Generalized $(\kappa,\mu)$-contact Metric Manifolds with $\xi\mu=0$

Abstract

This paper analytically describes the local geometry of a generalized $(\kappa,\mu )$-manifold $M(\eta,\xi,\phi,g)$ with $\kappa<1$ which satisfies the condition ``the function $\mu$ is constant along the integral curves of the characteristic vector field $\xi$''. This class of manifolds is especially rich, since it is possible to construct in $R^3$ two families of such manifolds, for any smooth function $\kappa$ ($\kappa<1$) of one variable. Every family is determined by two arbitrary functions of one variable.