On the Geometry of Orbits of Conformal Vector Fields

Narmanov, A. Ya.; Rajabov, E.O.

doi:10.7546/jgsp-51-2019-29-39

Cited by 5 publications

(5 citation statements)

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“…£ u g = σRic, (1) where £ u g is the Lie derivative of the metric g with respect to u, σ is a smooth function and Ric is the Ricci tensor of (N m , g). A σ-RVF is a generalization of conformal vector fields (known for their utility in studying geometry and relativity), on Einstein manifolds (see [1][2][3][4][5][6][7][8][9][10][11]). Moreover, it represents a Killing vector field, which is known to have a great influence on the geometry as well as topology on which it lives (see [12][13][14][15]).…”

Section: Introductionmentioning

confidence: 99%

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Ricci Vector Fields Revisited

Alohali,

Deshmukh,

Vîlcu

2024

Mathematics

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We continue studying the σ-Ricci vector field u on a Riemannian manifold (Nm,g), which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold(Nm,g), m>1, of positive scalar curvature τ, admits a closed σ-Ricci vector field u such that the vector u−∇σ is an eigenvector of T with eigenvalue τm−1, if and only if it is isometric to the m-sphere Sαm. In the second result, we show that if a compact and connected T-manifold(Nm,g), m>2, admits a σ-Ricci vector field u with σ≠0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature Ricu,u that has a suitable lower bound, then (Nm,g) is isometric to the m-sphere Sαm, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold (Nm,g) admits a σ-Ricci vector field u with σ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature Ricu,u has a lower bound depending on a positive constant α, if and only if (Nm,g) is isometric to the m-sphere Sαm.

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Section: Introductionmentioning

confidence: 99%

“…There is yet another important differential equation on a Riemannian manifold (N m , g) (cf. [7] and references therein), given by…”

Section: Introductionmentioning

confidence: 99%

Ricci Vector Fields Revisited

Alohali,

Deshmukh,

Vîlcu

2024

Mathematics

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“…F is singular, i.e., the module XF of smooth vector fields on M that are tangent at each point to the corresponding leaf acts transitively on each leaf. In other words, for each leaf L and each v ∈ T p L with foot point p, there is X ∈ XF with X(p) = v [1][2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

The Geometry of Vector Fields and Two Dimensional Heat Equation

Narmanov

ELDOR

2023

International Electronic Journal of Geometry

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The geometry of orbits of families of smooth vector fields was studied by many mathematicians due to its importance in applications in the theory of control systems, in dynamic systems, in geometry and in the theory of foliations. In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.

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“…If the dimension of L is maximal, it is called regular, otherwise L is called singular. It is known that orbits of vector fields generate singular foliation (see [1], [2], [3]).…”

Section: Introductionmentioning

confidence: 99%

Vector fields and differential equations

Narmanov

Rajabov

2022

J. Phys.: Conf. Ser.

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The geometry of orbits of families of smooth vector fields is an important object of mathematics due to its importance in applications, in the theory of dynamic systems and in the foliation theory. The paper devoted to the applications of the geometry of orbits of vector fields in four dimensional Euclidean space in theory of differential equations. It is shown that orbits generate singular foliation ever regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields. In this paper smoothness is smoothness of the class C ∞.

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On the Geometry of Orbits of Conformal Vector Fields

Cited by 5 publications

References 0 publications

Ricci Vector Fields Revisited

Ricci Vector Fields Revisited

The Geometry of Vector Fields and Two Dimensional Heat Equation

Vector fields and differential equations

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