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The core object of this paper is a pair (L,e)$(L, e)$, where L$L$ is a Nijenhuis operator and e$e$ is a vector field satisfying a specific Lie derivative condition, that is, LeL=Id$\mathcal {L}_{e}L=\operatorname{Id}$. 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$L$ has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for gl$\mathrm{gl}$‐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$F$‐manifolds. Specifically, we prove that the class of regular F$F$‐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$F$‐manifolds around singularities. [ABSTRACT FROM AUTHOR]

We describe all metrics geodesically compatible with a gl -regular Nijenhuis operator L. The set of such metrics is large enough so that a generic local curve γ is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of γ to be a g-geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g-geodesics is naturally equivalent to the Poisson action of R n on the cotangent bundle generated by the integrals coming from geodesic compatibility. [ABSTRACT FROM AUTHOR]

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and Stäckel matrices. [ABSTRACT FROM AUTHOR]

We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ⩽n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics. [ABSTRACT FROM AUTHOR]

By a (1 , 1) -tensor field K = K j i , we construct an explicit system of differential invariants that vanish if and only if there (locally) exists a metric for which K generates separation of variables. [ABSTRACT FROM AUTHOR]

This work is the first, and main, of a series of papers in progress dedicated to Nijenhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis torsion. It serves as an introduction to Nijenhuis Geometry that should be understood in much wider context than before: from local description at generic points to singularities and global analysis. The goal of the present paper is to introduce terminology, develop new important techniques (e.g., analytic functions of Nijenhuis operators, splitting theorem and linearisation), summarise and generalise basic facts (some of which are already known but we give new self-contained proofs), and more importantly, to demonstrate that the research programme proposed in the paper is realistic by proving a series of new, not at all obvious, results. [ABSTRACT FROM AUTHOR]

A field of endomorphisms R is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of R called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures, we state the linearisation problem for Nijenhuis operators and give an answer in terms of non-degenerate left-symmetric algebras. In particular, in dimension 2, we give classification of non-degenerate left-symmetric algebras for the smooth category and, with some small gaps, for the analytic one. These two cases, analytic and smooth, differ. We also obtain a complete classification of two-dimensional real left-symmetric algebras, which may be an interesting result on its own. [ABSTRACT FROM AUTHOR]