# Invariant tensors under the twin interchange of the pairs of the associated metrics on almost paracomplex pseudo-Riemannian manifolds

## TL;DR

This paper investigates invariant tensors on almost paracomplex pseudo-Riemannian manifolds with paired metrics, identifying invariant geometric objects under a twin interchange and providing explicit examples on a constructed Lie group.

## Contribution

It introduces invariant tensors under twin interchange in almost paracomplex pseudo-Riemannian manifolds and constructs explicit examples on a 4-dimensional Lie group.

## Key findings

- Identified tensors invariant under twin interchange
- Constructed explicit invariant objects on a Lie group
- Provided a new example of such manifolds

## Abstract

The object of study is almost paracomplex pseudo-Riemannian manifolds with a pair of metrics associated each other by the almost paracomplex structure. 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 associated metrics and the corresponding Levi-Civita connections. A Lie group depending on two real parameters is constructed as an example of a 4-dimensional manifold of the studied type and the mentioned invariant objects are found in an explicit form.