Generalized Goldberg Formula

TL;DR

This paper generalizes Goldberg's formula for the graded commutator involving the Hodge codifferential and wedge multiplication, providing new identities and quasi-isomorphism results in differential geometry.

Contribution

It extends Goldberg's formula to p-forms, introduces new identities on special manifolds, and establishes quasi-isomorphism conditions for subalgebras of differential forms.

Findings

New identities on locally conformally Kaehler manifolds

Results on quasi-Sasakian manifolds

Quasi-isomorphism of subalgebras to de Rham algebra

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 Kaehler 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.