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Y. Nikolayevsky

Publications and source records attributed to Y. Nikolayevsky.

At least 19 recordsLinked to original sources
arXiv2025

Non-singular weakly symmetric nilmanifolds

A Riemannian manifold $M$ is called weakly symmetric if any two points in $M$ can be interchanged by an isometry. The compact ones have been well understood, and the main remaining case is that of 2-step nilpotent Lie groups. We give a complete classification of simply connected non-singular weakly symmetric nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and a one-parameter family of dimensions 14. The classification is based on the authors classification of non-singular 2-step nilpotent Lie groups for which every geodesic is the image of a one parameter group of isometries.

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arXiv2022

Compact geodesic orbit spaces with a simple isotropy group

Let $M=G/H$ be a compact, simply connected, Riemannian homogeneous space, where $G$ is (almost) effective and $H$ is a simple Lie group. In this paper, we first classify all $G$-naturally reductive metrics on $M$, and then all $G$-geodesic orbit metrics on $M$.

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arXiv2018

Einstein extensions of Riemannian manifolds

Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on $N$ and $D$ which imply that the extension $M$ is Einstein. In particular, we show that in this case, $D$ has constant eigenvalues; moreover, they are all integer (up to scaling) if $\det D \ne 0$. They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of $D$ in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of $D$, including the complete classification for the case when $D$ has two eigenvalues, one of which is multiplicity free. In the most interesting case, the extension is obtained, by an explicit procedure, from an almost Kähler Ricci flat manifold (in particular, from a Calabi-Yau manifold). We also show that all Einstein extensions of dimension four are Einstein solvmanifolds. A similar result holds valid in the case when $N$ is a Lie group with a left-invariant metric, under some additional assumptions.

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arXiv2018

Geodesic orbit and naturally reductive nilmanifolds associated with graphs

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.

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arXiv2018

On cylindricity of submanifolds of nonnegative Ricci curvature in a Minkowski space

We consider Finsler submanifolds $M^n$ of nonnegative Ricci curvature in a Minkowski space $\mathbb{M}^{n+p}$ which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag curvature in a three-dimensional Minkowski space which is not locally convex.

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arXiv2017

On invariant Riemannian metrics on Ledger-Obata spaces

We study invariant metrics on Ledger-Obata spaces $F^m/\mathrm{diag}(F)$. We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case $m=3$, any invariant metric is naturally reductive. We prove that a Ledger-Obata space is a geodesic orbit space if and only if the metric is naturally reductive. We then show that a Ledger-Obata space is reducible if and only if it is isometric to the product of Ledger-Obata spaces (and give an effective method of recognising reducible metrics), and that the full connected isometry group of an irreducible Ledger-Obata space $F^m/\mathrm{diag}(F)$ is $F^m$. We deduce that a Ledger-Obata space is a geodesic orbit manifold if and only if it is the product of naturally reductive Ledger-Obata spaces.

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arXiv2015

The Duality principle for Osserman algebraic curvature tensors

We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan-Osserman condition implies the Rakić duality principle, and that the Osserman condition and the duality principle are equivalent in the diagonalisable case.

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arXiv2015

Curvature properties of metric nilpotent Lie algebras which are independent of metric

This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra $\mathfrak{g}$ (respectively, of the Grassmannian of two-planes of $\mathfrak{g}$) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on $\mathfrak{g}$. In the second part we study the subsets of $\mathfrak{g}$ which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closures of these subsets is the whole algebra $\mathfrak{g}$, apart from two exceptional cases: when $\mathfrak{g}$ is two-step nilpotent and when $\mathfrak{g}$ contains a codimension one abelian ideal.

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arXiv2015

Solvable extensions of negative Ricci curvature of filiform Lie groups

We give necessary and sufficient conditions of the existence of a left-invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra $\mathfrak{g}$ is a filiform Lie algebra $\mathfrak{n}$. It turns out that such a metric always exists, except for in the two cases, when $\mathfrak{n}$ is one of the algebras of rank two, $L_n$ or $Q_n$, and $\mathfrak{g}$ is a one-dimensional extension of $\mathfrak{n}$, in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.

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arXiv2013

Solvable Lie groups of negative Ricci curvature

We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, and discuss some open questions.

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arXiv2012

Totally geodesic hypersurfaces of homogeneous spaces

We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c) the twisted product of the line and a homogeneous space (with the warping/twisting function given explicitly). In the first case, F is also a Riemannian product; in the last two cases, it is a leaf of a totally geodesic homogeneous fibration. Case (c) can alternatively be characterised by the fact that M admits a Riemannian submersion onto the universal cover of the group SL(2) equipped with a particular left-invariant metric, and F is the preimage of the two-dimensional solvable totally geodesic subgroup.

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arXiv2009

Weyl homogeneous manifolds modeled on compact Lie groups

A Riemannian manifold is called Weyl homogeneous, if its Weyl tensors at any two points are "the same", up to a positive multiple. A Weyl homogeneous manifold is modeled on a homogeneous space $M_0$, if its Weyl tensor at every point is "the same" as the Weyl tensor of $M_0$, up to a positive multiple. We prove that a Weyl homogeneous manifold $M^n, n \ge 4$, modeled on an irreducible symmetric space $M_0$ of types II or IV (compact simple Lie group with a bi-invariant metric or its noncompact dual) is conformally equivalent to $M_0$.

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arXiv2009

Osserman manifolds and Weyl-Schouten Theorem for rank-one symmetric spaces

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in $\mathbb{R}^16$. As a consequence, we show that a Riemannian manifold having the same Weyl tensor as a rank-one symmetric space, is conformally equivalent to it.

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arXiv2008

Einstein solvmanifolds attached to two-step nilradicals

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a lack of classification results even for some well-studied classes of nilpotent Lie algebras, such as the two-step ones. In this paper, we give a classification of two-step nilpotent Einstein nilradicals in one of the rare cases when the complete set of affine invariants is known: for the two-step nilpotent Lie algebras with the two-dimensional center. Informally speaking, we prove that such a Lie algebra is an Einstein nilradical, if it is defined by a matrix pencil having no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity. We also discuss the connection between the property of a two-step nilpotent Lie algebra and its dual to be an Einstein nilradical.

math.DG↗
arXiv2008

Einstein solvmanifolds and the pre-Einstein derivation

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre-Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pre-Einstein derivation, we then give a variational characterization of Einstein nilradicals. As an application, we prove an easy-to-check convex geometry condition for a nilpotent Lie algebra with a nice basis to be an Einstein nilradical and also show that a typical two-step nilpotent Lie algebra is an Einstein nilradical.

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arXiv2007

Einstein solvmanifolds with a simple Einstein derivation

The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).

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arXiv2006

Einstein solvmanifolds with free nilradical

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the one-step (abelian) and the two-step ones, there are only six others: f(2,3), f(2,4), f(2,5), f(3,3), f(4,3), f(5,3) (where f(m,p) is a free p-step Lie algebra on m generators). The reason for that is the inequality-type restrictions on the eigenvalue type of an Einstein nilradical obtained in the paper.

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