Metric $f$-contact manifolds satisfying the $(\kappa ,\mu )$-nullity condition

Abstract

We prove that if the $f$-sectional curvature at any point $p$ of a $(2n+s)$-dimensional $f$-$(\kappa ,\mu )$ manifold with $n>1$ is independent of the $f$-section at $p$, then it is constant on the manifold. Moreover, we also prove that an $f$-$(\kappa ,\mu )$ manifold which is not an $S$-manifold is of constant $f$-sectional curvature if and only if $\mu =\kappa +1$ and we give an explicit expression for the curvature tensor field. Finally, we present some examples.