On Almost Paracontact Almost Paracomplex Riemannian Manifolds

TL;DR

This paper investigates the geometric properties of almost paracontact almost paracomplex Riemannian manifolds, deriving tensor components, characterizing low-dimensional cases, and exploring special structures like paracontact and para-Sasakian types.

Contribution

It provides a detailed analysis of tensor components and characterizations of these manifolds, including low-dimensional cases and special structures, with new examples.

Findings

Tensor components are explicitly derived for basic classes.

Low-dimensional (3D) cases are characterized using Nijenhuis tensor.

Examples illustrating theoretical results are constructed.

Abstract

Almost paracontact manifolds of an odd dimension having an almost paracomplex structure on the paracontact distribution are studied. The components of the fundamental (0,3)-tensor, derived by the covariant derivative of the structure endomorphism and the metric on the considered manifolds in each of the basic classes, are obtained. Then, the case of the lowest dimension 3 of these manifolds is considered. An associated tensor of the Nijenhuis tensor is introduced and the studied manifolds are characterized with respect to this pair of tensors. Moreover, cases of paracontact and para-Sasakian types are commented. A family of examples is given.