The foliated structure of contact metric $(\kappa,\mu)$-spaces

Abstract

In this note, we study the foliated structure of a contact metric $(\kappa,\mu)$-space. In particular, using the theory of Legendre foliations, we give a geometric interpretation of the Boeckx's classification of contact metric $(\kappa,\mu)$-spaces and we find necessary conditions for a contact manifold to admit a compatible contact metric $(\kappa,\mu)$-structure. Finally, we prove that any contact metric $(\kappa,\mu)$-space $M$ whose Boeckx invariant $I_M$ is different from $\pm1$ admits a compatible Sasakian or Tanaka–Webster parallel structure according to the circumstance that $|I_M|>1$ or $|I_M| \lt 1$, respectively.