Almost (para-) contact metric $(\kappa,\mu)$-manifolds. Part 1: Riemannian

TL;DR

This paper aims to classify three-dimensional para-contact and almost para-cosymplectic $(,)$-manifolds, exploring their role as fundamental components of higher-dimensional structures and proposing a unified framework.

Contribution

It introduces the first steps towards classifying para-contact and almost para-cosymplectic manifolds, and suggests a new class encompassing both structures.

Findings

Classification for contact and almost cosymplectic cases is established.

Proposes that these manifolds are building blocks for higher-dimensional structures.

Conjectures a unifying class for contact and para-contact manifolds.

Abstract

The author is planning if possible classify all three-dimensional $(\kappa,\mu)$-manifolds wether contact metric, almost cosymplectic, para-contact metric, almost para-cosymplectic. Of course classification in contact or almost cosymplectic cases already is provdied. Up to authors knowledge there is no classification for para-contact or almost para-cosymplectic $(\kappa,\mu)$. Conjecture is described by the author in coming paper structures provide classification. The main goal however is to show that these three dimensional manifolds are essentially building blocks of higher-dimensional manifolds. The other possiibilty is to introduce class of manifolds which contain both almost contact metric and almost para-contact metric manifolds as proper subclasses.