Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian-Norden Type

TL;DR

This paper introduces associated Nijenhuis tensors for almost hypercomplex structures with Hermitian-Norden metrics, explores their relations and conditions for vanishing, and provides an example of such a manifold.

Contribution

It defines and studies associated Nijenhuis tensors in the context of almost hypercomplex manifolds with Hermitian-Norden metrics, linking their vanishing to unique torsion connections.

Findings

Relations between six associated Nijenhuis tensors are established.

Vanishing of these tensors characterizes the existence of a unique torsion connection.

An explicit example of a 4-dimensional manifold with vanishing tensors is provided.

Abstract

An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. It is given a geometric interpretation of the vanishing of these tensors as a necessary and sufficient condition for the existence of a unique connection with totally skew-symmetric torsion tensor. Similar idea is used in the paper of T. Friedrich and S. Ivanov in Asian J. Math. (2002) for some other structures. Finally, an example of a 4-dimensional manifold of the considered type with vanishing associated Nijenhuis tensors is given.