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

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.

arXiv commentLarge part of paper is irrelevant in study of three dimensional manifolds. It is not quite evident but differential equation considered in paper for time-dependent matrix is satisfied identically - because of dimension. Yet classification remains valid and will be merged in coming paper with similar study concerning almost para-contact metric structures on three-dimensional Lie groups