arXiv subjects

Iva Dokuzova

Publications and source records attributed to Iva Dokuzova.

15 recordsLinked to original sources

arXiv2025

Three-dimensional Riemannian manifolds associated with locally conformal Riemannian product manifolds

A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose fourth power is the identity, is considered. This structure acts as an isometry with respect to the metric. A Riemannian almost product manifold associated with such a manifold is also studied. It turns out, that the almost product manifold belongs to the class of locally conformal Riemannian product manifolds of the Naveira classification. Conditions for the additional structures of the manifolds to be parallel with respect to the Levi-Civita connection of the metric were found. Classes of almost Einstein manifolds and Einstein manifolds are determined and some of their curvature properties are obtained. As examples of these manifolds, a hypersurface is considered.

math.DG↗

arXiv2023

Spheres and circles with respect to an indefinite metric on a Riemanian manifold with a skew-circulant structure

We study hyper-spheres, spheres and circles, with respect to an indefinite metric, in a tangent space on a 4-dimensional differentiable manifold. The manifold is equipped with a positive definite metric and an additional tensor structure of type (1, 1)11. The fourth power of the additional structure is minus identity and its components form a skew-circulant matrix in some local coordinate system. The both structures are compatible and they determine an associated indefinite metric on the manifold.

math.DG↗

arXiv2021

Curvature properties of Riemannian manifolds with skew-circulant structures

We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M, g, S), whose curvature tensors are invariant under S. For such manifolds we obtain properties of the Ricci tensor. Also we get expressions of the sectional curvatures of some special 2-planes in a tangent space of (M, g, S).

math.DG↗

arXiv2020

A Riemannian manifold with skew-circulant structures and an associated locally conformal Kähler manifold

A 4-dimensional Riemannian manifold M, equipped with an additional tensor structure S, whose fourth power is minus identity, is considered. The structure S has a skew-circulant matrix with respect to some basis of the tangent space at a point on M. Moreover, S acts as an isometry with respect to the metric g. A fundamental tensor is defined on such a manifold (M,g,S) by g and by the covariant derivative of S. This tensor satisfies a characteristic identity which is invariant to the usual conformal transformation. Some curvature properties of (M,g,S) are obtained. A Lie group as a manifold of the considered type is constructed. A Hermitian manifold associated with (M,g,S) is also considered. It turns out that it is a locally conformal Kähler manifold.

math.DG↗

arXiv2020

On 3-dimensional almost Einstein manifolds with circulant structures

A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e., these structures are circulant. An associated manifold, whose metric is expressed by both structures, is studied. Three classes of such manifolds are considered. Two of them are determined by special properties of the curvature tensor of the manifold. The third class is composed by manifolds whose structure is parallel with respect to the Levi-Civita connection of the metric. Some geometric characteristics of these manifolds are obtained. Examples of such manifolds are given.

math.DG↗

arXiv2019

Four-dimensional Riemannian product manifolds with circulant structures

A 4-dimensional Riemannian manifold equipped with an additional tensor structure, whose fourth power is the identity, is considered. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant, and it acts as an isometry with respect to the metric. The Riemannian product manifold associated with the considered manifold is studied. Conditions for the metric, which imply that the Riemannian product manifold belongs to each of the basic classes of Staikova-Gribachev's classification, are obtained. Examples of such manifolds are given.

math.DG↗

arXiv2018

On a Riemannian manifold with a circulant structure whose third power is the identity

It is studied a 3-dimensional Riemannian manifold equipped with a tensor structure of type (1,1), whose third power is the identity. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant. On such a manifold a fundamental tensor by the metric and by the covariant derivative of the circulant structure is defined. An important characteristic identity for this tensor is obtained. It is established that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity. A Lie group as a manifold of the considered type is constructed and some of its geometrical characteristics are found.

math.DG↗

arXiv2018

Two types of Lie Groups as 4-dimensional Riemannian manifolds with circulant structure

A 4-dimensional Riemannian manifold equipped with an endomorphism of the tangent bundle, whose fourth power is the identity, is considered. The matrix of this structure in some basis is circulant and the structure acts as an isometry with respect to the metric. Such manifolds are constructed on 4-dimensional real Lie groups with Lie algebras of two remarkable types. Some of their geometric characteristics are obtained.

math.DG↗

arXiv2017

Three-dimensional Riemannian manifolds with circulant structures

We consider a 3-dimensional Riemannian manifold M with two circulant structures -- a metric g and an endomorphism q whose third power is identity. The structure q is compatible with g such that an isometry is induced in any tangent space of M. We obtain some curvature properties of this manifold (M, g, q) and give an explicit example of such a manifold.

math.DG↗

arXiv2017

On a Three Dimensional Riemannian Manifold with an Additional Structure

We consider a 3-dimensional Riemannian manifold with additional structure q. We find a condition that the affine structure q is parallel with respect to the Riamannian connection.We prove the sectional curvatures of three 2-sections formed linearly independent vectors are equal among them.

math.DG↗

arXiv2016

Curvature properties of $4$-dimensional Riemannian manifolds with a circulant structure

We consider a $4$-dimensional Riemannian manifold $M$ equip\-ped with a circulant structure $q$, which is an isometry with respect to the metric $g$ and $q^{4}=\id$, $q^{2}\neq \pm \id$. For such a manifold $(M, g, q)$ we obtain some assertions for the sectional curvatures of $2$-planes. We construct an example of such a manifold on a Lie group and we find some of its geometric characteristics.

math.DG↗

arXiv2015

Riemannian manifolds with two circulant structures

We consider a three-dimensional Riemannian manifold equipped with two circulant structures - a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.

math.DG↗

arXiv2011

Almost conformal transformation in a four dimensional Riemannian manifold with an additional structure

We consider a four dimensional Riemannian manifold M with a metric g and affinor structure q. The local coordinates of these tensors are circulant matrices. Their first orders are (A, B, C, B), A, B, C\in FM and (0, 1, 0, 0), respectively. We construct another metric \tilde{g} on M. We find the conditions for \tilde{g} to be a positively defined metric, and for q to be a parallel structure with respect to the Riemannian connection of g. Further, let x be an arbitrary vector in T_{p}M, where p is a point on M. Let ϕand ϕbe the angles between x and qx, x and q^{2}x with respect to g. We express the angles between x and qx, x and q^{2}x with respect to $\tilde{g}$ with the help of the angles $ϕ$ and ϕ. Also,we construct two series {ϕ_{n}}and {ϕ_{n}}. We prove that every of it is an increasing one and it is converge.

math.DG↗

arXiv2010

On affine connections in a Riemannian manifold with a circulant metric and two circulant affinor structures

In the present paper it is considered a class V of 3-dimensional Riemannian manifolds M with a metric g and two affinor tensors q and S. It is defined another metric \bar{g} in M. The local coordinates of all these tensors are circulant matrices. It is found: 1)\ a relation between curvature tensors R and \bar{R} of g and \bar{g}, respectively; 2)\ an identity of the curvature tensor R of g in the case when the curvature tensor \bar{R} vanishes; 3)\ a relation between the sectional curvature of a 2-section of the type \{x, qx\} and the scalar curvature of M.

math.DG↗

arXiv2010

Almost conformal transformation in a class of Riemannian manifolds

We consider a 3-dimensional Riemannian manifold V with a metric g and an affinor structure q. The local coordinates of these tensors are circulant matrices. In V we define an almost conformal transformation. Using that definition we construct an infinite series of circulant metrics which are successively almost conformaly related. In this case we get some properties.

math.DG↗