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2010 | 8 | 5 | 950-965

Tytuł artykułu

A spectral estimate for the Dirac operator on Riemannian flows

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EN
We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.

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Bibliografia

  • [1] Alexandrov B., Grantcharov G., Ivanov S., An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifolds admitting a parallel one-form, J. Geom. Phys., 1998, 28(3–4), 263–270 http://dx.doi.org/10.1016/S0393-0440(97)00080-6
  • [2] Ammann B., Bär C., The Dirac operator on nilmanifolds and collapsing circle bundles, Ann. Global Anal. Geom., 1998, 16(3), 221–253 http://dx.doi.org/10.1023/A:1006553302362
  • [3] Bär C., Metrics with harmonic spinors, Geom. Funct. Anal., 1996, 6(6), 899–942 http://dx.doi.org/10.1007/BF02246994
  • [4] Bär C., Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom., 1998, 16(6), 573–596 http://dx.doi.org/10.1023/A:1006550532236
  • [5] Bär C., Gauduchon P., Moroianu A., Generalized cylinders in semi-Riemannian and spin geometry, Math. Z., 2005, 249(3), 545–580 http://dx.doi.org/10.1007/s00209-004-0718-0
  • [6] Belgun F.A., On the metric structure of non-Kähler complex surfaces, Math. Ann., 2000, 317(1), 1–40 http://dx.doi.org/10.1007/s002080050357
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  • [8] Boyer C.P., Galicki K., Matzeu P., On η-Einstein Sasakian geometry, Comm. Math. Phys., 2006, 262(1), 177–208 http://dx.doi.org/10.1007/s00220-005-1459-6
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  • [14] Ginoux N., Habib G., Geometric aspects of transversal Killing spinors on Riemannian flows, Abh. Math. Sem. Univ. Hamburg, 2008, 78(1), 69–90 http://dx.doi.org/10.1007/s12188-008-0006-8
  • [15] Habib G., Tenseur d’impulsion-énergie et feuilletages, Ph.D. thesis, Université Henri Poincaré - Nancy 1, 2006
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  • [17] Kim E.C., Friedrich T., The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys., 2000, 33(1–2), 128–172 http://dx.doi.org/10.1016/S0393-0440(99)00043-1
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Bibliografia

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bwmeta1.element.doi-10_2478_s11533-010-0060-1
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