Abstract
On a Riemannian spin manifold (M n, g), equipped with a non-integrable geometric structure and characteristic connection ▽c with parallel torsion ▽c T c = 0, we can introduce the Dirac operator D 1/3, which is constructed by lifting the affine metric connection with torsion 1/3 T c to the spin structure. D 1/3 is a symmetric elliptic differential operator, acting on sections of the spinor bundle and can be identified in special cases with Kostant’s cubic Dirac operator or the Dolbeault operator. For compact (M n, g), we investigate the first eigenvalue of the operator \({\left(D^{1/3} \right)^{2}}\) . As a main tool, we use Weitzenböck formulas, which express the square of the perturbed operator D 1/3 + S by the Laplacian of a suitable spinor connection. Here, S runs through a certain class of perturbations. We apply our method to spaces of dimension 6 and 7, in particular, to nearly Kähler and nearly parallel G 2-spaces.
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References
Agricola I.: Connections on naturally reductive spaces, their Dirac operators and homogeneous models in string theory. Commun. Math. Phys. 232, 535–563 (2003)
Agricola, I.: The Srni lectures on non-integrable geometries with torsion. Arch. Math. (Brno) 42, 5–84 (2006). With an appendix by Mario Kassuba
Agricola I., Friedrich T.: On the holonomy of connections with skew-symmetric torsion. Math. Ann. 328, 711–748 (2004)
Agricola, I., Friedrich, T.: 3-Sasakian manifolds in dimension 7, their spinors and G 2-structures. math.DG/0812.1651
Agricola I., Friedrich T., Kassuba M.: Eigenvalue estimates for Dirac operators with parallel characteristic torsion. Differ. Geom. Appl. 26, 613–624 (2008)
Alexandrov B., Friedrich T., Schoeman N.: Almost Hermitian 6-manifolds revisited. J. Geom. Phys. 53, 1–30 (2005)
Anderson M.T., Kronheimer P.B., LeBrun C.: Complete Ricci-flat Kähler manifolds of infinite topological type. Commun. Math. Phys. 125, 637–642 (2005)
Baum H., Friedrich T., Grunewald R., Kath I.: Twistors and Killing Spinors on Riemannian Manifolds. Teubner-Verlag, Leipzig/Stuttgart (1991)
Bismut J.-M.: A local index theorem for non Kähler manifolds. Math. Ann. 284, 681–699 (1989)
Friedrich T.: G 2-manifolds with parallel characteristic torsion. J. Differ. Geom. Appl. 25, 632–648 (2007)
Friedrich T., Grunewald R.: On the first eigenvalue of the Dirac operator on 6-dimensional manifolds. Ann. Glob. Anal. Geom. 3(3), 265–273 (1985)
Friedrich T., Ivanov S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6, 303–336 (2002)
Friedrich T., Kath I.: 7-dimensional compact Riemannian manifolds with Killing spinors. Commun. Math. Phys. 133, 543–561 (1990)
Friedrich T., Kath I., Semmelmann U., Moroianu A.: On nearly parallel G 2-structures. J. Differ. Geom. Phys. 23, 259–286 (1997)
Kobayashi S., Nomizu K.: Foundations of Differential Geometry II, p. 470. Wiley Classics Library, Wiley Inc., Princeton (1969)
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)
Kronheimer P.B.: The construction of ALE spaces as hyper-Kähler quotients. J. Differ. Geom. 29, 665–683 (1989)
Sulanke, S.: Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphäre und Untersuchungen zum ersten Eigenwert von D auf 5-dimensionalen Räumen konstanter positiver Schnittkrümmung. Doktorarbeit, Humboldt-Universität zu Berlin (1979)
Schoemann N.: Almost Hermitian structures with parallel torsion. J. Geom. Phys. 57, 2187–2212 (2007)
Takahashi T.: Deformations of Sasakian structures and its applications to the Brieskorn manifolds. Tôhoku Math. J. 30, 37–43 (1976)
Tanno S.: The topology of contact Riemannian manifolds. Illinois J. Math. 12, 700–717 (1968)
Vaisman I.: Locally conformal Kähler manifolds with parallel Lee form. Rend. Math. Roma 12, 263–284 (1979)
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Kassuba, M. Eigenvalue estimates for Dirac operators in geometries with torsion. Ann Glob Anal Geom 37, 33–71 (2010). https://doi.org/10.1007/s10455-009-9172-x
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DOI: https://doi.org/10.1007/s10455-009-9172-x