The foliated structure of contact metric $(κ,μ)$-spaces

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

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