Abstract
We consider a generalization of Einstein–Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain a new hypo-contact structure on S 2 × T 3.
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References
Bär C.: Real Killing spinors and holonomy. Comm. Math. Phys. 154(3), 509–521 (1993)
Bär C., Gauduchon P., Moroianu A.: Generalized cylinders in semi-Riemannian and spin geometry. Math. Z. 249, 545–580 (2005)
Boyer C.P., Galicki K., Matzeu P.: On eta-Einstein Sasakian geometry. Comm. Math. Phys. 262(1), 177–208 (2006)
Bryant, R.L.: Nonembedding and nonextension results in special holonomy. In: Proceedings of the August 2006 Madrid conference in honor of Nigel Hitchin’s 60th Birthday. Oxford University Press (2006)
Bryant R.L.: Calibrated embeddings in the special Lagrangian and coassociative cases. Ann. Global Anal. Geom. 18, 405–435 (2000)
CoCoA Team. CoCoA: a system for doing Computations in Commutative Algebra. http://cocoa.dima.unige.it
Conti D.: Invariant forms, associated bundles and Calabi-Yau metrics. J. Geom. Phys. 57(12), 2483–2508 (2007)
Conti, D.: Embedding into manifolds with torsion. arXiv:0812.4186 (2008)
Conti D., Salamon S.: Generalized Killing spinors in dimension 5. Trans. Amer. Math. Soc. 359(11), 5319–5343 (2007)
Cvetič M., Lü H., Page D.N., Pope C.N.: New Einstein–Sasaki spaces in five and higher dimensions. Phys. Rev. Lett. 95, 071101 (2005)
de Andres L.C., Fernandez M., Fino A., Ugarte L.: Contact 5-manifolds with SU(2)-structure. Q. J. Math. 60, 429–459 (2009)
de Graaf W.A.: Classification of solvable Lie algebras. Experiment. Math. 14(1), 15–25 (2005)
Fino A., Tomassini A.: Generalized G 2-manifolds and SU(3)-structures. Internat. J. Math. 19, 1147–1165 (2008)
Friedrich T., Kath I.: Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator. J. Differential Geom. 29, 263–279 (1989)
Friedrich T., Kim E.C.: The Einstein–Dirac equation on Riemannian spin manifolds. J. Geom. Phys. 33, 128–172 (2000)
Gauntlett J.P., Martelli D., Sparks J., Waldram D.: Sasaki–Einstein metrics on S 2 × S 3. Adv. Theor. Math. Phys. 8, 711 (2004)
Geiges H.: Constructions of contact manifolds. Math. Proc. Cambridge Philos. Soc. 121(3), 455–464 (1997)
Grantcharov G., Ornea L.: Reduction of Sasakian manifolds. J. Math. Phys. 42(8), 3809–3816 (2001)
Hitchin, N.: Stable forms and special metrics. In: Global Differential Geometry: The Mathematical Legacy of Alfred Gray. Contemp. Math., vol. 288, pp. 70–89. American Math. Soc. (2001)
Hitchin N.J., Karlhede A., Lindström U., Roček M.: Hyper-Kähler metrics and supersymmetry. Comm. Math. Phys. 108(4), 535–589 (1987)
Lerman E.: Contact toric manifolds. J. Symplectic Geom. 1(4), 785–828 (2003)
Morel, B.: The energy-momentum tensor as a second fundamental form. DG/0302205 (2003)
Salamon S.: A tour of exceptional geometry. Milan J. Math. 71, 59–94 (2003)
Tomassini A., Vezzoni L.: Contact Calabi-Yau manifolds and special Legendrian submanifolds. Osaka J. Math. 45, 127–147 (2008)
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Conti, D., Fino, A. Calabi-Yau cones from contact reduction. Ann Glob Anal Geom 38, 93–118 (2010). https://doi.org/10.1007/s10455-010-9202-8
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DOI: https://doi.org/10.1007/s10455-010-9202-8