Abstract
We consider a two-parameter generalization \(D_{ab}\) of the Riemann Dirac operator \(D\) on a closed Sasakian spin manifold, focusing attention on eigenvalue estimates for \(D_{ab}\). We investigate a Sasakian version of twistor spinors and Killing spinors, applying it to establish a new connection deformation technique that is adapted to fit with the Sasakian structure. Using the technique and the fact that there are two types of eigenspinors of \(D_{ab}\), we prove several eigenvalue estimates for \(D_{ab}\) which improve Friedrich’s estimate (Friedrich, Math Nachr 97, 117–146, 1980).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agricola, I.: The Srni lectures on non-integrable geometries with torsion. Arch. Math. Brno 42, 5–84 (2006)
Agricola, I., Becker-Bender, J., Kim, H.: Twistorial eigenvalue estimates for generalized Dirac operators with torsion. Adv. Math. 243, 296–329 (2013)
Agricola, I., Friedrich, Th., Kassuba, M.: Eigenvalue estimates for Dirac operators with parallel characteristic torsion. Differ. Geom. Appl. 26, 613–624 (2008)
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975)
Baum, H., Friedrich, Th., Grunewald, R., Kath, I.: Twistors and Killing Spinors on Riemannian Manifolds. Teubner, Leipzig (1991)
Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97, 117–146 (1980)
Friedrich, Th.: The second Dirac eigenvalue of a nearly parallel \(G_2\)-manifold. Adv. Appl. Clifford Algebr. 22, 301–311 (2012)
Friedrich, Th., Kim, E.C.: The Einstein–Dirac equation on Riemannian spin manifolds. J. Geom. Phys. 33, 128–172 (2000)
Ginoux, N.: The Dirac Spectrum. Lecture Notes in Mathematics. Springer, Berlin (2009)
Kim, E.C.: The \(\hat{A}\)-genus and symmetry of the Dirac spectrum on Riemannian product manifolds. Differ. Geom. Appl. 25, 309–321 (2007)
Kim, E.C.: Dirac eigenvalues estimates in terms of divergencefree symmetric tensors. Bull. Korean Math. Soc. 46, 949–966 (2009)
Kim, E.C.: Estimates of small Dirac eigenvalues on 3-dimensional Sasakian manifolds. Differ. Geom. Appl. 28, 648–655 (2010)
Kim, E.C.: The first positive eigenvalue of the Dirac operator on 3-dimensional Sasakian manifolds. Bull. Korean Math. Soc. 50, 431–440 (2013)
Pilca, M.: Kählerian twistor spinors. Math. Z. 268, 223–255 (2011)
Kirchberg, K.-D.: Eigenvalue estimates for the Dirac operator on Kähler-Einstein manifolds of even complex dimension. Ann. Global Anal. Geom. 38, 273–284 (2010)
Kirchberg, K.-D., Semmelmann, U.: Complex contact structures and the first eigenvalue of the dirac operator on Kähler manifolds. Geom. Funct. Anal. 5, 604–618 (1995)
Moroianu, A.: Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz. Ann. Inst. Fourier 49, 1637–1659 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, E.C. Eigenvalue estimates for generalized Dirac operators on Sasakian manifolds. Ann Glob Anal Geom 45, 67–93 (2014). https://doi.org/10.1007/s10455-013-9388-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-013-9388-7