Metric $f$-contact manifolds satisfying the $(κ,μ)$-nullity condition

We prove that if the $f$-sectional curvature at any point $p$ of a $(2n+s)$-dimensional $f$-$(κ,μ)$ 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 $f$-$(κ,μ)$ manifold which is not an $S$-manifold is of constant $f$-sectional curvature if and only if $μ=κ+1$ and we give an explicit expression for the curvature tensor field. Finally, we present some examples.

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