Nijenhuis operators with a unity and F $F$ -manifolds

The core object of this paper is a pair $(L,e)$, where $L$ is a Nijenhuis operator and $e$ is a vector field satisfying a specific Lie derivative condition, that is, $L_{e}L=Id$. Our research unfolds in two parts. In the first part, we establish a splitting theorem for Nijenhuis operators with a unity, offering an effective reduction of their study to cases where $L$ has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for $\mathrm{gl}$-regular Nijenhuis operators with a unity around algebraically generic points, along with seminormal forms for dimensions 2 and 3. In the second part, we establish the relationship between Nijenhuis operators with a unity and $F$-manifolds. Specifically, we prove that the class of regular $F$-manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding $F$-manifolds around singularities.