scholarly journals Three-dimensional discrete curvature flows and discrete Einstein metrics

2017 ◽  
Vol 287 (1) ◽  
pp. 49-70 ◽  
Author(s):  
Huabin Ge ◽  
Xu Xu ◽  
Shijin Zhang
Keyword(s):  
Einstein Metrics ◽  
Three Dimensional ◽  
Discrete Curvature

scholarly journals On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

2021 ◽  
pp. 86-90
Author(s):  
D.V. Vylegzhanin ◽  
P.N. Klepikov ◽  
E.D. Rodionov ◽  
O.P. Khromova
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Natural Generalization ◽  
Ricci Solitons ◽  
Tensor Fields ◽  
Left Invariant Riemannian Metric ◽  
Riemannian Case

Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

scholarly journals Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

2021 ◽  
Vol 60 (1) ◽  
pp. 23-29
Author(s):  
Pavel N. Klepikov ◽  
Evgeny D. Rodionov ◽  
Olesya P. Khromova
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Riemannian Metric ◽  
Natural Generalization ◽  
Tensor Fields ◽  
Parallel Transfer ◽  
Left Invariant Riemannian Metric

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

Kähler–Einstein metrics with SU(2) action

1994 ◽  
Vol 115 (3) ◽  
pp. 513-525 ◽  
Author(s):  
Andrew S. Dancer ◽  
Ian A. B. Strachan
Keyword(s):  
Einstein Metrics ◽  
Complex Structure ◽  
Three Dimensional ◽  
Isometric Action ◽  
Real Dimension ◽  
Image Position ◽  
Kähler Einstein Metrics

The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined byis closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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