A prolongation of the conformal-Killing operator on quaternionic-Kahler manifolds

TL;DR

This paper extends the conformal-Killing operator on quaternionic-Kähler manifolds by constructing a flat prolongation connection linked to the vanishing of the quaternionic-Weyl tensor, revealing new geometric insights.

Contribution

It introduces a prolongation connection for compatible 2-forms and characterizes its flatness via the quaternionic-Weyl tensor, advancing understanding of quaternionic-Kähler geometry.

Findings

D is flat iff the quaternionic-Weyl tensor vanishes

Constructed a skew-symmetric multiplication on conformal-Killing 2-forms

Analyzed properties of compatible conformal-Killing 2-forms

Abstract

A 2-form on a quaternionic-Kahler manifold (M, g) is called compatible (with the quaternionic structure) if it is a section of the direct sum bundle S^2(H) \oplus S^2(E). We construct a connection D on S^2(H) \oplus S^2(E)\oplus TM, which is a prolongation of the conformal-Killing operator acting on compatible 2-forms. We show that D is flat if and only if the quaternionic-Weyl tensor of (M,g) is zero. Consequences of this result are developed. We construct a skew-symmetric multiplication on the space of conformal-Killing 2-forms on (M,g) and we study its properties in connection with the subspace of compatible conformal-Killing 2-forms.