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The Weitzenböck machine

Part of: Local differential geometry Partial differential equations on manifolds; differential operators Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  23 February 2010

Uwe Semmelmann  and
Gregor Weingart
Uwe Semmelmann
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany (email: uwe.semmelmann@math.uni-koeln.de)
Gregor Weingart
Affiliation:
Instituto de Matematicas (Unidad Cuernavaca), Universidad Nacional Autonoma de Mexico, Avenida Universidad s/n, Colonia Lomas de Chamilpa, 62210 Cuernavaca, Morelos, Mexico (email: gw@matcuer.unam.mx)
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Abstract

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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.

Keywords

53B20generalized gradientsWeitzenböck formulasCasimir elements

MSC classification

Secondary: 53B20: Local Riemannian geometry 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.) 17B35: Universal enveloping (super)algebras
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 507 - 540
Copyright
Copyright © Foundation Compositio Mathematica 2010

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