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

##### Authors
Montano, Beniamino Cappelletti
##### Description
In this paper 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 to 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 $\pm 1$ admits a compatible Sasakian or Tanaka-Webster parallel structure according to the circumstance that $|I_M|>1$ or $|I_M|<1$, respectively.
Comment: accepted for publication on Illinois Journal of Mathematics
##### Keywords
Mathematics - Differential Geometry, Mathematics - Symplectic Geometry, 53C12