manuscripta mathematica

, Volume 158, Issue 1–2, pp 273–293 | Cite as

The standard Laplace operator

  • Uwe Semmelmann
  • Gregor WeingartEmail author


The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stressing the functorial aspects of the standard Laplace operator \(\Delta \) with respect to the category of geometric vector bundles we show that the standard Laplace operator commutes not only with all homomorphisms, but also with a large class of natural first order differential operators between geometric vector bundles. Several examples are included to highlight the conclusions of this article.

Mathematics Subject Classification

53C21 53C26 58A14 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Besse, A.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 10. Springer, Berlin (1987)Google Scholar
  2. 2.
    Calderbank, D., Gauduchon, P., Herzlich, M.: Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal. 173(1), 214–255 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Galaev, A.: Decomposition of the covariant derivative of the curvature tensor of a pseudo-Kählerian manifold. Ann. Glob. Anal. Geom. 51(3), 245–265 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gauduchon, P.: Structures de Weyl et theoremes d’annulation sur une variete conforme autoduale. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18(4), 563–629 (1991)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gray, A.: Compact Kähler manifolds with nonnegative sectional curvature. Invent. Math. 41, 33–43 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Heil, K., Moroianu, A., Semmelmann, U.: Killing and conformal Killing tensors. J. Geom. Phys. 106, 383–400 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Homma, Y.: Casimir Elements and Bochner Identities on Riemannian Manifolds, Progress in Mathematical Physics, vol. 34. Birkhäuser, Boston (2004)zbMATHGoogle Scholar
  8. 8.
    Homma, Y.: Twisted Dirac operators and generalized gradients. Ann. Glob. Anal. Geom. 50(2), 101–127 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ivanov, S.: Geometry of quaternionic Kähler connections with torsion. J. Geom. Phys. 41(3), 235–257 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Joyce, D.: Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)Google Scholar
  11. 11.
    Lichnerowicz, A.: Propagateurs et commutateurs en relativite generale. Inst. Hautes Études Sci. Publ. Math. 10, 5–56 (1961)CrossRefzbMATHGoogle Scholar
  12. 12.
    Moroianu, A., Semmelmann, U.: The Hermitian Laplace operator on nearly Kähler manifolds. Commun. Math. Phys. 294(1), 251–272 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Salamon, S.: Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics Series, vol. 201. Wiley, New York (1989)Google Scholar
  14. 14.
    Semmelmann, U., Weingart, G.: Vanishing theorems for quaternionic Kähler manifolds. J. Reine Angew. Math. 544, 111–132 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Semmelmann, U., Weingart, G.: The Weitzenböck machine. Compos. Math. 146(2), 507–540 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Verbitsky, M.: Manifolds with parallel differential forms and Kähler identities for \({{\bf G}}_2\)-manifolds. J. Geom. Phys. 61(6), 1001–1016 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Weingart, G.: Differential Forms on Quaternionic Kähler Manifolds, Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lectures in Mathematics and Theoretical Physics, vol. 16, pp. 15–37. European Mathematical Society, Zurich (2010)CrossRefGoogle Scholar
  18. 18.
    Weingart, G.: Moments of sectional curvatures. (2017)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Geometrie und Topologie, Fachbereich MathematikUniversität StuttgartStuttgartGermany
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de MexicoCuernavacaMexico

Personalised recommendations