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Yuri Nikolayevsky

Publications and source records attributed to Yuri Nikolayevsky.

At least 19 recordsLinked to original sources
arXiv2026

Quadratic Killing tensors on some symmetric spaces of higher rank

All Killing tensor fields on the spaces of constant curvature and on the complex projective space are decomposable, that is, can be represented as the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial integral of the geodesic flow is a polynomial in the linear integrals). This is no longer true for quadratic Killing tensor fields on the quaternionic projective spaces $\mathbb{H} P^n, \, n \ge 3$, and on the Cayley projective plane $\mathbb{O} P^2$. We prove that for the real Grassmannians and for the spaces $\mathrm{SL}(n)/\mathrm{SO}(n)$, all quadratic Killing tensor fields are decomposable.

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arXiv2026

Killing tensors on projective spaces

A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A (contravariant) Killing tensor field is called \emph{decomposable} if it is a polynomial in Killing vector fields. While all Killing tensor fields on the spaces of constant curvature and on the complex projective space are decomposable, there is an explicitly constructed family of indecomposable quadratic Killing tensor fields on the quaternionic projective spaces $\mathbb{H}P^n, \, n \ge 3$. We prove that the algebra of Killing tensor fields on the quaternionic projective space is generated by Killing vector fields and these indecomposable quadratic Killing tensor fields. We also give another proof of the fact that the algebra of Killing tensor fields on the complex projective space is generated by Killing vector fields.

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arXiv2026

Quadratic Killing tensors on classical Lie groups are decomposable

A Killing tensor field on a Riemannian manifold $(M,g)$ is a covariant symmetric tensor field whose contraction with the velocity vector along a geodesic produces a homogeneous polynomial first integral of the geodesic flow. Such a tensor is called \emph{decomposable} if it lies in the subalgebra generated by Killing vector fields; equivalently, the corresponding polynomial integral is then a polynomial in the linear integrals coming from infinitesimal isometries. On spaces of constant sectional curvature and on the complex projective space, every Killing tensor field is decomposable. By contrast, the quaternionic projective spaces and the Cayley projective plane admit indecomposable quadratic Killing tensor fields. We prove that every quadratic Killing tensor field on the compact classical Lie groups $\mathrm{SO}(n)$, $\mathrm{Spin}(n)$, $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$, equipped with a bi-invariant Riemannian metric, is decomposable; equivalently, every quadratic first integral of the geodesic flow on these groups is a quadratic polynomial in the linear first integrals.

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arXiv2026

On Killing tensors on Riemannian symmetric spaces

A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A Killing tensor field is called decomposable if it is a polynomial in Killing vector fields. In this paper, we first prove that the study of Killing tensor fields on symmetric spaces can be reduced to the case of compact irreducible ones. Then we introduce the class of top slot Killing tensor fields. We obtain an explicit and elegant description of such tensor fields and prove that the quadratic Killing tensor fields are spanned by the top-slot ones. We also show that quadratic Killing tensor fields on the quaternionic projective space and on the Cayley projective space are spanned by the indecomposable ones constructed in our earlier paper and the decomposable ones. This completes the classification of quadratic Killing tensor fields on Riemannian symmetric spaces of rank one.

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arXiv2025

Non-singular geodesic orbit nilmanifolds

A Riemannian manifold is called a geodesic orbit manifolds, GO for short, if any geodesic is an orbit of a one-parameter group of isometries. By a result of C.Gordon, a non-flat GO nilmanifold is necessarily a two-step nilpotent Lie group with a left-invariant metric. We give a complete classification of non-singular GO nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and two one-parameter families of dimensions 14 and 15.

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arXiv2024

Three Theorems on Negami's Planar Cover Conjecture

A long-standing Conjecture of S. Negami states that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It is known that the Conjecture is equivalent to the fact that \emph{the graph $K_{1,2, 2, 2}$ has no finite planar cover}. We prove three theorems showing that the graph $K_{1,2, 2, 2}$ admits no planar cover with certain structural properties, and that the minimal planar cover of $K_{1,2, 2, 2}$ (if it exists) must be $4$-connected.

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arXiv2024

Weakly Einstein hypersurfaces in space forms

A Riemannian manifold $(M,g)$ is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We give a complete classification of weakly Einstein hypersurfaces in the spaces of nonzero constant curvature (the classification in a Euclidean space has been previously known). The main result states that such a hypersurface can only be the product of two spaces of constant curvature or a rotation hypersurface.

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arXiv2024

On weakly Einstein Lie groups

A Riemannian manifold is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We consider weakly Einstein Lie groups with a left-invariant metric which are weakly Einstein. We prove that there exist no weakly Einstein non-abelian $2$-step nilpotent Lie groups and no weakly Einstein non-abelian nilpotent Lie groups whose dimension is at most $5$. We also prove that an almost abelian Lie group is weakly Einstein if and only if at the Lie algebra level it is defined by a normal operator whose square is a multiple of the identity.

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arXiv2024

Killing tensors on reducible spaces

We prove that on the product of two Riemannian manifolds one of which is compact, any Killing tensor is reducible, that is, is the sum of products of Killing tensors on the factors. The same is true for the lifts to the universal cover of Killing tensors on a compact manifold with reducible holonomy. We give a local description of Killing tensors on product manifolds and present an example of a complete product manifold whose factors are locally irreducible which admits an irreducible Killing tensor field.

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arXiv2024

$K_{1,2,2,2}$ has no $n$-fold planar cover graph for $n<14$

S. Negami conjectured in $1988$ that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It follows from the works of D. Archdeacon, M. Fellows, P. Hliněný, and S. Negami that this conjecture is true if the graph $K_{1, 2, 2, 2}$ has no finite planar cover. We prove a number of structural results about putative finite planar covers of $K_{1,2,2,2}$ that may be of independent interest. We then apply these results to prove that $K_{1, 2, 2, 2}$ has no planar cover of fold number less than $14$.

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arXiv2023

Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields

Every Killing tensor field on the space of constant curvature and on the complex projective space can be decomposed into the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial in the velocities integral of the geodesic flow is a polynomial in the linear integrals). This fact led to the natural question on whether this property is shared by Killing tensor fields on all Riemannian symmetric spaces. We answer this question in the negative by constructing explicit examples of quadratic Killing tensor fields which are not quadratic forms in the Killing vector fields on the quaternionic projective spaces $\mathbb{H} P^n, n \ge 3$, and on the Cayley projective plane $\mathbb{O} P^2$.

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arXiv2023

Pseudo-Riemannian geodesic orbit nilmanifolds of signature $\boldsymbol{(n-2,2)}$

The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds $M = G/H$ of signature $(n-2,2)$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. For that we suppose that there is a reductive decomposition $\g = \h \oplus \n \text{ (vector space direct sum) with } [\h,\n] \subset \n$ and $\n$ nilpotent. When the metric is nondegenerate on $[\n,\n]$ we show that $\n$ is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on $[\n,\n]$ we show that $\n$ is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature.

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arXiv2023

On triangular biregular degree sequences

A simple graph is called triangular if every edge of it belongs to a triangle. We conjecture that any graphical degree sequence all terms of which are greater than or equal to 4 has a triangular realisation, and establish this conjecture for a class of biregular graphical degree sequences.

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arXiv2022

The Structure of Geodesic Orbit Lorentz Nilmanifolds

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $M = G/H$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. Suppose that there is a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{n}$ (vector space direct sum) with $\mathfrak{n}$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case we construct examples to show that the number of nilpotency steps is unbounded.

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arXiv2022

Respectful decompositions of Lie algebras

One of Pierre Molino's principal mathematical achievements was his theory of Riemannian foliations. One of his last papers, published in 2001, showed that his theory could be extended to a large class of non-integrable distributions. The key example here is that of a \emph{respectful decomposition} of a Lie algebra $\mathfrak{g}$; this is vector space decomposition $\mathfrak{g}=H+V$ such that $[V,H]\subseteq H$. This paper will examine the basic properties of respectful decompositions.

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arXiv2022

Stability of geodesic vectors in low-dimensional Lie algebras

A naturally parameterised curve in a Lie group with a left invariant metric is a geodesic, if its tangent vector left-translated to the identity satisfies the Euler equation $\dot{Y}=\operatorname{ad}^t_YY$ on the Lie algebra $\mathfrak{g}$ of $G$. Stationary points (equilibria) of the Euler equation are called geodesic vectors: the geodesic starting at the identity in the direction of a geodesic vector is a one-parameter subgroup of $G$. We give a complete classification of Lyapunov stable and unstable geodesic vectors for metric Lie algebras of dimension $3$ and for unimodular metric Lie algebras of dimension $4$.

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arXiv2021

Einstein hypersurfaces in irreducible symmetric spaces

We show that if $M$ is an Einstein hypersurface in an irreducible Riemannian symmetric space $\overline{M}$ of rank greater than $1$ (the classification in the rank-one case was previously known), then either $\overline{M}$ is of noncompact type and $M$ is a codimension one Einstein solvmanifold, or $\overline{M}=\mathrm{SU}(3)/\mathrm{SO}(3)$ (respectively, $\overline{M}=\mathrm{SL}(3)/\mathrm{SO}(3)$) and $M$ is foliated by totally geodesic spheres (respectively, hyperbolic planes) of $\overline{M}$, with the space of leaves parametrised by a special Legendrian surface in $S^5$ (respectively, by a proper affine sphere in $\mathbb{R}^3$).

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