Homogeneous spin Riemannian manifolds with the simplest Dirac operator
-
P. M. Gadea
,
J. C. González-Dávila
Abstract
We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds (M, g) which are traceless cyclic with respect to some quotient expression M = G/K and reductive decomposition 𝔤 = 𝔨 ⊕ 𝔪. Using transversally symmetric fibrations of noncompact type, we give a list of them.
Acknowledgements
We are indebted to the referee. The helpful suggestions in his/her report have contributed to improve several aspects of the manuscript.
Funding: The first and second authors have been supported by DGI (Spain) Project MTM2013-46961-P.
References
[1] B. Ammann, C. Bär, The Dirac operator on nilmanifolds and collapsing circle bundles. Ann. Global Anal. Geom. 16 (1998), 221–253. MR1626659 Zbl 0911.5803710.1023/A:1006553302362Search in Google Scholar
[2] C. Bär, The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces. Arch. Math. (Basel) 59 (1992), 65–79. MR1166019 Zbl 0786.5303010.1007/BF01199016Search in Google Scholar
[3] A. Borel, F. Hirzebruch, Characteristic classes and homogeneous spaces. I. Amer. J. Math. 80 (1958), 458–538. MR0102800 Zbl 0097.3640110.2307/2372795Search in Google Scholar
[4] M. Cahen, S. Gutt, Spin structures on compact simply connected Riemannian symmetric spaces. In: Proceedings of the Workshop on Clifford Algebra, Clifford Analysis and their Applications in Mathematical Physics (Ghent, 1988), volume 62, 209–242, 1988. MR976428 Zbl 0677.53057Search in Google Scholar
[5] P. M. Gadea, J. C. González-Dávila, J. A. Oubiña, Cyclic metric Lie groups. Monatsh. Math. 176 (2015), 219–239. MR3302156 Zbl 1321.5305710.1007/s00605-014-0692-5Search in Google Scholar
[6] P. M. Gadea, J. C. González-Dávila, J. A. Oubiña, Cyclic homogeneous Riemannian manifolds. Ann. Mat. Pura Appl. (4) 195 (2016), 1619–1637. MR3537965 Zbl 1361.5304310.1007/s10231-015-0534-7Search in Google Scholar
[7] J. C. González-Dávila, Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion. Rev. Mat. Iberoam. 30 (2014), 247–275. MR3186939 Zbl 1292.5800910.4171/RMI/777Search in Google Scholar
[8] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, volume 80 of Pure and Applied Mathematics. Academic Press 1978. MR514561 Zbl 0451.53038Search in Google Scholar
[9] A. Ikeda, Formally self adjointness for the Dirac operator on homogeneous spaces. Osaka J. Math. 12 (1975), 173–185. MR0376962 Zbl 0317.58019Search in Google Scholar
[10] S. Kobayashi, K. Nomizu, Foundations of differential geometry. Vol. II. Interscience Publ. 1969. MR0238225 Zbl 0175.48504Search in Google Scholar
[11] H. B. Lawson, Jr., M.-L. Michelsohn, Spin geometry. Princeton Univ. Press 1989. MR1031992 Zbl 0688.57001Search in Google Scholar
[12] F. Tricerri, L. Vanhecke, Homogeneous structures on Riemannian manifolds. Cambridge Univ. Press 1983. MR712664 Zbl 0509.5304310.1017/CBO9781107325531Search in Google Scholar
[13] J. A. Wolf, Spaces of constant curvature. AMS Chelsea Publishing, Providence, RI 2011. MR2742530 Zbl 1216.53003Search in Google Scholar
© 2018 Walter de Gruyter GmbH Berlin/Boston
Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with â…“ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with â…“ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
