Invariant Tensors under the Twin Interchange of Norden Metrics on Almost Complex Manifolds

TL;DR

This paper investigates almost complex manifolds with paired Norden metrics, identifying invariant tensors under twin interchange and providing explicit forms, including an example involving a four-parameter Lie group.

Contribution

It introduces invariant tensors under twin interchange of Norden metrics and explicitly constructs these objects on a specific 4-dimensional Lie group example.

Findings

Invariant tensors are explicitly derived for the twin interchange scenario.

A 4-dimensional Lie group example illustrates the theoretical constructs.

Explicit forms of invariant objects are provided for the studied manifolds.

Abstract

The object of study are almost complex manifolds with a pair of Norden metrics, mutually associated by means of the almost complex structure. More precisely, a torsion-free connection and tensors with geometric interpretation are found which are invariant under the twin interchange, i.e. the swap of the counterparts of the pair of Norden metrics and the corresponding Levi-Civita connections. A Lie group depending on four real parameters is considered as an example of a 4-dimensional manifold of the studied type. The mentioned invariant objects are found in an explicit form.