When a (1,1)(1,1)-tensor generates separation of variables of a certain metric

By a $(1,1)$-tensor field $K= K^i_j$, 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.Comment: 16 pages, no figure

Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures

We consider multicomponent local Poisson structures of the form $\mathcal P_3 + \mathcal P_1$, under the assumption that the third order term $\mathcal P_3$ is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. of linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular we show that each such Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with edges labeled by numbers $\lambda_\alpha$'s and vertices labeled by natural numbers whose sum is the dimension of the manifold. These pencils are naturally related to certain (polynomial, in the most nondegenerate case) pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the Nijenhuis pencil.Comment: In Version v2, Theorem 4 and its proof are improved, and a mistake in Theorem 5 is corrected. In Version v3 we changed the signs in certain formulas for cosmetical reasons, to avoid multiple use of $(-1)^n$, and to make the paper better compatible with arXiv:2212.01605, and also updated the reference

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl-regular Nijenhuis operators. We prove the splitting Theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl-regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl-regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl-regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e., its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature

Viktor Nesmelov: Personalistic Spiritualism or Neo-Leibnizianism

The figure of Viktor Ivanovich Nesmelov, a Kazan based theologian, is of a special importance among the variety of Russian theological movements. His anthropological views take stem from an intention to reconcile a traditional view for the Christian Orthodox theology — a biblical narrative of the creation of man, as well as the patristic anthropological teaching with psychological, philosophical data and the data of natural sciences contemporary with Nesmelov’s active years. That intention was gradually unfolding into an original theological and philosophical teaching, which was based on the category of person. However, in spite of the authenticity of the author’s anthropological ideas, some views of another philosophical movement show themselves in particular moments of V. I. Nesmelov’s teaching. By the time of V. I. Nesmelov’s active years that movement had already been grounded in Russian thought and had formed, albeit somewhat incoherent, a very traceable intellectual tradition. It was the ideas of Neo-Leibnizianism movement representatives. The latter had Leibniz’s Monadology as their ideological source, although, ideological diversity, in general, was their peculiarity. Another characteristic of that movement was the personalistic emphasis in their ideas, which in some cases was of a key role. Some researchers note that Neo-Leibnizianists in most cases tended to disassociate themselves from the abovementioned tradition. This to some extent unites the former with V. I. Nesmelov, taking both him and Neo-Leibnizianists from the frames of a fixed intellectual tradition into a field of diversity and originality of their teachings. In this article an effort is taken to express the personalistic teaching of V. I. Nesmelov as well as to compare his ideas with the views of some representatives of Neo-Leibnizianism. The ideological vagueness of the latter movement gives some space to undertake such comparison, which is localized, in this case, in the particular moments of V. I. Nesmelov’s teaching. An effort is taken as well to define the author’s ideas, taking into account his personal methodological emphasis

Orthogonal separation of variables for spaces of constant curvature

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 the St\"ackel matrices.Comment: 28 pages, one figure. Comments are welcom

Nijenhuis Geometry

This work is the first, and main, of the series of papers in progress dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis tensor. 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

Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and 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 $\le n+2$ 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 generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.Comment: comments are welcom

Applications of Nijenhuis Geometry V: geodesically equivalent metrics and finite-dimensional reductions of certain integrable quasilinear systems

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 $\gamma$ 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 $\gamma$ 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 $\mathbb{R}^n$ on the cotangent bundle generated by the integrals coming from geodesic compatibility