scholarly journals Killing vector fields of constant length on compact homogeneous Riemannian manifolds

2015 ◽  
Vol 48 (4) ◽  
pp. 305-330 ◽  
Author(s):  
Yuriĭ Gennadievich Nikonorov
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Constant Length ◽  
Homogeneous Riemannian Manifolds

scholarly journals Killing vector fields of constant length on Riemannian manifolds

2008 ◽  
Vol 49 (3) ◽  
pp. 395-407 ◽  
Author(s):  
V. N. Berestovskii ◽  
Yu. G. Nikonorov
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Constant Length

scholarly journals Investigation of conformally killing vector fields on 5-dimensional 2-symmetric lorentzian manifolds

2021 ◽  
Vol 60 (1) ◽  
pp. 17-22
Author(s):  
Tatiana A. Andreeva ◽  
Dmitry N. Oskorbin ◽  
Evgeny D. Rodionov
Keyword(s):  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Lorentzian Manifolds ◽  
Killing Vector Fields ◽  
Homogeneous Riemannian Manifolds ◽  
Locally Homogeneous ◽  
Riemannian Case ◽  
Riemannian Symmetric Spaces

Conformally Killing fields play an important role in the theory of Ricci solitons and also generate an important class of locally conformally homogeneous (pseudo) Riemannian manifolds. In the Riemannian case, V. V. Slavsky and E.D. Rodionov proved that such spaces are either conformally flat or conformally equivalent to locally homogeneous Riemannian manifolds. In the pseudo-Riemannian case, the question of their structure remains open. Pseudo-Riemannian symmetric spaces of order k, where k 2, play an important role in research in pseudo-Riemannian geometry. Currently, they have been investigated in cases k=2,3 by D.V. Alekseevsky, A.S. Galaev and others. For arbitrary k, non-trivial examples of such spaces are known: generalized Kachen - Wallach manifolds. In the case of small dimensions, these spaces and Killing vector fields on them were studied by D.N. Oskorbin, E.D. Rodionov, and I.V. Ernst with the helpof systems of computer mathematics. In this paper, using the Sagemath SCM, we investigate conformally Killing vector fields on five-dimensional indecomposable 2- symmetric Lorentzian manifolds, and construct an algorithm for their computation.

scholarly journals Riemannian manifolds with many Killing vector fields

1980 ◽  
Vol 105 (3) ◽  
pp. 241-247 ◽  
Author(s):  
Ann Stehney ◽  
Richard Millman
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields

scholarly journals Clifford–Wolf translations of Finsler spaces

2014 ◽  
Vol 26 (5) ◽  
Author(s):  
Shaoqiang Deng ◽  
Ming Xu
Keyword(s):  
Vector Fields ◽  
Killing Vector ◽  
Finsler Spaces ◽  
Killing Vector Fields ◽  
Constant Length ◽  
Killing Field ◽  
Sufficient And Necessary Conditions ◽  
Group Of Isometries ◽  
Special Case

AbstractIn this paper, we study Clifford–Wolf translations of Finsler spaces. We give a characterization of those Clifford–Wolf translations generated by Killing vector fields. In particular, we show that there is a natural interrelation between the local one-parameter groups of Clifford–Wolf translations and the Killing vector fields of constant length. In the special case of homogeneous Randers spaces, we give some explicit sufficient and necessary conditions for a Killing vector field to have a constant length, in which case the local one-parameter group of isometries generated by the Killing field consist of Clifford–Wolf translations. Finally, we construct explicit examples to explain some of the results of this paper.

scholarly journals Conformally Killing Fields on 2-Symmetric Five-Dimensional Lorentzian Manifolds

2021 ◽  
pp. 68-71
Author(s):  
T.A. Andreeva ◽  
V.V. Balashchenko ◽  
D.N. Oskorbin ◽  
E.D. Rodionov
Keyword(s):  
General Solution ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Killing Vector ◽  
Ricci Soliton ◽  
Lorentzian Manifolds ◽  
Killing Vector Fields ◽  
Soliton Equation ◽  
Killing Fields ◽  
Einstein Constant

The papers of many mathematicians are devoted to the study of conformally Killing vector fields. Being a natural generalization of the concept of Killing vector fields, these fields generate a Lie algebra corresponding to the Lie group of conformal transformations of the manifold. Moreover, they generate the class of locally conformally homogeneous (pseudo) Riemannian manifolds studied by V.V. Slavsky and E.D. Rodionov. Ricci solitons, which R. Hamilton first considered, are another important area of research. Ricci solitons are a generalization of Einstein's metrics on (pseudo) Riemannian manifolds. The Ricci soliton equation has been studied on various classes of manifolds by many mathematicians. In particular, a general solution of the Ricci soliton equation was found on 2-symmetric Lorentzian manifolds of low dimension, and the solvability of this equation in the class of 3-symmetric Lorentzian manifolds was proved. The Killing vector fields make it possible to find the general solution of the Ricci soliton equation in the case of the constancy of the Einstein constant in the Ricci soliton equation. However, the role of the Killing fields is played by conformally Killing vector fields for different values of the Einstein constant. In this paper, we investigate conformal Killing vector fields on 5-dimensional 2-symmetric Lorentzian manifolds. The general solution of the conformal analog of the Killing equation on five-dimensional locally indecomposable 2-symmetric Lorentzian manifolds is described in local coordinates, discovered by A.S. Galaev and D.V. Alekseevsky.

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