Twistorial eigenvalue estimates for generalized Dirac operators with torsion. (English) Zbl 1287.58015
The authors use twistor theory to examine the spectrum of the Dirac operator defined by a metric connection with torsion on a compact Riemannian spin manifold \((M^n,g)\). Let \(D^g\) be the Riemannian Dirac operator and let \(\nabla\) be a metric connection with skew-symmetric torsion \(T\in\bigwedge^3(M^n)\). The operator under consideration then takes the form \(D_T:=D^g+\frac14T\). The authors determine an optimal lower bound for \(D_T\) generalizing the classical estimate in the Riemannian setting of Friedrich. They also determine novel twister and Killing equations and use these equations to determine when the bound is sharp.
§1 provides an introduction to the problem, §2 reviews the universal eigenvalue estimate, §3 discusses the twistorial eigenvalue estimate, §4 presents further material concerning the twistorial estimate, §5 treats Killing and twistor spinors with torsion, §6 presents the twistor equation in dimension 6, §7 deals with twistor estimates for manifolds with reducible holonomy.
The paper contains three appendices. Appendix A gives a proof and application of the integrability condition. Appendix B treats curvature properties for families of connections. Appendix C has a compilation of important formulas.
There is an extensive and very helpful bibliography to the subject.
§1 provides an introduction to the problem, §2 reviews the universal eigenvalue estimate, §3 discusses the twistorial eigenvalue estimate, §4 presents further material concerning the twistorial estimate, §5 treats Killing and twistor spinors with torsion, §6 presents the twistor equation in dimension 6, §7 deals with twistor estimates for manifolds with reducible holonomy.
The paper contains three appendices. Appendix A gives a proof and application of the integrability condition. Appendix B treats curvature properties for families of connections. Appendix C has a compilation of important formulas.
There is an extensive and very helpful bibliography to the subject.
Reviewer: Peter B. Gilkey (Eugene)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 53C28 | Twistor methods in differential geometry |
Keywords:
Dirac operator; eigenvalue estimate; metric connection with torsion; twistor operator; Killing spinorReferences:
| [1] | Agricola, I., Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Comm. Math. Phys., 232, 535-563 (2003) · Zbl 1032.53041 |
| [2] | Agricola, I., The Srní lectures on non-integrable geometries with torsion, Arch. Math. (Brno), 42, 5-84 (2006), With an appendix by Mario Kassuba · Zbl 1164.53300 |
| [3] | Agricola, I.; Friedrich, Th., On the holonomy of connections with skew-symmetric torsion, Math. Ann., 328, 711-748 (2004) · Zbl 1055.53031 |
| [4] | Agricola, I.; Friedrich, Th., The Casimir operator of a metric connection with skew-symmetric torsion, J. Geom. Phys., 50, 188-204 (2004) · Zbl 1080.53043 |
| [5] | Agricola, I.; Friedrich, Th.; Kassuba, M., Eigenvalue estimates for Dirac operators with parallel characteristic torsion, Differential Geom. Appl., 26, 613-624 (2008) · Zbl 1161.53031 |
| [6] | Alexandrov, B., \(S p(n) U(1)\)-connections with parallel totally skew-symmetric torsion, J. Geom. Phys., 57, 323-337 (2006) · Zbl 1107.53012 |
| [7] | Alexandrov, B., The first eigenvalue of the Dirac operator on locally reducible Riemannian manifolds, J. Geom. Phys., 57, 467-472 (2007) · Zbl 1116.58028 |
| [8] | Alexandrov, B.; Friedrich, Th.; Schoemann, N., Almost Hermitian 6-manifolds revisited, J. Geom. Phys., 53, 1-30 (2005) · Zbl 1075.53036 |
| [9] | Alexandrov, B.; Grantcharov, G.; Ivanov, S., An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form, J. Geom. Phys., 28, 263-270 (1998) · Zbl 0934.58026 |
| [10] | Alexandrov, B.; Ivanov, S., Dirac operators on Hermitian spin surfaces, Ann. Global Anal. Geom., 18, 529-539 (2000) · Zbl 0991.53029 |
| [11] | Baum, H.; Friedrich, Th.; Grunewald, R.; Kath, I., Twistors and Killing spinors on Riemannian manifolds, (Teubner-Texte zur Mathematik. Teubner-Texte zur Mathematik, Band, vol. 124 (1991), Teubner-Verlag, Stuttgart: Teubner-Verlag, Stuttgart Leipzig) · Zbl 0734.53003 |
| [13] | Besse, A., Einstein manifolds, (Ergebnisse der Mathematik und ihrer Grenzgebiete Bd., vol. 10 (1987), Springer-Verlag: Springer-Verlag Berlin-Heidelberg) · Zbl 0613.53001 |
| [14] | Bismut, J. M., A local index theorem for non-Kählerian manifolds, Math. Ann., 284, 681-699 (1989) · Zbl 0666.58042 |
| [15] | Dalakov, P.; Ivanov, S., Harmonic spinors of Dirac operator of connection with torsion in dimension 4, Classical Quantum Gravity, 18, 253-265 (2001) · Zbl 0976.83036 |
| [16] | Friedrich, Th., Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr., 97, 117-146 (1980) · Zbl 0462.53027 |
| [17] | Friedrich, Th., On the conformal relations between twistor and Killing spinors, Suppl. Rend. Circ. Mat. Palermo, 59-75 (1989) · Zbl 0703.53012 |
| [18] | Friedrich, Th., \(G_2\)-manifolds with parallel characteristic torsion, Differential Geom. Appl., 25, 632-648 (2007) · Zbl 1141.53019 |
| [19] | Friedrich, Th.; Grunewald, R., On the first eigenvalue of the Dirac operator on 6-dimensional manifolds, Ann. Global Anal. Geom., 3, 265-273 (1985) · Zbl 0577.58034 |
| [20] | Friedrich, Th.; Ivanov, S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math., 6, 303-336 (2002) · Zbl 1127.53304 |
| [21] | Friedrich, Th.; Ivanov, S., Almost contact manifolds, connections with torsion and parallel spinors, J. Reine Angew. Math., 559, 217-236 (2003) · Zbl 1035.53058 |
| [22] | Friedrich, Th.; Kath, I.; Moroianu, A.; Semmelmann, U., On nearly parallel \(G_2\)-structures, J. Geom. Phys., 23, 256-286 (1997) · Zbl 0898.53038 |
| [23] | Gauduchon, P., Hermitian connections and Dirac operators, Boll. Unione Mat. Ital. Ser. VII, 2, 257-289 (1997) · Zbl 0876.53015 |
| [24] | Gauduchon, P.; Ornea, L., Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier, 48, 1107-1127 (1998) · Zbl 0917.53025 |
| [25] | Gray, A., Weak holonomy groups, Math. Z., 123, 290-300 (1971) · Zbl 0222.53043 |
| [26] | Habermann, K., The twistor equation on Riemannian spin manifolds, J. Geom. Phys., 469-488 (1990) · Zbl 0732.53030 |
| [27] | Hijazi, O., Twistor operators and eigenvalues of the Dirac operator, (Gentili, G.; etal., Quaternionic Structures in Mathematics and Physics. Proceedings of the Meeting, Trieste, Italy, September 5-9, 1994 (1998), International School for Advanced Studies (SISSA): International School for Advanced Studies (SISSA) Trieste), 151-174 · Zbl 0947.53026 |
| [28] | Hijazi, O.; Lichnerowicz, A., Spineurs harmoniques, spineurs-twisteurs et géométrie conforme, C. R. Acad. Sci. Paris, 307, Série I, 833-838 (1988) · Zbl 0659.53016 |
| [29] | Hitchin, N., Harmonic spinors, Adv. Math., 14, 1-55 (1974) · Zbl 0284.58016 |
| [30] | Hitchin, N., The geometry of three-forms in six and seven dimensions, J. Differential Geom., 55, 547-576 (2000) · Zbl 1036.53042 |
| [31] | Houri, T.; Kubiznak, D.; Warnick, C.; Yasui, Y., Symmetries of the Dirac operator with skew-symmetric torsion, Classical Quantum Gravity, 27, 185019 (2010) · Zbl 1200.83102 |
| [32] | Jensen, G., Imbeddings of Stiefel manifolds into Grassmannians, Duke Math. J., 42, 397-407 (1975) · Zbl 0335.53042 |
| [34] | Kassuba, M., Eigenvalue estimates for Dirac operators in geometries with torsion, Ann. Global Anal. Geom., 37, 33-71 (2010) · Zbl 1190.53044 |
| [35] | Kath, I., Pseudo-Riemannian \(T\)-duals of compact Riemannian homogeneous spaces, Transform. Groups, 5, 157-179 (2000) · Zbl 0965.53037 |
| [36] | Kim, E. C., Lower bounds of the Dirac eigenvalues on Riemannian product manifolds |
| [37] | Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry I, 1991 (1963), Wiley Classics Library, Wiley Inc.: Wiley Classics Library, Wiley Inc. Princeton · Zbl 0119.37502 |
| [38] | Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry II, 1996 (1969), Wiley Classics Library, Wiley Inc.: Wiley Classics Library, Wiley Inc. Princeton · Zbl 0175.48504 |
| [39] | Kostant, B., A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J., 100, 447-501 (1999) · Zbl 0952.17005 |
| [40] | Lichnerowicz, A., Spin manifolds, Killing spinors and universality of the Hijazi inequality, Lett. Math. Phys., 13, 331-344 (1987) · Zbl 0624.53034 |
| [41] | Lichnerowicz, A., Killing spinors, twistor-spinors and Hijazi inequality, J. Geom. Phys., 5, 2-18 (1988) · Zbl 0678.53018 |
| [42] | Salamon, S., Riemannian geometry and holonomy groups, (Pitman Research Notes in Mathematical Series, vol. 201 (1989), Jon Wiley & Sons) · Zbl 0685.53001 |
| [44] | Schoemann, Nils, Almost Hermitian structures with parallel torsion, J. Geom. Phys., 57, 2187-2212 (2007) · Zbl 1137.53014 |
| [45] | Swann, A. F., Weakening holonomy, ESI preprint No. 816 (2000), (Marchiafava, S.; etal., Proc. of the Second Meeting on Quaternionic Structures in Mathematics and Physics, Roma 6-10 September 1999 (2001), World Scientific: World Scientific Singapore), 405-415 · Zbl 1028.53051 |
| [46] | Vaisman, I., Locally conformal Kähler manifolds with parallel Lee form, Rend. Math. Roma, 12, 263-284 (1979) · Zbl 0447.53032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
