Curvature of almost quaternion-Hermitian manifolds

TL;DR

This paper analyzes the curvature tensor decomposition of almost quaternion-Hermitian manifolds, revealing how intrinsic torsion influences curvature components and their relations, with a focus on Ricci curvatures.

Contribution

It provides a detailed decomposition of the Riemannian curvature tensor under the structure group, linking it to intrinsic torsion and its derivatives in almost quaternion-Hermitian manifolds.

Findings

Most curvature components are determined by intrinsic torsion and its covariant derivative.

Relations between decompositions of torsion tensor, its derivative, and curvature are established.

Special attention is given to the behavior of Ricci and q-Ricci curvatures.

Abstract

We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the intrinsic torsion \xi and its covariant derivative \widetilde\nabla\xi and determine relations between the decompositions of \xi\otimes\xi, \widetilde\nabla\xi and R. We pay particular attention to the behaviour of the Ricci curvature and the q-Ricci curvature.