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Yuri Nikolayevsky

Publications and source records attributed to Yuri Nikolayevsky.

At least 37 records · Page 2Linked to original sources
arXiv2021

Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimension

Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. In this situation, the metric on $M$ induces a bilinear form $\langle\cdot,\cdot\rangle$ on the Lie algebra $\mathfrak{g}$ of $G$ which is nil-invariant, a property closely related to invariance. We study such spaces $M$ in three important cases. First, we assume $\langle\cdot,\cdot\rangle$ is invariant, in which case the Einstein property requires that $G$ is either solvable or semisimple. Next, we investigate the case where $G$ is solvable. Here, $M$ is compact and $M=G/Γ$ for a lattice $Γ$ in $G$. We show that in dimensions less or equal to $7$, compact quotients $M=G/Γ$ exist only for nilpotent groups $G$. We conjecture that this is true for any dimension. In fact, this holds if Schanuel's Conjecture on transcendental numbers is true. Finally, we consider semisimple Lie groups $G$, and find that $M$ is covered by a pseudo-Riemannian product of Einstein manifolds corresponding to the compact and the non-compact factors of $G$.

math.DG↗
arXiv2021

Einstein hypersurfaces of Damek-Ricci spaces

Einstein hypersurfaces are ``very rare" in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.

math.DG↗
arXiv2019

Bad drawings of small complete graphs

We show that for $K_5$ (resp.~ $K_{3,3}$) there is a drawing with $i$ independent crossings, and no pair of independent edges cross more than once, provided $i$ is odd with $1\le i\le 15$ (resp.~ $1\le i\le 17$). Conversely, using the deleted product cohomology, we show that for $K_5$ and $K_{3,3}$, if $A$ is any set of pairs of independent edges, and $A$ has odd cardinality, then there is a drawing in the plane for which each element in $A$ cross an odd number of times, while each pair of independent edges not in $A$ cross an even number of times. For $K_6$ we show that there is a drawing with $i$ independent crossings, and no pair of independent edges cross more than once, if and only if $3\le i\le 40$.

math.CO↗
arXiv2019

Totally geodesic submanifolds of Damek-Ricci spaces and Einstein hypersurfaces of the Cayley projective plane

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces. We also show that if a $2$-stein space admits a $2$-stein hypersurface, then both are of constant curvature, under some additional conditions.

math.DG↗
arXiv2018

Annular and pants thrackles

A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for thrackle drawings all of whose vertices lie on the boundaries of $d \le 3$ connected domains in the complement of the drawing. We also give a detailed description of thrackle drawings corresponding to the cases when $d=2$ (annular thrackles) and $d=3$ (pants thrackles).

math.CO↗
arXiv2018

On planar bipartite biregular degree sequences

A pair of sequences of natural numbers is called planar if there exists a simple, bipartite, planar graph for which the given sequences are the degree sequences of its parts. For a pair to be planar, the sums of the sequences have to be equal and Euler's inequality must be satisfied. Pairs that verify these two necessary conditions are called Eulerian. We prove that a pair of constant sequences is planar if and only if it is Eulerian (such pairs can be easily listed) and is different from $(3^5 \, | \, 3^5)$ and $(3^{25} \, | \, 5^{15})$.

math.CO↗
arXiv2016

$H$-contact unit tangent sphere bundles of Riemannian manifolds

A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein.

math.DG↗
arXiv2016

Thrackles containing a standard musquash

A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a planar thrackle drawing of a graph cannot have more edges than vertices, which is equivalent to saying that no connected component of the graph contains more than one cycle. We prove that a thrackle drawing containing a standard musquash (standard $n$-gonal thrackle) cannot contain any other cycle of length three or five.

math.CO↗
arXiv2016

Cohomology of $\mathbb{N}$-graded Lie algebras of maximal class over $\mathbb{Z}_2$

We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings $H^*(\mathfrak{m}_0)$ and $H^*(\mathfrak{m}_2)$ are isomorphic, in contrast with the case of the ground field of characteristic zero, and we obtain a complete description of them. In the finite-dimensional case, we find the first three Betti numbers of $\mathfrak{m}_0(n)$ and $\mathfrak{m}_2(n)$ over $\mathbb{Z}_2$.

math.RA↗
arXiv2015

Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties

We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is closed. We further prove a de Rham type splitting theorem which states that if such a manifold is analytic, it is isometric to the Riemannian product of a Euclidean space and an incomplete manifold.

math.DG↗
arXiv2014

Spherical Thrackles

We establish Conway's thrackle conjecture in the case of spherical thrackles; that is, for drawings on the unit sphere where the edges are arcs of great circles.

math.CO↗
arXiv2014

A counterexample to Belgun-Moroianu conjecture

We construct an example of a closed manifold with a nonflat reducible locally metric connection such that it preserves a conformal structure and such that it is not the Levi-Civita connection of a Riemannian metric.

math.DG↗
arXiv2014

Piecewise linear periodic maps of the plane with integer coefficients

We study periodic, piecewise linear maps on the plane starting with the Mort Brown's map. We show that if the number of pieces is two, there is only a short list of possible periods (this fact can be seen as the crystallographic restriction for this class of maps). Otherwise, without the restriction on the number of pieces, a map can have any period. We show how to construct such maps using binary trees and so called admissible sequences.

math.DS↗
arXiv2013

Geodesic Bases for Lie Algebras

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algebras having a codimension one ideal of a particular kind, nonunimodular algebras of dimension \$\leq 4\$, and unimodular algebras of dimension 5.

math.DG↗