Basic Gravitational Currents and Killing-Yano Forms

TL;DR

This paper explores the properties of gravitational currents associated with Killing-Yano forms in (pseudo)Riemannian manifolds, revealing their conservation laws and implications for scalar curvature and eigenforms of the Laplace-Beltrami operator.

Contribution

It introduces two basic gravitational currents linked to Killing-Yano forms and proves their conservation, providing new insights into geometric invariants and eigenform properties on Einstein and constant curvature manifolds.

Findings

Conservation of currents implies scalar curvature is a flow invariant.

KY-forms and their duals are eigenforms of the Laplace-Beltrami operator in constant curvature manifolds.

On Einstein manifolds, only certain KY-forms are eigenforms of the Laplace-Beltrami operator.

Abstract

It has been shown that for each Killing-Yano (KY)-form accepted by an $n$-dimensional (pseudo)Riemannian manifold of arbitrary signature, two basic gravitational currents can be defined. Conservation of the currents are explicitly proved by showing co-exactness of the one and co-closedness of the other. Some general geometrical facts implied by these conservation laws are also elucidated. In particular, the conservation of the one-form currents implies that the scalar curvature of the manifold is a flow invariant for all of its Killing vector fields. It also directly follows that, while all KY-forms and their Hodge duals on a constant curvature manifold are the eigenforms of the Laplace-Beltrami operator, for an Einstein manifold this is certain only for KY 1-forms, $(n-1)$-forms and their Hodge duals.