Some remarks on the generalized Tanaka-Webster connection of a contact metric manifold

We find necessary and sufficient conditions for the bi-Legendrian connection $\mathrm{\nabla }$ associated to a bi-Legendrian structure $(\mathcal{F},\mathcal{G})$ on a contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$ being a metric connection and then we give conditions ensuring that $\mathrm{\nabla }$ coincides with the (generalized) Tanaka-Webster connection of $(M,\varphi ,\xi ,\eta ,g)$. Using these results, we give some interpretations of the Tanaka-Webster connection and we study the interactions between the Tanaka-Webster, the bi-Legendrian and the Levi Civita connection in a Sasakian manifold.