Isometric Flows and Killing Vector Fields on Riemannian Manifolds

2020 ◽  
pp. 117-165
Author(s):  
Valerii Berestovskii ◽  
Yurii Nikonorov
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields

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 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 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.

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.

On Locally Nonhomogeneous Pseudo–Riemannian Manifolds with Locally Homogeneous Levi–Civita Connections

2003 ◽  
Vol 14 (05) ◽  
pp. 559-572 ◽  
Author(s):  
Oldřich Kowalski ◽  
Zdeněk Vlášek ◽  
Barbara Opozda
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Killing Vector ◽  
Higher Dimensions ◽  
Killing Vector Fields ◽  
The Difference ◽  
Locally Homogeneous ◽  
Full Classification

In this paper we make the first steps to a classification of (pseudo-) Riemannian manifolds which are not locally homogeneous but their Levi–Civita connections are homogeneous. The full classification is given for dimension n = 2; in higher dimensions we prove some substantial partial results. In more generality, we are also interested in the difference between the dimension of the algebra of affine Killing vector fields and that of the algebra of metric Killing vector fields (without any homogeneity properties).

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