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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.

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 welcome

We construct a new series of multicomponent integrable PDE systems that contain as particular example (with appropriately chosen parameters) many famous integrable systems including KdV, coupled KdV, Harry Dym, coupled Harry Dym, Camassa-Holm, multicomponent Camassa-Holm, Kaup-Boussinesq systems. The series contains also integrable systems with no low-component analogues.

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.

We show that two-dimensional conformally related Douglas metrics are Randers, Comment: are welcome

We show that for any complete connected K��hler manifold the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the K��hler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini-Study metric. This establishes a stronger version of the recently proved Yano-Obata conjecture for complete K��hler manifolds., 24 pages; a few more minor corrections; to appear in Annals of Global Analysis and Geometry

Two K\"ahler metrics on a complex manifold are called c-projectively equivalent if their $J$-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent K\"ahler metrics of arbitrary signature and use this description to prove the classical Yano-Obata conjecture: we show that on a closed connected K\"ahler manifold of arbitrary signature, any c-projective vector field is an affine vector field unless the manifold is $CP^n$ with (a multiple of) the Fubini-Study metric. As a by-product, we prove the projective Lichnerowicz conjecture for metrics of Lorentzian signature: we show that on a closed connected Lorentzian manifold, any projective vector field is an affine vector field., Comment: comments are welcome

A vector field on a K��hler manifold is called c-projective if its flow preserves the J-planar curves. We give a complete local classification of K��hler real 4-dimensional manifolds that admit an essential c-projective vector field. An important technical step is a local description of 4-dimensional c-projectively equivalent metrics of arbitrary signature. As an application of our results we prove the natural analog of the classical Yano-Obata conjecture in the pseudo-Riemannian 4-dimensional case., 33 pages, 2 figures

We show that the second greatest possible dimension of the group of (local) almost isometries of a Finsler metric is $\frac{n^2 -n}{2} +1$ for $n= dim(M)\ne 4 $ and $\frac{n^2 -n}{2} +2 =8$ for $n=4$. If a Finsler metric has the group of almost isometries of dimension greater than $\frac{n^2 -n}{2} +1$, then the Finsler metric is Randers, i.e., $F(x,y)= \sqrt{g_x(y,y)} + \tau(y)$. Moreover, if $n\ne 4$, the Riemannian metric $g$ has constant sectional curvature and, if in addition $n\ne 2$, the 1-form $\tau$ is closed, so (locally) the metric admits the group of local isometries of the maximal dimension $\frac{n(n+1)}2$., Comment: simple result; comments are welcome

We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian K\"ahler manifold consists of isometries unless the metric has constant positive holomorphic curvature., Comment: 28 pages, 3 pictures. The (v2) version contains in addition a relation between h-projectively equivalent metrics and hamiltonian 2-forms studies in math.DG/0202280, math.DG/0401320, math.DG/0511118, math.DG/0511119. In the (v3) version, we also added an appendix with the h-projectively invariant formulation of the main equations and, as their application, an alternative proof of Lemma 7

One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $��$ such that $d��$-lemma holds. We first show that such derivation $��$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $��$ (which is a natural, essentially necessary condition to get a $d��$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $d��$-lemma holds: we consider the cases when $R$ is self adjoint with respect to a Riemannian metric or compatible with an almost symplectic structure. Finally, we show that if $R$ is scew-symmetric with respect to a Riemannian metric, has constant determinant, and if its Nijenhujis torsion vanishes, then the orthogonal component of $R$ in its polar decomposition is a complex structure compatible with the metric, which gives us a new characterization of K��hler structures, 9 pages; no figures

We show that $PQ^��$-projectivity of two Riemannian metrics introduced in \cite{Top2003} implies affine equivalence of the metrics unless $��\in\{0,-1,-3,-5,-7,...\}$. Moreover, we show that for $��=0$, $PQ^��$-projectivity implies projective equivalence., 6 pages, 2 figures

I prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture stating that on a closed manifold different from the round sphere every projective (i.e., geodesic-preserving) vector field is Killing., Comment: This paper has been withdrawn by the author. this paper, together with the paper arXiv:0911.3521, is now a part of the paper arXiv:1002.3934

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, Comment: This paper has been withdrawn by the author. This paper is now, together with the paper arXiv:1002.0133, a part of the paper arXiv:1002.3934

Let Riemannian metrics $g$ and $\bar g$ on a connected manifold $M^n$ have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the famous Levi-Civita's Theorem, the metrics have a certain standard form near the point. Our main result is a generalization of Levi-Civita's Theorem for the points where the eigenvalues of one metric with respect to the other bifurcate., Comment: 28 pages; no figures

In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F. As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; this statement is a "Berwald" version of Hilbert's 4th problem. Further, we investigate geodesic equivalence of Berwald metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwald metric admits a Riemannian metric that is (nontrivially) geodesically equivalent to it. The system of PDE is linear and of Cauchy-Frobenius type, i.e., the derivatives of unknown functions are explicit expressions of the unknown functions. As an application (Corollary 2), we obtain that geodesic equivalence of an essentially Berwald metric and a Riemannian metric is always affine equivalence provided both metrics are complete., Comment: 11 pages; two .eps figures. Only grammatic and stylistic corrections with respect to (v1). The paper is accepted to Publicationes Mathematicae Debrecen and may appear in volume 74 (2009)

We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on $M^{n\ge 3}$ share the same unparametrized geodesics, and two of them (say, $g$ and $\bar g$) are strictly nonproportional (i.e., the minimal polynomial of $g^{i\alpha} \bar g_{\alpha j}$ coincides with the characteristic polynomial) at least at one point, then they have constant curvature.

A gap in the proof of the finsler "unicorn" conjecture in the paper "Regular Landsberg metrics are always Berwald" by Z. I. Szabo is pointed out, Comment: 3 pages, no figures

Suppose the Riemannian metrics $g$ and $\bar g$ on a closed connected manifold $M^n$ are geodesically equivalent and strictly non-proportional at least at one point. Then the topological entropy of the geodesic flow of $g$ vanishes., Comment: This is a slightly extended version of the paper submitted to ETDS. 16 pages, one .eps figure

This paper is a review on recently found connection between geodesically equivalent metrics and integrable geodesic flows. Suppose two different metrics on one manifold have the same geodesics. We show that then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution and vice versa. The quantum version of this result is also true. As a corollary we have that the topology of the manifold (with two different metrics with the same geodesics) must be sufficiently simple. Also we have that the non-proportionality of the metrics at a point induces the non-proportionality of the metrics at almost all points., Comment: 12 pages, LaTeX