Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 072, 7 pages      arXiv:2411.03882      https://doi.org/10.3842/SIGMA.2025.072

Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index

Thomas Tony
Institute of Mathematics, University of Potsdam, Germany

Received May 19, 2025, in final form August 21, 2025; Published online August 25, 2025

Abstract
Every closed connected Riemannian spin manifold of non-zero $\hat{A}$-genus or non-zero Hitchin invariant with non-negative scalar curvature admits a parallel spinor, in particular is Ricci-flat. In this note, we generalize this result to closed connected spin manifolds of non-vanishing Rosenberg index. This provides a criterion for the existence of a parallel spinor on a finite covering and yields that every closed connected Ricci-flat spin manifold of dimension $\geq 2$ with non-vanishing Rosenberg index has special holonomy.

Key words: Ricci-flat manifolds; special holonomy; parallel spinor; scalar curvature; higher index theory.

pdf (347 kb)   tex (18 kb)  

References

  1. Berger M., Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279-330.
  2. Besse A.L., Einstein manifolds, Ergeb. Math. Grenzgeb. (3), Vol. 10, Springer, Berlin, 1987.
  3. Cheeger J., Gromoll D., The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971), 119-128.
  4. Ebert J., Elliptic regularity for Dirac operators on families of noncompact manifolds, arXiv:1608.01699.
  5. Fischer A.E., Wolf J.A., The structure of compact Ricci-flat Riemannian manifolds, J. Differential Geometry 10 (1975), 277-288.
  6. Friedrich T., Dirac operators in Riemannian geometry, Grad. Stud. Math., Vol. 25, American Mathematical Society, Providence, RI, 2000.
  7. Hanke B., Schick T., Enlargeability and index theory, J. Differential Geom. 74 (2006), 293-320, arXiv:math.GT/0403257.
  8. Hanke B., Schick T., Enlargeability and index theory: infinite covers, $K$-Theory 38 (2007), 23-33, arXiv:math.GT/0604540.
  9. Hatcher A., Algebraic topology, Cambridge University Press, Cambridge, 2002.
  10. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
  11. Kazdan J.L., Warner F.W., Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113-134.
  12. Lance E.C., Hilbert $C^*$-modules. A toolkit for operator algebraists, London Math. Soc. Lecture Note Ser., Vol. 210, Cambridge University Press, Cambridge, 1995.
  13. Lawson Jr. H.B., Michelsohn M.L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  14. Mishchenko A.S., Fomenko A.T., The index of elliptic operators over $C^{\ast} $-algebras, Math. USSR. Izv. 15 (1980), 87-112.
  15. Ramras D., Willett R., Yu G., A finite-dimensional approach to the strong Novikov conjecture, Algebr. Geom. Topol. 13 (2013), 2283-2316, arXiv:1203.6168.
  16. Roe J., Lectures on K-theory and operator algebras, AMS Open Math. Notes, 2017, https://www.ams.org/open-math-notes/omn-view-listing?listingId=110719.
  17. Rosenberg J., $C^{\ast} $-algebras, positive scalar curvature, and the Novikov conjecture, Publ. Math. Inst. Hautes Etudes Sci. 58 (1983), 409-424.
  18. Schick T., Wraith D.J., Non-negative versus positive scalar curvature, J. Math. Pures Appl. (9) 146 (2021), 218-232, arXiv:1607.00657.
  19. Tony T., Scalar curvature rigidity and the higher mapping degree, J. Funct. Anal. 288 (2025), 110744, 41 pages, arXiv:2402.05834.
  20. Wang M.Y., Parallel spinors and parallel forms, Ann. Global Anal. Geom. 7 (1989), 59-68.

Previous article  Next article  Contents of Volume 21 (2025)