Quaternionic $k$-vector fields on quaternionic K\"{a}hler manifolds

TL;DR

This paper introduces a new differential operator on quaternionic Kähler manifolds, establishing a correspondence between quaternionic $k$-vector fields and holomorphic $k$-vector fields on the twistor space, and computes their dimension on quaternionic projective space.

Contribution

It defines a modified Dirac operator to study quaternionic $k$-vector fields and links these fields to holomorphic vector fields on the twistor space, providing explicit dimension formulas.

Findings

Quaternionic $k$-vector fields correspond to holomorphic $k$-vector fields on twistor space.

Dimension of quaternionic $k$-vector fields on $\,\mathbb{H}P^n$ is computed.

New differential operator introduced for studying quaternionic structures.

Abstract

In this paper, we define a differential operator as a modified Dirac operator. Using the operator, we introduce a quaternionic $k$-vector field on a quaternionic K\"{a}hler manifold and show that any quaternionic $k$-vector field corresponds to a holomorphic $k$-vector field on the twistor space. We calculate the dimension of the space of quaternionic $k$-vector fields on $\mathbb{H}P^n$.