Four-Dimensional Almost Einstein Manifolds with Skew-Circulant Structres

2020 ◽  
Author(s):  
Iva Dokuzova ◽  
Dimitar Razpopov
Keyword(s):  
Tangent Space ◽  
Local Coordinate System ◽  
Hermitian Manifold ◽  
Circulant Matrices ◽  
Fourth Power ◽  
Einstein Manifolds ◽  
Riemannian Connection ◽  
Skew Circulant ◽  
Sectional Curvatures ◽  
Characteristic 2

We consider a four-dimensional Riemannian manifold M with an additional structure S, whose fourth power is minus identity. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of Einstein and almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some characteristic 2-planes in a tangent space of M. We consider a Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a K\"{a}hler manifold. We construct some examples of the considered manifolds on Lie groups.

scholarly journals Four-Dimensional Almost Einstein Manifolds with Skew-Circulant Structres

2019 ◽  
Author(s):  
Iva Dokuzova ◽  
Dimitar Razpopov
Keyword(s):  
Tangent Space ◽  
Local Coordinate System ◽  
Complex Structure ◽  
Hermitian Manifold ◽  
Circulant Matrices ◽  
Fourth Power ◽  
Einstein Manifolds ◽  
Almost Hermitian Manifold ◽  
Riemannian Connection ◽  
Skew Circulant

We consider a four-dimensional Riemannian manifold M equipped with an additional tensor structure S, whose fourth power is minus identity and the second power is an almost complex structure. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of an Einstein manifolds and an almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some special 2-planes in a tangent space of M. We consider an almost Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a Kähler manifold. We construct some examples of the considered manifolds on Lie groups.

scholarly journals The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jin-jiang Yao ◽  
Zhao-lin Jiang
Keyword(s):  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Skew Circulant ◽  
The Relationship

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

scholarly journals Skew Circulant Type Matrices Involving the Sum of Fibonacci and Lucas Numbers

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Zhaolin Jiang ◽  
Yunlan Wei
Keyword(s):  
Differential Equations ◽  
Research Area ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Frobenius Norm ◽  
Lucas Numbers ◽  
Skew Circulant ◽  
Fibonacci And Lucas Numbers

Skew circulant and circulant matrices have been an ideal research area and hot issue for solving various differential equations. In this paper, the skew circulant type matrices with the sum of Fibonacci and Lucas numbers are discussed. The invertibility of the skew circulant type matrices is considered. The determinant and the inverse matrices are presented. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, the maximum row sum matrix norm, and bounds for the spread of these matrices are given, respectively.

scholarly journals Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhaolin Jiang ◽  
Tingting Xu ◽  
Fuliang Lu
Keyword(s):  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Cyclic Group ◽  
Functional Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Linear Vector ◽  
Skew Circulant ◽  
Linear Vector Space

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra ofn×ncomplex skew-circulant matrices are displayed in this paper.

On Homotopic Harmonic Maps

1967 ◽  
Vol 19 ◽  
pp. 673-687 ◽  
Author(s):  
Philip Hartman
Keyword(s):  
Coordinate System ◽  
Harmonic Maps ◽  
Local Coordinate System ◽  
Arc Length ◽  
Christoffel Symbols ◽  
Riemann Manifolds ◽  
Local Coordinates ◽  
Sectional Curvatures ◽  
Image Position

Let M, M′ be C∞ Riemann manifolds such that(1.0) M is compact;(1.1) M′ is complete and its sectional curvatures are non-positive.In terms of local coordinates x = (x1, … , xn) on M and y = (y1, … , ym) on M′, let the respective Riemann elements of arc-length beand Γijk, Γ′αβγ be the corresponding Christoffel symbols. When there is no danger of confusion, x (or y) will represent a point of M (or M′) or its coordinates in some local coordinate system.

scholarly journals Totally real surfaces of the six-dimensional sphere

1991 ◽  
Vol 33 (1) ◽  
pp. 83-87
Author(s):  
M. A. Bashir
Keyword(s):  
Division Algebra ◽  
Normal Bundle ◽  
Hermitian Manifold ◽  
Dimensional Sphere ◽  
Real Submanifolds ◽  
Almost Hermitian Manifold ◽  
Riemannian Connection ◽  
Totally Real ◽  
Natural Way

An almost Hermitian manifold (, J, g) with Riemannian connection is called nearly Kaehlerian if (xJ)X = 0 for any . The typical example is the sphere S6. The nearly Kaehlerian structure J for S6 is constructed in a natural way by making use of Cayley division algebra [3]. It is because of this nearly Kaehler, non-Kaehler, structure that S6 has attracted attention. Different classes of submanifolds of S6 have been considered by A. Gray [4], K. Sekigawa [5] and N. Ejiri [2]. In this paper we study 2-dimensional totally real submanifolds of S6. These are submanifolds with the property that for every x є M, J (Tx M) belongs to the normal bundle v. For this class we have obtained the following result.

scholarly journals A system of PDEs for Riemannian spaces

2007 ◽  
Vol 82 (2) ◽  
pp. 249-262
Author(s):  
Padma Senarath ◽  
Gillian Thornley ◽  
Bruce van Brunt
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Constant Curvature ◽  
Tangent Space ◽  
Finsler Space ◽  
System Of Equations ◽  
Finsler Spaces ◽  
Projectively Flat ◽  
Riemannian Connection ◽  
Riemannian Spaces

AbstractMatsumoto [10] remarked that some locally projectively flat Finsler spaces of non-zero constant curvature may be Riemannian spaces of non-zero constant curvature. The Riemannian connection, however, must be metric compatible, and this requirement places restrictions on the geodesic coefficients for the Finsler space in the form of a system of partial differential equations. In this paper, we derive this system of equations for the case where the geodesic coefficients are quadratic in the tangent space variables yi, and determine the solutions. We recover two standard Remannian metrics of non-zero constant curvature from this class of solutions.

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