A note on static spaces

Deshmukh, Sharief; Turki, Nasser Bin; Vîlcu, Gabriel-Eduard

doi:10.1016/j.rinp.2021.104519

Cited by 5 publications

(4 citation statements)

References 29 publications

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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%

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%

Ricci Vector Fields Revisited

Alohali,

Deshmukh,

Vîlcu

2024

Mathematics

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“…where Ric is the Ricci tensor, τ is the scalar curvature, h is the Hessian tensor and ∆ is the Laplacian operator [11,21]. The concept of static perfect fluid space plays an important role in both general relativity and differential geometry.…”

Section: Static Perfect Fluid Spacetimes On Twisted Productsmentioning

confidence: 99%

“…The next purpose of this paper is to study and explore some characteristics of static perfect fluid spacetimes on twisted product manifolds. Static perfect fluids spacetimes are special global solutions of Einstein's equations that show the relationship between matter content and spacetime in general relativity, [10,11,21].…”

Section: Introductionmentioning

confidence: 99%

Geometry of Twisted Products and Applications on Static Perfect Fluid Spacetimes

GÜLER,

DE,

ÜNAL

2023

International Electronic Journal of Geometry

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In this paper, first we study the harmonicity of the functions and forms on the twisted products, and then we determine its sectional curvature. We explore some characteristics of static perfect fluid and static vacuum spacetimes on twisted product manifolds by proving the existence and obstructions on Ricci curvature. Finally, we study the problem of the existence static perfect fluid spacetime associated with the twisted generalized Robertson-Walker and standard static spacetime metrics.

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“…for a 1-form ω on M. Torse-forming vector fields play a role in physics (cf. [9,[12][13][14][15][16][17][18]). Chen, in [19], considered a specific torse-forming vector field called a torqued vector field.…”

Section: Introductionmentioning

confidence: 99%

On an Anti-Torqued Vector Field on Riemannian Manifolds

2021

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A torqued vector field ξ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative.

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A note on static spaces

Cited by 5 publications

References 29 publications

Ricci Vector Fields Revisited

Ricci Vector Fields Revisited

Geometry of Twisted Products and Applications on Static Perfect Fluid Spacetimes

On an Anti-Torqued Vector Field on Riemannian Manifolds

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