Journal of the Mathematical Society of Japan

Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds

Craig van COEVERING

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus in contrast to the case of parallel spinors the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 53-91.

Dates
First available in Project Euclid: 18 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1484730018

Digital Object Identifier
doi:10.2969/jmsj/06910053

Mathematical Reviews number (MathSciNet)
MR3597547

Zentralblatt MATH identifier
1373.53066

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx]

Keywords
Killing spinor Sasaki–Einstein 3-Sasakian Einstein deformation

Citation

van COEVERING, Craig. Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds. J. Math. Soc. Japan 69 (2017), no. 1, 53--91. doi:10.2969/jmsj/06910053. https://projecteuclid.org/euclid.jmsj/1484730018


Export citation

References

  • B. Alexandrov and U. Semmelmann, Deformations of nearly parallel ${\rm G}_2$-structures, Asian J. Math., 16 (2012), 713–744.
  • C. Bär, Real Killing spinors and holonomy, Comm. Math. Phys., 154 (1993), 509–521.
  • H. Baum, T. Friedrich, R. Grunewald and I. Kath, Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 124, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991, With German, French and Russian summaries.
  • M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry, 3 (1969), 379–392.
  • A. L. Besse. Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10, Springer-Verlag, Berlin, 1987.
  • J.-P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys., 144 (1992), 581–599.
  • C. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., VI, Int. Press, Boston, MA, 1999, pp.123–184.
  • C. P. Boyer and K. Galicki, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
  • C. P. Boyer, K. Galicki, B. M. Mann and E. G. Rees, Compact $3$-Sasakian $7$-manifolds with arbitrary second Betti number, Invent. Math., 131 (1998), 321–344.
  • T. Br öcker and T. T. Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, 1985.
  • M. J. Duff, B. E. W. Nilsson and C. N. Pope, Kaluza–Klein supergravity, Phys. Rep., 130 (1986), 1–142.
  • D. G. Ebin, The manifold of Riemannian metrics, Global Analysis (Proc. Sympos. Pure Math., XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970 pp.11–40.
  • A. El Kacimi Alaoui and B. Gmira, Stabilité du caractère kählérien transverse, Israel J. Math., 101 (1997), 323–347.
  • A. El Kacimi-Alaoui and M. Nicolau, Déformations des feuilletages transversalement holomorphes à type différentiable fixe, Publ. Mat., 33 (1989), 485–500.
  • A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math., 73 (1990), 57–106.
  • Th. Friedrich, I. Kath, A. Moroianu and U. Semmelmann, On nearly parallel $G_2$-structures, J. Geom. Phys., 23 (1997), 259–286.
  • S. Gallot, Équations différentielles caractéristiques de la sphère, Ann. Sci. École Norm. Sup. (4), 12 (1979), 235–267.
  • J. Girbau, A versality theorem for transversely holomorphic foliations of fixed differentiable type, Illinois J. Math., 36 (1992), 428–446.
  • P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978, Pure and Applied Mathematics.
  • N. Koiso, Einstein metrics and complex structures, Invent. Math., 73 (1983), 71–106.
  • N. Koiso, Rigidity and stability of Einstein metrics–-the case of compact symmetric spaces, Osaka J. Math., 17 (1980), 51–73.
  • M. Kuranishi, New proof for the existence of locally complete families of complex structures, In Proc. Conf. Complex Analysis, (Minneapolis, 1964), Springer, Berlin, 1965, pp.142–154.
  • H. B. Lawson Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, 38, Princeton University Press, Princeton, NJ, 1989.
  • C. LeBrun, A rigidity theorem for quaternionic-Kähler manifolds, Proc. Amer. Math. Soc., 103 (1988), 1205–1208.
  • C. LeBrun, Quaternionic-Kähler manifolds and conformal geometry, Math. Ann., 284 (1989), 353–376.
  • C. LeBrun, Fano manifolds, contact structures, and quaternionic geometry, Internat. J. Math., 6 (1995), 419–437.
  • Y. Matsushima, Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne, Nagoya Math. J., 11 (1957), 145–150.
  • A. Moroianu, P.-A. Nagy and U. Semmelmann, Deformations of nearly Kähler structures, Pacific J. Math., 235 (2008), 57–72.
  • A. Moroianu and U. Semmelmann, The Hermitian Laplace operator on nearly Kähler manifolds, Comm. Math. Phys., 294 (2010), 251–272.
  • A. Moroianu and U. Semmelmann, Infinitesimal Einstein deformations of nearly Kähler metrics, Trans. Amer. Math. Soc., 363 (2011), 3057–3069.
  • P.-A. Nagy, Nearly Kähler geometry and Riemannian foliations, Asian J. Math., 6 (2002), 481–504.
  • T. Nitta and M. Takeuchi, Contact structures on twistor spaces, J. Math. Soc. Japan, 39 (1987), 139–162.
  • J. Nordstr öm, Ricci-flat deformations of metrics with exceptional holonomy, Bull. Lond. Math. Soc., 45 (2013), 1004–1018.
  • B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459–469.
  • H. Pedersen and Y. S. Poon, A note on rigidity of $3$-Sasakian manifolds, Proc. Amer. Math. Soc., 127 (1999), 3027–3034.
  • S. M. Salamon, Quaternionic structures and twistor spaces, Global Riemannian geometry, (Durham, 1983), Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1984, pp.65–74.
  • S. M. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4), 19 (1986), 31–55.
  • C. van Coevering, Sasaki–Einstein 5-manifolds associated to toric 3-Sasaki manifolds, New York J. Math., 18 (2012), 555–608.
  • C. van Coevering, Stability of Sasaki-extremal metrics under complex deformations, Int. Math. Res. Not. IMRN, (2012), no. 24, 5527–5570.
  • C. van Coevering and C. Tipler, Deformations of constant scalar curvature Sasakian metrics and K-stability, Int. Math. Res. Not. IMRN, 2015, no. 22, 11566–11604.
  • M. Y. Wang, Parallel spinors and parallel forms, Ann. Global Anal. Geom., 7 (1989), 59–68.
  • M. Y. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. J., 40 (1991), 815–844.