A Calabi operator for Riemannian locally symmetric spaces

TL;DR

This paper generalizes the Calabi operator to Riemannian locally symmetric spaces, providing conditions for the range of the Killing operator, with a focus on irreducible cases and specific product spaces.

Contribution

It extends the Calabi operator to a broader class of spaces and characterizes when this generalization is applicable.

Findings

The generalized operator always works in irreducible cases.

Identifies specific product spaces where the operator fails.

Provides linear second order local integrability conditions.

Abstract

On a Riemannian manifold of constant curvature, the Calabi operator is a second order linear differential operator that provides local integrability conditions for the range of the Killing operator. We generalise this operator to provide linear second order local integrability conditions on Riemannian locally symmetric spaces, whenever this is possible. Specifically, we show that this generalised operator always works in the irreducible case and we identify precisely those products for which it fails.