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# Generalized Goldberg Formula

Published:2016-02-12
Printed: Sep 2016
• Antonio De Nicola,
CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
• Ivan Yudin,
CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
 Format: LaTeX MathJax PDF

## Abstract

In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of application we obtain new identities on locally conformally Kähler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a CDGA to the full de Rham algebra.
 Keywords: graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasi-Sasakian manifold
 MSC Classifications: 53C25 - Special Riemannian manifolds (Einstein, Sasakian, etc.) 53D35 - Global theory of symplectic and contact manifolds [See also 57Rxx]

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