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

The standard Laplace operator

  • Uwe Semmelmann
  • Gregor WeingartEmail author
Article
  • 37 Downloads

Abstract

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 

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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

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