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

TL;DR

This paper investigates conditions under which the $f$-sectional curvature of $f$-$( abla, abla)$ manifolds with $( abla, abla)$-nullity is constant, providing characterizations and explicit curvature tensor expressions.

Contribution

It establishes conditions for constant $f$-sectional curvature in $f$-$( abla, abla)$ manifolds satisfying the $( abla, abla)$-nullity condition, including explicit curvature tensor formulas.

Findings

$f$-sectional curvature is constant if independent of the $f$-section at each point.

Characterization of $f$-$( abla, abla)$ manifolds with constant $f$-sectional curvature.

Explicit expression for the curvature tensor in these manifolds.

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.