2023/11/08 by Evgenii I. Antonov, Antonov, Evgenii I., Andrey Yu. Konyaev +1 · 2 citations
Mathematics · #32B05 #32B10 #32G99 #37K25 #37K30 #37K50 #53A45 #53A55 #53B25 #53B99 #53D45 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2311.04624
openalex publication_date 2023/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, i.e., LieeL=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 gl-regular Nijenhuis operators with a unity around algebraically generic points, along with semi-normal forms for dimensions two and three. 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. By extending our results from dimension three, we reveal semi-normal forms for corresponding F-manifolds around singularities.