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N. K. Smolentsev

Publications and source records attributed to N. K. Smolentsev.

14 recordsLinked to original sources
arXiv2024

Left-invariant Kähler and semi-para-Kähler structures on some six-dimensional unsolvable Lie groups

In this article studies questions about the existence of left-invariant Kähler and semi-para-Kähler structures on six-dimensional unsolvable Lie groups whose Lie algebras are semidirect products. According to the classification results, there are four such Lie algebras. It is shown that one of these four Lie algebras admits left-invariant Kähler metrics, and the other three admit left-invariant semi-para-Kähler and semi-Kähler structures. The paper also shows that a six-dimensional symplectic Lie algebra must be solvable except in one case.

math.DG↗
arXiv2024

On left-invariant semi-Kähler structures on six-dimensional nilpotent nonsymplectic Lie groups

It is known that there are 34 classes of six-dimensional nilpotent Lie groups, many of which admit left-invariant symplectic and complex structures. Among them there are three classes of groups on which there are no left-invariant symplectic structures, but there are complex structures. In this work, semi-Kähler and almost para-semi-Kähler structures are defined on such Lie groups and their geometric properties are studied.

math.DG↗
arXiv2020

On the classification of left-invariant para-Kähler structures on four-dimensional Lie groups

A first classification of para-Kähler structures on four-dimensional Lie algebras was obtained by D. Calvaruzo in 2015. In this paper, we propose another classification based on the classification of symplectic Lie algebras. For each four-dimensional symplectic Lie algebra, compatible para-complex structures and the corresponding pseudo-Riemannian metrics are found in explicit form. This leads to the classification of Sasaki para-contact structures on five-dimensional contact Lie algebras with a nontrivial center.

math.DG↗
arXiv2018

Wavelet analysis in problems of classification of ECG signals

In this paper, the wavelet analysis is used to study the ECG signal. We show that the high-frequency wavelet components of the ECG signal contain information on the functioning of the heart and can be used in diagnosis. We describe the automated classification system that separates the ECG of sick and healthy persons using only a high-frequency ECG component.

eess.SP↗
arXiv2018

Almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculated. In contrast to the usual Riemann sphere $\mathbb{S}^6$, there exist (integrable) complex structures and para-complex structures on the pseudospheres under consideration.

math.DG↗
arXiv2014

Bi-invariant metric on contact diffeomorphisms group

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the contact diffeomorphisms group $\mathcal{D}_θ$ of a contact Riemannian manifold $(M,g,θ)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_θ$ and calculate the sectional curvature of $\mathcal{D}_θ$. In a case $\dim M =3$ connection between the bi-invariant metric on $\mathcal{D}_θ$ and the bi-invariant metric on volume-preserving diffeomorphisms group $\mathcal{D}_μ$ of $M^3$ is discover.

math.DG↗
arXiv2014

Bi-invariant metric on symplectic diffeomorphisms group

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the symplectic diffeomorphisms group $\mathcal{D}_ω$ of a symplectic Riemannian manifold $(M,g,ω)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_ω$ and calculate the sectional curvature of $\mathcal{D}_ω(T^n)$.

math.DG↗
arXiv2014

Bi-invariant metric on volume-preserving diffeomorphisms group of a three-dimensional manifold

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the volume-preserving diffeomorphism group of a three-dimensional manifold and study its properties. Despite the fact that the space $\mathcal{D}_μ(M^3)$ is infinite-dimensional, we succeed in defining the signature of the bi-invariant quadric form. It is equal to the $η$-invariant of the manifold $M^3$.

math.DG↗
arXiv2013

Canonical almost pseudo-Kähler structures on six-dimensional nilpotent Lie groups

It is known that there are 34 classes of isomorphic connected simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes suppose left-invariant symplectic structures \cite{Goze-Khakim-Med}. In \cite{CFU2} it is shown that 14 classes of symplectic six-dimensional nilpotent Lie groups suppose compatible complex structures and, therefore, define pseudo-Kähler metrics. In this paper we show that on the remaining 12 classes of six-dimensional nilpotent symplectic Lie groups there are left-invariant almost pseudo-Kähler metrics, and we study their geometrical properties.

math.DG↗
arXiv2013

Canonical pseudo-Kähler structures on six-dimensional nilpotent Lie groups

In this paper we consider left-invariant pseudo-Kähler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated pseudo-Kähler metrics are investigated. It is proved that the associated pseudo-Kähler metric is Ricci-flat, that the curvature tensor has zero pseudo-Riemannian norm, and that the curvature tensor has some non-zero components that depend only on two or, at most, three parameters. The pseudo-Kähler structures obtained give basic models of pseudo-Kähler six-dimensional nilmanifolds.

math.DG↗
arXiv2007

Construction of some types wavelets with coefficient of scaling N

In this paper it is shown, that B-splines are scaling functions for any natural N. For any natural N construction of Haar wavelets, Kotelnikov-Shannon wavelets, nonorthogonal wavelets based on B-splines is given. Examples of construction of filters of N-channel decomposition of signal and filters of reconstruction based on B-splines are given

math.FA↗
arXiv2002

On the space of almost complex structures

The space ${\mathcal A}$ of almost complex structures on a closed manifold $M$ is studied. A natural parametrization of the space ${\mathcal A}$ is defined. It is shown, that ${\mathcal A}$ is a infinite dimensional complex weak Pseudo-Riemannian manifold. A curvature of the space ${\mathcal A}$ is found. The space ${\mathcal A}_ω$ of associated almost complex structures on a symplectic manifold $M,ω$ and space ${\mathcal AO}$ of orthogonal almost complex structures on a Riemannian manifold $M,g_0$ are considered in more detail.

math.DG↗
arXiv2001

The space of associated metrics on a symplectic manifold

The spaces of Riemannian metrics on a closed manifold $M$ are studied. On the space ${\mathcal M}$ of all Riemannian metrics on $M$ the various weak Riemannian structures are defined and the corresponding connections are studied. The space ${\mathcal AM}$ of associated metrics on a symplectic manifold $M,ω$ is considered in more detail. A natural parametrization of the space ${\mathcal AM}$ is defined. It is shown, that ${\mathcal AM}$ is a complex manifold. A curvature of the space ${\mathcal AM}$ and quotient space ${\mathcal AM}/{\mathcal D}_ω$ is found. The finite dimensionality of the space of associated metrics of a constant scalar curvature with Hermitian Ricci tensor is shown.

math.DG↗