On the flatness of a class of metric f-manifolds

Abstract

We consider a metric $f$--structure on a manifold $M$ of dimension $2n+s$. We suppose that its kernel is paralellizable by global orthonormal vector fields $\xi_1,\dots,\xi_s$ and that the dual 1--forms satisfy $d\eta^k=F$ where $F$ is the associated Sasaki 2--form and $k=1,\dots,s$. We prove that if $n$ is greater than one then $M$ cannot be flat. This is a generalization of a result by D.E.Blair proved for contact metric manifolds. We also give a counterexample in the case $n=1$.