scholarly journals A Survey on Geometric Dynamics of 4-Walker Manifold

2011 ◽  
Vol 02 (11) ◽  
pp. 1318-1323
Author(s):  
Mehmet Tekkoyun
Keyword(s):  
Geometric Dynamics ◽  
Walker Manifold

Para-Kähler–Einstein structures on Walker 4-manifolds

2016 ◽  
Vol 13 (02) ◽  
pp. 1650006
Author(s):  
Murat Iscan ◽  
Gulnur Caglar
Keyword(s):  
Riemannian Manifold ◽  
Goldberg Conjecture ◽  
Walker Manifold

A 4-dimensional Walker manifold [Formula: see text] is a semi-Riemannian manifold [Formula: see text] of signature (++––) (or neutral), which admits a field of null 2-plane. The goal of this paper is to study certain almost paracomplex structures [Formula: see text] on 4-dimensional Walker manifolds. We discuss when these structures are integrable and when the para-Kähler forms are symplectic. We show that such a Walker 4-manifold can carry a class of indefinite para-Kähler–Einstein 4-manifolds, examples of indefinite para-Kähler 4-manifolds, and also almost indefinite para-Hermitian–Einstein 4-manifold. Finally, we give a counterexample for the almost para-Hemitian version of Goldberg conjecture.

scholarly journals Geometric Dynamics on Riemannian Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 79 ◽  
Author(s):  
Constantin Udriste ◽  
Ionel Tevy
Keyword(s):  
Differential Equations ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Second Order ◽  
Time Dynamics ◽  
Outer Planets ◽  
Time Motion ◽  
Euclidean Metrics ◽  
Economical System ◽  
Geometric Dynamics

The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows.

Geometric Spectral Algorithms for the Simulation of Rigid Bodies

2019 ◽  
Vol 14 (12) ◽  
Author(s):  
Yiqun Li ◽  
Razikhova Meiramgul ◽  
Jiankui Chen ◽  
Zhouping Yin
Keyword(s):  
Lie Group ◽  
Spectral Methods ◽  
Rigid Bodies ◽  
Geometric Control ◽  
Homogeneous Manifolds ◽  
Practical Algorithms ◽  
Group Methods ◽  
Dynamics And Control ◽  
And Control ◽  
Geometric Dynamics

Abstract Lie group methods are an excellent choice for simulating differential equations evolving on Lie groups or homogeneous manifolds, as they can preserve the underlying geometric structures of the corresponding manifolds. Spectral methods are a popular choice for constructing numerical approximations for smooth problems, as they can converge geometrically. In this paper, we focus on developing numerical methods for the simulation of geometric dynamics and control of rigid body systems. Practical algorithms, which combine the advantages of Lie group methods and spectral methods, are given and they are tested both in a geometric dynamic system and a geometric control system.

Almost Kähler eight-dimensional Walker manifold

2018 ◽  
Vol 48 (1) ◽  
pp. 129-141
Author(s):  
Abdoul Salam Diallo ◽  
Silas Longwap ◽  
Fortuné Massamba
Keyword(s):  
Almost Kähler ◽  
Walker Manifold

scholarly journals Jet theoretical Yang-Mills energy in the geometric dynamics of two-dimensional monolayer

2013 ◽  
Vol 54 (3) ◽  
pp. 031508 ◽  
Author(s):  
M. Neagu ◽  
N. G. Krylova ◽  
H. V. Grushevskaya
Keyword(s):  
Two Dimensional ◽  
Yang Mills ◽  
Geometric Dynamics

Geometric Dynamics

2000 ◽  
Vol 24 (2) ◽  
pp. 313-322 ◽  
Author(s):  
Constantin Udrişte
Keyword(s):  
Geometric Dynamics

A NOTE ON THE GOLDBERG CONJECTURE OF WALKER MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 925-928 ◽  
Author(s):  
A. A. SALIMOV
Keyword(s):  
Compact Manifold ◽  
Complex Structure ◽  
Additional Restriction ◽  
Almost Complex Structure ◽  
Almost Complex ◽  
Walker Metric ◽  
Almost Kähler ◽  
Goldberg Conjecture ◽  
Walker Manifold

This paper is concerned with Goldberg conjecture. Using the ϕφ-operator we prove the following result. Let (M, φ, w g) be an almost Kähler–Walker–Einstein compact manifold with the proper almost complex structure φ. The proper almost complex structure φ on Walker manifold (M, w g) is integrable if ϕφgN+ = 0, where gN+ is the induced Norden–Walker metric on M. This resolves a conjecture of Goldberg under the additional restriction on Norden–Walker metric (gN+ ∈ Ker ϕφ).

scholarly journals Geometric dynamics of optimization

2013 ◽  
Vol 11 (1) ◽  
pp. 163-231 ◽  
Author(s):  
François Gay-Balmaz ◽  
Darryl D. Holm ◽  
Tudor S. Ratiu
Keyword(s):  
Geometric Dynamics
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