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14 pages, 271 KiB

Open AccessArticle

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

by Hanan Alohali, Sharief Deshmukh, Bang-Yen Chen and Hemangi Madhusudan Shah

Mathematics 2024, 12(17), 2628; https://doi.org/10.3390/math12172628 - 24 Aug 2024

Cited by 1 | Viewed by 1044

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and [...] Read more.

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and also,

ζ

admits the Hodge decomposition

ζ = \bar{ζ} + \nabla ρ

, where

\bar{ζ}

satisfies

div \bar{ζ} = 0

, which is called the Hodge vector, and

ρ

is the Hodge potential of

ζ

. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on

M^{m}

. The first result of this article states that a compact Riemannian m-manifold

M^{m}

is an m-sphere

S^{m} (c)

if and only if (1) for a nonzero constant c, the function

- σ / c

is a solution of the Poisson equation

Δ ρ = m σ

, and (2) the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

. The second result states that if

M^{m}

has constant scalar curvature

τ = m (m - 1) c > 0

, then it is an

S^{m} (c)

if and only if the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

and the Hodge potential

ρ

satisfies a certain static perfect fluid equation. The third result provides another new characterization of

S^{m} (c)

using the affinity tensor of the Hodge vector

\bar{ζ}

of a conformal vector field

ζ

on a compact Riemannian manifold

M^{m}

with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold

M^{m}

m > 2

, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field

ζ

whose affinity tensor vanishes identically and

ζ

annihilates its associated tensor

φ

. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

30 pages, 2037 KiB

Open AccessArticle

Probing Modified Gravity Theories with Scalar Fields Using Black-Hole Images

by Georgios Antoniou, Alexandros Papageorgiou and Panagiota Kanti

Universe 2023, 9(3), 147; https://doi.org/10.3390/universe9030147 - 11 Mar 2023

Cited by 11 | Viewed by 2797

Abstract

We study a number of well-motivated theories of modified gravity with the common overarching theme that they predict the existence of compact objects, such as black holes and wormholes endowed with scalar hair. We compute the shadow radius of the resulting compact objects [...] Read more.

^{*}

or the more recent SgrA

^{*}

by the Event Horizon Telescope (EHT) collaboration, could provide a powerful way to constrain deviations of the metric functions from what is expected from general relativity (GR) solutions. We focus our attention on Einstein-scalar-Gauss–Bonnet (EsGB) theory with three well-motivated couplings, including the dilatonic and

Z_{2}

symmetric cases. We then analyze the shadow radius of black holes in the context of the spontaneous scalarization scenario within EsGB theory with an additional coupling to the Ricci scalar (EsRGB). Finally, we turn our attention to spontaneous scalarization in the Einstein–Maxwell-Scalar (EMS) theory and demonstrate the impact of the parameters on the black hole shadow. Our results show that black hole imaging is an important tool for constraining black holes with scalar hair, and, for some part of the parameter space, black hole solutions with scalar hair may be marginally favored compared to solutions of GR. Full article

(This article belongs to the Section Gravitation)

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Figure 1

16 pages, 319 KiB

Open AccessArticle

Improved Chen’s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms

by Yanlin Li, Mohan Khatri, Jay Prakash Singh and Sudhakar K. Chaubey

Axioms 2022, 11(7), 324; https://doi.org/10.3390/axioms11070324 - 1 Jul 2022

Cited by 23 | Viewed by 2370

Abstract

In this article, we derive Chen’s inequalities involving Chen’s

δ

-invariant

δ_{M}

, Riemannian invariant

δ (m_{1}, \dots, m_{k})

, Ricci curvature, Riemannian invariant

Θ_{k} (2 \leq k \leq m)

, the scalar [...] Read more.

In this article, we derive Chen’s inequalities involving Chen’s

δ

-invariant

δ_{M}

, Riemannian invariant

δ (m_{1}, \dots, m_{k})

, Ricci curvature, Riemannian invariant

Θ_{k} (2 \leq k \leq m)

, the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we first derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application)

12 pages, 292 KiB

Open AccessArticle

Geodesic Mappings onto Generalized m-Ricci-Symmetric Spaces

by Volodymyr Berezovski, Yevhen Cherevko, Irena Hinterleitner and Patrik Peška

Mathematics 2022, 10(13), 2165; https://doi.org/10.3390/math10132165 - 21 Jun 2022

Cited by 2 | Viewed by 1352

Abstract

In this paper, we study geodesic mappings of spaces with affine connections onto generalized 2-, 3-, and m-Ricci-symmetric spaces. In either case, the main equations for the mappings are obtained as a closed system of linear differential equations of the Cauchy type in the covariant derivatives. For the systems, we have found the maximum number of essential parameters on which the solutions depend. These results generalize the properties of geodesic mappings onto symmetric, recurrent, and also 2-, 3-, and m-(Ricci-)symmetric spaces with affine connections. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

12 pages, 297 KiB

Open AccessEditor’s ChoiceArticle

Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces

by Volodymyr Berezovski, Yevhen Cherevko, Josef Mikeš and Lenka Rýparová

Mathematics 2021, 9(4), 437; https://doi.org/10.3390/math9040437 - 22 Feb 2021

Cited by 10 | Viewed by 2363

Abstract

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

13 pages, 260 KiB

Open AccessArticle

Geodesic Mappings of Spaces with Affine Connections onto Generalized Symmetric and Ricci-Symmetric Spaces

by Volodymyr Berezovski, Yevhen Cherevko, Irena Hinterleitner and Patrik Peška

Mathematics 2020, 8(9), 1560; https://doi.org/10.3390/math8091560 - 11 Sep 2020

Cited by 9 | Viewed by 2181

Abstract

In the paper, we consider geodesic mappings of spaces with an affine connections onto generalized symmetric and Ricci-symmetric spaces. In particular, we studied in detail geodesic mappings of spaces with an affine connections onto 2-, 3-, and m- (Ricci-) symmetric spaces. These spaces play an important role in the General Theory of Relativity. The main results we obtained were generalized to a case of geodesic mappings of spaces with an affine connection onto (Ricci-) symmetric spaces. The main equations of the mappings were obtained as closed mixed systems of PDEs of the Cauchy type in covariant form. For the systems, we have found the maximum number of essential parameters which the solutions depend on. Any m- (Ricci-) symmetric spaces (

m \geq 1

) are geodesically mapped onto many spaces with an affine connection. We can call these spaces projectivelly m- (Ricci-) symmetric spaces and for them there exist above-mentioned nontrivial solutions. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

8 pages, 235 KiB

Open AccessArticle

Conformal and Geodesic Mappings onto Some Special Spaces

by Volodymyr Berezovski, Yevhen Cherevko and Lenka Rýparová

Mathematics 2019, 7(8), 664; https://doi.org/10.3390/math7080664 - 25 Jul 2019

Cited by 6 | Viewed by 3283

Abstract

In this paper, we consider conformal mappings of Riemannian spaces onto Ricci-2-symmetric Riemannian spaces and geodesic mappings of spaces with affine connections onto Ricci-2-symmetric spaces. The main equations for the mappings are obtained as a closed system of Cauchy-type differential equations in covariant derivatives. We find the number of essential parameters which the solution of the system depends on. A similar approach was applied for the case of conformal mappings of Riemannian spaces onto Ricci-m-symmetric Riemannian spaces, as well as geodesic mappings of spaces with affine connections onto Ricci-m-symmetric spaces. Full article

(This article belongs to the Special Issue Differential Geometry of Special Mappings)

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