scholarly journals A Note on Geodesic Vector Fields

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1663
Author(s):  
Sharief Deshmukh ◽  
Josef Mikeš ◽  
Nasser Bin Turki ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Vector Fields ◽  
Euclidean Spaces ◽  
Geodesic Vector

The concircularity property of vector fields implies the geodesicity property, while the converse of this statement is not true. The main objective of this note is to find conditions under which the concircularity and geodesicity properties of vector fields are equivalent. Moreover, it is shown that the geodesicity property of vector fields is also useful in characterizing not only spheres, but also Euclidean spaces.

scholarly journals Geodesic Vector Fields on a Riemannian Manifold

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 137 ◽  
Author(s):  
Sharief Deshmukh ◽  
Patrik Peska ◽  
Nasser Bin Turki
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Vector Fields ◽  
Euclidean Spaces ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Geodesic Vector ◽  
Integral Curves

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.

Geodesic vector fields and Eikonal equation on a Riemannian manifold

2019 ◽  
Vol 30 (4) ◽  
pp. 542-552 ◽  
Author(s):  
Sharief Deshmukh ◽  
Viqar Azam Khan
Keyword(s):  
Riemannian Manifold ◽  
Vector Fields ◽  
Eikonal Equation ◽  
Geodesic Vector

scholarly journals A correction to the paper “Injective mappings and solvable vector fields of Euclidean spaces”

2016 ◽  
Vol 204 ◽  
pp. 256-265
Author(s):  
Francisco Braun ◽  
José Ruidival dos Santos Filho
Keyword(s):  
Vector Fields ◽  
Euclidean Spaces

scholarly journals Quotient manifolds of flows

2017 ◽  
Vol 26 (02) ◽  
pp. 1740005 ◽  
Author(s):  
Robert E. Gompf
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Topological Manifold ◽  
Fundamental Groups ◽  
Smooth Manifolds ◽  
Euclidean Spaces ◽  
Orbit Spaces ◽  
Quotient Maps ◽  
Fixed Dimension

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

On geodesic vector fields in a compact orientableRiemannian space

1961 ◽  
Vol 35 (1) ◽  
pp. 55-64 ◽  
Author(s):  
Kentaro Yano ◽  
Tadashi Nagano
Keyword(s):  
Vector Fields ◽  
Geodesic Vector

scholarly journals Slant Curves in Contact Lorentzian Manifolds with CR Structures

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 46
Author(s):  
Ji-Eun Lee
Keyword(s):  
Vector Fields ◽  
Lorentzian Manifold ◽  
Space Forms ◽  
Lorentzian Space ◽  
Geodesic Vector ◽  
Magnetic Curve ◽  
Cr Structures ◽  
Necessary And Sufficient ◽  
Jacobi Equations ◽  
Slant Curves

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .

Spheres and Euclidean Spaces Via Concircular Vector Fields

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Sharief Deshmukh ◽  
Kazim Ilarslan ◽  
Hana Alsodais ◽  
Uday Chand De
Keyword(s):  
Vector Fields ◽  
Euclidean Spaces

Characterizing spheres and Euclidean spaces by conformal vector fields

2017 ◽  
Vol 196 (6) ◽  
pp. 2135-2145 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Vector Fields ◽  
Euclidean Spaces ◽  
Conformal Vector ◽  
Conformal Vector Fields

scholarly journals Some remarks on Yamabe solitons

2019 ◽  
Vol 13 (06) ◽  
pp. 2050120
Author(s):  
Debabrata Chakraborty ◽  
Shyamal Kumar Hui ◽  
Yadab Chandra Mandal
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Geodesic Vector ◽  
Yamabe Solitons ◽  
Yamabe Soliton

The evolution of some geometric quantities on a compact Riemannian manifold [Formula: see text] whose metric is Yamabe soliton is discussed. Using these quantities, lower bound on the soliton constant is obtained. We discuss about commutator of soliton vector fields. Also, the condition of soliton vector field to be a geodesic vector field is obtained.

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