The purpose of this chapter is to introduce some special directions that belong to the contact subbundle of a contact metric manifold with negative sectional curvature for plane sections containing the characteristic vector field ξ or more generally when the operator h admits an eigenvalue greater than 1; these directions were introduced by the author in [1998]. We also discuss in this chapter some questions concerning Anosov and conformally Anosov flows. For simplicity we will often refer to the sectional curvature of plane sections containing the characteristic vector field ξ as ξ-sectional curvature.