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

Abstract The core object of this paper is a pair , where is a Nijenhuis operator and is a vector field satisfying a specific Lie derivative condition, that is, . 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 has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for ‐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 ‐manifolds. Specifically, we prove that the class of regular ‐manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding ‐manifolds around singularities.