A study of three-dimensional paracontact ((kappa)over-tilde, (mu)over-tilde, (nu)over-tilde)-spaces

Abstract

This paper is a study of three-dimensional paracontact metric ((kappa) over tilde, (mu) over tilde, (nu) over tilde)-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field xi is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric ((kappa) over tilde, (mu) over tilde, (nu) over tilde)-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases (kappa) over tilde > -1, (kappa) over tilde = -1, (kappa) over tilde < -1 and construct new examples of such manifolds for each case. We also show the existence of paracontact metric (-1, <(mu)over tilde> not equal 0, (nu) over tilde not equal 0) spaces with dimension greater than 3, such that (h) over tilde (2) = 0 but (h) over tilde not equal 0.