On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Oldrich Kowalski
Keyword(s):  
Three Dimensional ◽  
Lorentzian Manifolds ◽  
Ricci Operator ◽  
Locally Homogeneous

AbstractWe determine the admissible forms for the Ricci operator of three-dimensional locally homogeneous Lorentzian manifolds.

scholarly journals Eigenvalues of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with a Four-Dimensional Isotropy Subgroup

2021 ◽  
pp. 93-96
Author(s):  
D.V. Vylegzhanin ◽  
P.N. Klepikov ◽  
O.P. Khromova
Keyword(s):  
Riemannian Manifolds ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Isotropy Subgroup ◽  
Lorentzian Manifolds ◽  
Ricci Operator ◽  
Homogeneous Riemannian Manifolds ◽  
Left Invariant Riemannian Metric ◽  
Locally Homogeneous ◽  
Unique Eigenvalue

The problem of restoring a (pseudo)Riemannian manifold  from a given Ricci operator was studied in the papers of many mathematicians. This problem was solved by O. Kowalski and S. Nikcevic for the case of three-dimensional locally homogeneous Riemannian manifolds. The work of G. Calvaruso and O. Kowalski contains the answer to the question above for the case of three –dimensional locally homogeneous Lorentzian manifolds. For the four-dimensional case, similar studies were carried out only in the case of Lie groups with a left-invariant Riemannian metric. The works of A.G. Kremlyov and Yu.G. Nikonorov presented the possible signatures of the eigenvalues of the Ricci operator. However, the question of recovering a four-dimensional Lie group with a left-invariant Riemannian metric from a given Ricci operator remains open. This paper is devoted to the study of the eigenvalues of the Ricci operator on four-dimensional locally homogeneous (pseudo)Riemannian manifolds with a four-dimensional isotropy subgroup. An algorithm for calculating the eigenvalues of the Ricci operator is presented. A theorem on the restoration of such manifolds from a given Ricci operator is proved. It is established that such possibility can happen only in the case when the prescribed operator is diagonalizable and has a unique eigenvalue of multiplicity four.

scholarly journals Mathematical Modeling in the Study of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with Isotropic Weyl Tensor

2020 ◽  
pp. 92-95
Author(s):  
S.V. Klepikova ◽  
T.P. Makhaeva
Keyword(s):  
Riemannian Manifolds ◽  
Weyl Tensor ◽  
Three Dimensional ◽  
Isotropy Subgroup ◽  
Ricci Operator ◽  
One Dimensional ◽  
3 Dimensional ◽  
Homogeneous Riemannian Manifolds ◽  
Locally Homogeneous ◽  
The One

It is known that a locally homogeneous manifold can be obtained from a locally conformally homogeneous (pseudo)Riemannian manifolds by a conformal deformation if the Weyl tensor (or the Schouten-Weyl tensor in the three-dimensional case) has a nonzero squared length. Thus, the problem arises of studying (pseudo)Riemannian locally homogeneous and locally conformally homogeneous manifolds, the Weyl tensor of which has zero squared length, and itself is not equal to zero (in this case, the Weyl tensor is called isotropic). One of the important aspects in the study of such manifolds is the study of the curvature operators on them, namely, the problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator. The problem of the prescribed values of the Ricci operator on 3-dimensional locally homogeneous Riemannian manifolds has been solved by O. Kowalski and S. Nikcevic. Analogous results for the one-dimensional and sectional curvature operators were obtained by D.N. Oskorbin, E.D. Rodionov, and O.P Khromova. This paper is devoted to the description of an example of studying the problem of the prescribed Ricci operator for four-dimensional locally homogeneous (pseudo) Riemannian manifolds with a nontrivial isotropy subgroup and isotropic Weyl tensor.

Three-Dimensional Naturally Reductive Homogeneous Lorentzian Manifolds

2008 ◽  
Vol 5 (1) ◽  
pp. 113-131 ◽  
Author(s):  
Nadjia Haouari ◽  
Wafaa Batat ◽  
Noureddine Rahmani ◽  
Salima Rahmani
Keyword(s):  
Three Dimensional ◽  
Lorentzian Manifolds ◽  
Naturally Reductive

scholarly journals Trans-Sasakian 3-Manifolds with Reeb Flow Invariant Ricci Operator

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 246
Author(s):  
Yan Zhao ◽  
Wenjie Wang ◽  
Ximin Liu
Keyword(s):  
Sectional Curvature ◽  
Three Dimensional ◽  
Sasakian Manifold ◽  
Constant Sectional Curvature ◽  
Ricci Operator ◽  
Cosymplectic Manifold

Let M be a three-dimensional trans-Sasakian manifold of type ( α , β ) . In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an α -Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Applying this, we give a new characterization of proper trans-Sasakian 3-manifolds.

CERTAIN RESULTS ON THREE-DIMENSIONAL CONTACT METRIC MANIFOLDS

2010 ◽  
Vol 03 (04) ◽  
pp. 577-591 ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Ricci Tensor ◽  
Three Dimensional ◽  
Tensor Fields ◽  
Contact Metric Manifold ◽  
Ricci Operator ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Parallel Ricci Tensor

In this paper we study 3-dimensional contact metric manifolds satisfying certain conditions on the tensor fields *-Ricci tensorS*, h(= ½Lξφ), τ(= Lξg = 2hφ) and the Ricci operator Q. First, we prove that a 3-dimensional non-Sasakian contact metric manifold satisfies. [Formula: see text] (where ⊕X,Y,Z denotes the cyclic sum over X,Y,Z) if and only if M is a generalized (κ, μ)-space. Next, we prove that a 3-dimensional contact metric manifold with vanishing *-Ricci tensor is a generalized (κ, μ)-space. Finally, some results on 3-dimensional contact metric manifold with cyclic η-parallel h or cyclic η-parallel τ or η-parallel Ricci tensor are presented.

LORENTZIAN 3-MANIFOLDS WITH COMMUTING CURVATURE OPERATORS

2008 ◽  
Vol 05 (04) ◽  
pp. 557-572 ◽  
Author(s):  
EDUARDO GARCÍA-RÍO ◽  
ALI HAJI-BADALI ◽  
M. ELENA VÁZQUEZ-ABAL ◽  
RAMÓN VÁZQUEZ-LORENZO
Keyword(s):  
Three Dimensional ◽  
Lorentzian Manifolds

Three-dimensional Lorentzian manifolds with commuting curvature operators are studied. A complete description is given at the algebraic level. Consequences are obtained at the differentiable setting for manifolds which additionally are assumed to be locally symmetric or homogeneous.

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