scholarly journals Tangent bundle geometry from dynamics: Application to the Kepler problem

2017 ◽  
Vol 14 (03) ◽  
pp. 1750047 ◽  
Author(s):  
J. F. Cariñena ◽  
J. Clemente-Gallardo ◽  
J. A. Jover-Galtier ◽  
G. Marmo
Keyword(s):  
Vector Field ◽  
Harmonic Oscillator ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Kepler Problem ◽  
Second Order ◽  
Affine Spaces ◽  
Conformal Vector ◽  
Conformal Vector Fields

In this paper, we consider a manifold with a dynamical vector field and enquire about the possible tangent bundle structures which would turn the starting vector field into a second-order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second-order and apply the procedure to vector fields conformally related with the harmonic oscillator ([Formula: see text]-oscillators). We select one which covers the vector field describing the Kepler problem.

scholarly journals REDUCTION AND UNFOLDING: THE KEPLER PROBLEM

2005 ◽  
Vol 02 (01) ◽  
pp. 83-109 ◽  
Author(s):  
ANTONELLA D'AVANZO ◽  
GIUSEPPE MARMO
Keyword(s):  
Harmonic Oscillator ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Kepler Problem ◽  
Second Order ◽  
Lagrangian Formalism ◽  
Isotropic Harmonic Oscillator

In this paper, we show, in a systematic way, how to relate the Kepler problem to the isotropic harmonic oscillator. Unlike previous approaches, our constructions are carried over in the Lagrangian formalism dealing, with second order vector fields. We therefore provide a tangent bundle version of the Kustaanheimo-Stiefel map.

scholarly journals A classification of conformal vector fields on the tangent bundle

2020 ◽  
Vol 72 (5) ◽  
Author(s):  
Zohre Raei ◽  
Dariush Latifi
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Conformal Transformations ◽  
Conformal Vector ◽  
Conformal Vector Fields

UDC 514.7 Let ( M , g ) be a Riemannian manifold and T M be its tangent bundle equipped with a Riemannian (or pseudo-Riemannian) lift metric derived from g .  We give a classification of infinitesimal fibre-preserving conformal transformations on the tangent bundle.

scholarly journals Closed conformal vector fields on pseudo-Riemannian manifolds

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
D. A. Catalano
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Geometric Proof ◽  
Conformal Vector Field ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Local Coordinates

We give here a geometric proof of the existence of certain local coordinates on a pseudo-Riemannian manifold admitting a closed conformal vector field.

scholarly journals On the zeros of a conformal vector field

1974 ◽  
Vol 55 ◽  
pp. 1-3 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Short Note ◽  
Killing Vector ◽  
Connected Components ◽  
Conformal Vector Field ◽  
Totally Geodesic Submanifolds ◽  
Conformal Vector ◽  
Conformal Vector Fields

In [1] S. Kobayashi showed that the connected components of the set of zeros of a Killing vector field on a Riemannian manifold (Mn,g) are totally geodesic submanifolds of (Mn,g) of even codimension including the case of isolated singular points. The purpose of this short note is to give a simple proof of the corresponding result for conformal vector fields on compact Riemannian manifolds. In particular we prove the following

scholarly journals Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 863
Author(s):  
Amira Ishan ◽  
Sharief Deshmukh ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Vector Field ◽  
Nontrivial Solution ◽  
Vector Fields ◽  
Dimensional Sphere ◽  
Conformal Vector Field ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
De Rham

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere Sm(c) of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation.

Conformal Vector Fields and Ricci Soliton Structures on Natural Riemann Extensions

2021 ◽  
Vol 18 (2) ◽  
Author(s):  
Mohamed Tahar Kadaoui Abbassi ◽  
Noura Amri ◽  
Cornelia-Livia Bejan
Keyword(s):  
Vector Fields ◽  
Ricci Soliton ◽  
Conformal Vector ◽  
Conformal Vector Fields

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

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