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

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.