A spectral estimate for the Dirac operator on Riemannian flows

• Autorzy: Nicolas Ginoux , Georges Habib
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Foliations; Sasakian manifolds; Spin geometry; Spectral geometry; Estimation of eigenvalues - upper and lower bounds

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 5
• Strony: 950-965

Daty

wydano

2010-10-01

online

2010-09-28

Twórcy

autor

Nicolas Ginoux

• Universität Regensburg,

autor

Georges Habib

• Lebanese University,

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
• [7] Boyer C.P., Galicki K., On Sasakian-Einstein geometry, Internat. J. Math., 2000, 11(7), 873–909 http://dx.doi.org/10.1142/S0129167X00000477
• [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
• [9] Carrière Y., Flots riemanniens, In: Structure transverse des feuilletages, Toulouse 1982, Astérisque, 1984, 116, 31–52
• [10] Chavel I., Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115, Academic Press, Orlando, 1984
• [11] Friedrich T., Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr., 1980, 97(1), 117–146 http://dx.doi.org/10.1002/mana.19800970111
• [12] Friedrich T., Zur Abhängigkeit des Dirac-Operators von der Spin-Struktur, Colloq. Math., 1984, 48, 57–62
• [13] Geiges H., Normal contact structures on 3-manifolds, Tôhoku Math. J., 1997, 49(3), 415–422 http://dx.doi.org/10.2748/tmj/1178225112
• [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
• [16] Hijazi O., Lower bounds for the eigenvalues of the Dirac operator, J. Geom. Phys., 1995, 16(1), 27–38 http://dx.doi.org/10.1016/0393-0440(94)00019-Z
• [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
• [18] Okumura M., Some remarks on space with a certain contact structure, Tôhoku Math. J., 1962, 14(2), 135–145 http://dx.doi.org/10.2748/tmj/1178244168
• [19] O’Neill B., The fundamental equations of a submersion, Michigan Math. J., 1966, 13(4), 459–469 http://dx.doi.org/10.1307/mmj/1028999604
• [20] Pfäffle F., The Dirac spectrum of Bieberbach manifolds, J. Geom. Phys., 2000, 35(4), 367–385 http://dx.doi.org/10.1016/S0393-0440(00)00005-X
• [21] Reinhart B.L., Foliated manifolds with bundle-like metrics, Ann. of Math., 1959, 69(1), 119–132 http://dx.doi.org/10.2307/1970097
• [22] Tanno S., The topology of contact Riemannian manifolds, Illinois J. Math., 1968, 12(4), 700–717
• [23] Tondeur P., Foliations on Riemannian Manifolds, Springer, Berlin, 1988
• [24] Trautman A., Spinors and the Dirac operator on hypersurfaces. I: General theory, J. Math. Phys., 1992, 33(12), 4011–4019 http://dx.doi.org/10.1063/1.529852

Identyfikatory

DOI

10.2478/s11533-010-0060-1