Contact metric $(\kappa,\mu)$-spaces as bi-Legendrian manifolds

Abstract

We regard a contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(\kappa,\mu)$-spaces in terms of a canonical connection which can be naturally defined on them.