. Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), no. 3, 509-521. MR 1224089 (94i:53042)
- 4.
- C. Bär, P. Gauduchon, and A. Moroianu, Generalized cylinders in semi-Riemannian and spin geometry, Math. Z. 249 (2005), 545-580. MR 2121740 (2006g:53065)
- 5.
- H. Baum, T. Friedrich, R. Grunewald, and I. Kath, Twistor and Killing spinors on Riemannian manifolds, Teubner-Verlag Leipzig/Stuttgart, 1991. MR 1164864 (94a:53077)
- 6.
- D.E. Blair, Riemannian geometry of contact and symplectic manifolds, Birkhäuser, 2002. MR 1874240 (2002m:53120)
- 7.
- C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surv. Differ. Geom. 7 (1999), 123-184.
- 8.
- R.L. Bryant, Calibrated embeddings in the special Lagrangian and coassociative cases, Ann. Global Anal. Geom. 18 (2000), 405-435. MR 1795105 (2002j:53063)
- 9.
- R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, and P.A. Griffiths, Exterior differential systems, Springer-Verlag, 1991. MR 1083148 (92h:58007)
- 10.
- R.L. Bryant, private communication.
- 11.
- S. Chiossi and A. Fino, Conformally parallel
 structures on a class of solvmanifolds, Math. Z. 252 (2006), no. 4, 825-848. MR 2206629 (2007a:53098)
- 12.
- S. Chiossi and A. Swann,
 -structures with torsion from half-integrable nilmanifolds, J.Geom.Phys. 54 (2005), 262-285. MR 2139083 (2006a:53054)
- 13.
- D. Conti, Special holonomy and hypersurfaces, Ph.D. thesis, Scuola Normale Superiore, Pisa, 2005.
- 14.
- T. Friedrich, On the spinor representation of surfaces in Euclidean 3-space, J.Geom.Phys. 28 (1998), 143-157. MR 1653146 (99i:53057)
- 15.
- T. Friedrich and I.Kath, Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator, J. Differential Geom. 29 (1989), 263-279. MR 982174 (90e:58158)
- 16.
- T. Friedrich and E.C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys. 33 (2000), 128-172. MR 1738150 (2001e:58046)
- 17.
- D. Giovannini, Special structures and symplectic geometry, Ph.D. thesis, Università degli Studi di Torino, 2003.
- 18.
- R. Harvey and H.B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47-157. MR 666108 (85i:53058)
- 19.
- N. Hitchin, Stable forms and special metrics, Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Contemp. Math., vol. 288, American Math. Soc., 2001, pp. 70-89. MR 1871001 (2003f:53065)
- 20.
- D. Joyce, Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications, J. Differential Geom. 63 (2003), 279-348. MR 2015549 (2005e:53076)
- 21.
- B. Morel, The energy-momentum tensor as a second fundamental form, DG/0302205, 2003.
- 22.
- S. Salamon, Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics, vol. 201, Longman, Harlow, 1989. MR 1004008 (90g:53058)
- 23.
- -, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333. MR 1812058 (2002g:53089)
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Additional Information
Diego Conti
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Address at time of publication:
Dipartimento di Matematica e Applicazioni, Università di Milano – Bicocca, Via Cozzi 53, 20125 Milano, Italy
Email:
diego.conti@unimib.it
Simon Salamon
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email:
simon.salamon@polito.it
DOI:
http://dx.doi.org/10.1090/S0002-9947-07-04307-3
PII:
S 0002-9947(07)04307-3
Received by editor(s):
September 2, 2005
Published electronically:
May 1, 2007
Additional Notes:
This research was carried out as part of the programme “Proprietà geometriche delle varietà reali e complesse” (Cofin 2002) funded by the Italian MIUR
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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