Uniformity in the orbit structure of a 3-dimensional vector field induced by first integrals and symmetry vectors

1996 ◽  
Vol 111 (10) ◽  
pp. 1247-1251 ◽  
Author(s):  
F. González-Gascón
Keyword(s):  
Vector Field ◽  
First Integrals ◽  
Dimensional Vector ◽  
3 Dimensional ◽  
Orbit Structure

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

scholarly journals New Formulas and Results for 3-Dimensional Vector Fields

2021 ◽  
Vol 12 (11) ◽  
pp. 1058-1096
Author(s):  
Sadanand D. Agashe
Keyword(s):  
Vector Fields ◽  
Dimensional Vector ◽  
3 Dimensional

scholarly journals On trans-Sasakian $3$-manifolds as $\eta$-Einstein solitons

2021 ◽  
Vol 13 (2) ◽  
pp. 460-474
Author(s):  
D. Ganguly ◽  
S. Dey ◽  
A. Bhattacharyya
Keyword(s):  
Vector Field ◽  
Sasakian Manifold ◽  
Sasakian Manifolds ◽  
3 Dimensional ◽  
Codazzi Type

The present paper is to deliberate the class of $3$-dimensional trans-Sasakian manifolds which admits $\eta$-Einstein solitons. We have studied $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds where the Ricci tensors are Codazzi type and cyclic parallel. We have also discussed some curvature conditions admitting $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds and the vector field is torse-forming. We have also shown an example of $3$-dimensional trans-Sasakian manifold with respect to $\eta$-Einstein soliton to verify our results.

scholarly journals Ricci Solitons in α-Sasakian Manifolds

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Gurupadavva Ingalahalli ◽  
C. S. Bagewadi
Keyword(s):  
Vector Field ◽  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional

We study Ricci solitons in α-Sasakian manifolds. It is shown that a symmetric parallel second order-covariant tensor in a α-Sasakian manifold is a constant multiple of the metric tensor. Using this, it is shown that if ℒVg+2S is parallel where V is a given vector field, then (g,V,λ) is Ricci soliton. Further, by virtue of this result, Ricci solitons for n-dimensional α-Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional α-Sasakian manifolds are discussed with an example.

scholarly journals On Calculation of the First Integrals for Three-dimensional ODE Systems

2019 ◽  
pp. 11-21
Author(s):  
A. V. Kavinov
Keyword(s):  
Vector Field ◽  
General Solution ◽  
Nonlinear Dynamical Systems ◽  
First Integral ◽  
First Integrals ◽  
Third Order ◽  
The Third ◽  
Ode System ◽  
Special Cases ◽  
Ode Systems

The search for solutions of nonlinear stationary systems of ordinary differential equations (ODE) is sometimes very complicated. It is not always possible to obtain a general solution in an analytical form. As a consequence, a qualitative theory of nonlinear dynamical systems has been developed. Its methods allow us to investigate the properties of solutions without finding a general solution. Numerical methods of investigation are also widely used.In the case when it is impossible to find an analytically general solution of the ODE system, sometimes, nevertheless, it is possible to find its first integral. There is a number of known results that make it possible to obtain the first integral for certain special cases.The article deals with the method for obtaining the first integrals of ODE systems of the third order, based on the fact of integrability of the involutive distribution.The method proposed in the paper allows us to obtain the first integral of a nonlinear ODE system of the third order in the case when a vector field, which generates an involutive distribution of dimension 2 together with the vector field of the right-hand side of a given ODE system, is known. In this case, the solution of a certain sequence of Cauchy problems allows us to construct a level surface of the function of the first integral containing the given point of the state space of the system. Using the method of least squares, in a number of cases it is possible to obtain an analytic expression for the first integral.The article gives examples of the method application to two ODE systems, namely to a simple nonlinear third-order system and to the Lorentz system with special parameter values. The article shows how the first integrals can be obtained analytically using the method developed for the two systems mentioned above.

Sign in / Sign up

Export Citation Format

Share Document