Hostname: page-component-5d59c44645-dknvm Total loading time: 0 Render date: 2024-02-28T06:18:41.900Z Has data issue: false hasContentIssue false

The Weitzenböck machine

Published online by Cambridge University Press:  23 February 2010

Uwe Semmelmann
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany (email:
Gregor Weingart
Instituto de Matematicas (Unidad Cuernavaca), Universidad Nacional Autonoma de Mexico, Avenida Universidad s/n, Colonia Lomas de Chamilpa, 62210 Cuernavaca, Morelos, Mexico (email:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Weitzenböck formulas are an important tool in relating local differential geometry to global topological properties by means of the so-called Bochner method. In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non-symmetric holonomy groups. We explicitly construct a basis of the space of Weitzenböck formulas. This classification allows us to find customized Weitzenböck formulas for applications such as eigenvalue estimates or Betti number estimates.

Research Article
Copyright © Foundation Compositio Mathematica 2010


[1]Branson, T. and Hijazi, O., Bochner–Weitzenböck formulas associated with the Rarita–Schwinger operator, Internat. J. Math. 13 (2002), 137182.Google Scholar
[2]Calderbank, D., Gauduchon, P. and Herzlich, M., Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal. 173 (2000), 214255.Google Scholar
[3]Fegan, H. D., Conformally invariant first order differential operators, Q. J. Math. Oxford (2) 27 (1976), 371378.Google Scholar
[4]Gauduchon, P., Structures de Weyl et theoremes d’annulation sur une variete conforme autoduale, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 18 (1991), 563629.Google Scholar
[5]Homma, Y., Casimir elements and Bochner identities on Riemannian manifolds, Progress in Mathematical Physics, vol. 34 (Birkhäuser, Boston, MA, 2004).Google Scholar
[6]Homma, Y., Bochner–Weitzenböck formulas and curvature actions on Riemannian manifolds, Trans. Amer. Math. Soc. 358 (2006), 87114.Google Scholar
[7]Semmelmann, U. and Weingart, G., Vanishing theorems for quaternionic Kähler manifolds, J. Reine Angew. Math. 544 (2002), 111132.Google Scholar
[8]Semmelmann, U., Killing forms on G2- and Spin(7)-manifolds, J. Geom. Phys. 56 (2006), 17521766.Google Scholar