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Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds


Author: Yasushi Homma
Journal: Trans. Amer. Math. Soc. 358 (2006), 87-114
MSC (2000): Primary 53B20, 58J60, 17B35
DOI: https://doi.org/10.1090/S0002-9947-05-04068-7
Published electronically: August 25, 2005
MathSciNet review: 2171224
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Abstract: Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.


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Additional Information

Yasushi Homma
Affiliation: Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, 2641 Noda, Chiba, 278-8510, Japan
Email: homma_yasushi@ma.noda.tus.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-05-04068-7
Keywords: Invariant operators, Bochner-Weitzenb\"ock formulas, $\mathrm{SO}(n)$-modules, Casimir elements
Received by editor(s): July 3, 2003
Published electronically: August 25, 2005
Additional Notes: The author was supported by the Grant-in-Aid for JSPS Fellows for Young Scientists.
Article copyright: © Copyright 2005 American Mathematical Society

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