Article
Mathematics, Applied
Rodrigo Marino-Villar
Summary: A comprehensive study is conducted on locally conformally flat metrics satisfying weakly-Einstein conditions, which are shown to be either a product Mn(c) x Mn(-c) or a warped product R x f Rn-1 with a specific warping function. Additionally, some conditions on locally conformally flat fixed points for the RG2 flow are highlighted.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Mathematics, Applied
Levi Rosa Adriano, Ilton Ferreira de Menezes, Mauricio Donizetti Pieterzack, Romildo da Silva Pina
Summary: This paper examines the existence conditions of metrics in pseudo-Euclidean spaces with specific metric components and tensor forms, and constructs an example of a static perfect fluid spacetime using the obtained results. Similar problems are also considered for locally conformally flat manifolds.
RESULTS IN MATHEMATICS
(2021)
Article
Mathematics
Wei Xiao, Yong He, Shuwen Li, Qihui Ni
Summary: In this study, the characterization and necessary and sufficient condition for the locally conformally flat property of doubly twisted product complex Finsler manifolds are investigated.
JOURNAL OF MATHEMATICS
(2022)
Article
Physics, Mathematical
Parvane Atashpeykar, Amirhesam Zaeim, Ali Haji-badali
Summary: This paper classifies Lorentzian manifolds of dimension n = 3 that satisfy the Codazzi equation, and applies this classification to characterize three-dimensional weakly-Einstein Lorentzian manifolds in the conformal class of flat metrics.
REPORTS ON MATHEMATICAL PHYSICS
(2023)
Article
Mathematics
Abdul Haseeb, Mohd Bilal, Sudhakar K. Chaubey, Abdullah Ali H. Ahmadini
Summary: In this paper, we study m-dimensional zeta-conformally flat LP-Kenmotsu manifolds (abbreviated as (LPK)(m)) equipped with Ricci-Yamabe solitons (RYS) and gradient Ricci-Yamabe solitons (GRYS). It is proven that the scalar curvature r of an (LPK)(m) with an RYS satisfies the Poisson equation delta r=4(m-1)/delta{beta(m-1)+rho}+2(m-3)r - 4m(m-1)(m-2), where rho, delta(&NOTEQUexpressionL; 0) is an element of R. Furthermore, the condition for the scalar curvature of an (LPK)(m) with an RYS to satisfy the Laplace equation is established. An affirmative answer is also given for the existence of a GRYS on an (LPK)(m). Finally, a non-trivial example of a four-dimensional LP-Kenmotsu manifold (LPK) is constructed to validate some of the results.
Article
Astronomy & Astrophysics
Marc Mars, Carlos Peon-Nieto
Summary: This paper presents a coordinate independent algorithm for determining the class of conformal Killing vectors in locally conformally flat n-metric gamma. The algorithm is based on endomorphisms in the pseudo-orthogonal Lie algebra and the group O. The explicit classification is provided for the Riemannian gamma case. As an application, the paper shows the one-to-one correspondence between a set of five-dimensional vacuum metrics analyzed in a previous study and the metrics in the Kerr-de Sitter-like class. The equivalence between these seemingly unrelated metric classes suggests interesting connections between the algebraically special spacetime and the conformal geometry.
Article
Mathematics, Applied
Amir Babak Aazami
Summary: It is proven in this article that there are no closed Lorentzian 3-manifolds whose Ricci tensor satisfies certain conditions. There is no such obstruction when lambda is negative, and when lambda is 0, the manifold must be isometric to a specific Riemannian manifold.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Astronomy & Astrophysics
Marc Mars, Carlos Peon-Nieto
Summary: By using the asymptotic characterization results of spacetimes at conformal infinity, this study proves the one-to-one correspondence between Kerr-Schild-de Sitter spacetimes and spacetimes in the Kerr-de Sitter-like class with conformally flat I.
Article
Mathematics, Applied
Ali Haji-Badali, Amirhesam Zaeim, Parvane Atashpeykar
Summary: In this paper, weakly-Einstein conditions are studied on four-dimensional conformally flat pseudo-Riemannian algebraic curvature models. Specifically, the locally conformally flat examples of Walker metrics satisfying weakly-Einstein conditions are presented.
UNIVERSITY POLITEHNICA OF BUCHAREST SCIENTIFIC BULLETIN-SERIES A-APPLIED MATHEMATICS AND PHYSICS
(2023)
Article
Mathematics, Applied
R. Diogenes, T. Gadelha, E. Ribeiro Jr
Summary: In this paper, the classification results of compact quasi-Einstein manifolds with boundary, nonnegative quasi-Einstein curvature, and zero radial Weyl tensor are proven. A new example is provided to justify the assumptions. The case of dimension 3 is also discussed.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Pavel Osipov
Summary: A statistical manifold is a mathematical concept with specific properties, and this paper investigates locally conformally Hessian manifolds and proves some structure theorems.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Mathematics, Applied
Simone Calamai
Summary: In this study, the space of projectively flat metrics is divided into three classes based on the sign of the Chern scalar curvature. It is shown that the class of negative projectively flat metrics is empty, while the positive class consists of locally conformally flat-Kahler metrics. Additionally, the characterization and properties of zero projectively flat metrics are reviewed, and it is proven that projectively flat astheno-Kahler metrics are actually Kahler and globally conformally flat.
PURE AND APPLIED MATHEMATICS QUARTERLY
(2021)
Article
Mathematics
Stefan Deaconu, Victor Vuletescu
Summary: We prove that Oeljeklaus-Toma manifolds X (K, U) with certain conditions do not admit any locally conformally Kahler metric, which completely solves the problem of existence of locally conformally Uhler metrics on these manifolds, in combination with earlier works by Dubickas and Oeljeklauss and Toma.
MANUSCRIPTA MATHEMATICA
(2023)
Article
Mathematics
Yuze Ren, Xiaoling Zhang, Lili Zhao
Summary: This paper studies the conformal transformation that preserves Einstein metrics on Finsler warped product manifolds. Sufficient and necessary conditions for a conformal transformation preserving Einstein metrics are obtained. Nontrivial examples of conformal transformations are provided. Furthermore, Einstein Riemannian warped product metrics are completely classified and the existence of a nontrivial conformal transformation preserving Einstein metrics is obtained.
Article
Mathematics
Duc Thoan Pham, Van Khien Tran, Thi Hong Nguyen
Summary: In this paper, we investigate the vanishing theorems for harmonic p-forms on a locally conformally flat Riemannian manifold. Specifically, we present a vanishing theorem for them without the need for scalar curvature conditions, when the integral of the traceless Ricci tensor satisfies a suitable bound. Furthermore, we provide another theorem under the condition of nonpositive scalar curvature, which improves and extends previous results.
COMMUNICATIONS IN MATHEMATICS AND STATISTICS
(2023)
Article
Mathematics, Applied
M. Brozos-Vazquez, E. Garcia-Rio, S. Caeiro-Oliveira
Summary: The research demonstrates the existence of non-Einstein homogeneous critical metrics in three-dimensional space for any quadratic curvature functional.
ANNALI DI MATEMATICA PURA ED APPLICATA
(2021)
Article
Mathematics
Miguel Brozos-Vazquez, Sandro Caeiro-Oliveira, Eduardo Garcia-Rio
Summary: This study demonstrates the existence of non-Einstein cones critical for all quadratic curvature functionals.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2021)
Article
Astronomy & Astrophysics
M. Brozos-Vazquez, S. Caeiro-Oliveira, E. Garcia-Rio
Summary: This article investigates the critical three-dimensional Brinkmann waves for quadratic curvature functionals. It is found that if a metric is critical for some functional, it is critical for all of them. Additionally, four special functionals are identified that do not share critical metrics with any other quadratic functional. Furthermore, it is shown that these metrics provide explicit solutions for different massive gravity models.
CLASSICAL AND QUANTUM GRAVITY
(2022)
Article
Astronomy & Astrophysics
M. Brozos-Vazquez, D. Mojon-Alvarez
Summary: In this paper, a weighted Einstein tensor is defined on a smooth metric measure spacetime, and its applications in isotropic solutions are studied. By investigating the nilpotent Ricci operator, specific types of manifold are obtained. In dimension 3, all isotropic solutions can be expressed as plane waves or Kundt spacetimes.
CLASSICAL AND QUANTUM GRAVITY
(2022)
Article
Mathematics, Applied
M. Ferreiro-Subrido, E. Garcia-Rio, R. Vazquez-Lorenzo
Summary: This article investigates non-Einstein Ricci solitons on four-dimensional Lorentzian Lie groups with left-invariant soliton vector fields. In addition to pp-wave and plane wave Lie groups, there are four families of Lorentzian metrics on semi-direct extensions R-3 (sic) R and E(1, 1) (sic) R. It is shown that some of these Ricci solitons are conformally Einstein and can be expanding, steady, or shrinking.
ANALYSIS AND MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
M. Brozos-Vazquez, S. Caeiro-Oliveira, E. Garcia-Rio
Summary: We investigate three-dimensional homogeneous and 1-curvature homogeneous Lorentzian metrics critical to a quadratic curvature functional. We demonstrate that any quadratic curvature functional can have distinct non-Einstein homogeneous critical metrics and that there are homogeneous metrics critical to all quadratic curvature functionals without being Einstein.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2023)
Article
Computer Science, Software Engineering
M. Brozos-Vazquez, M. J. Pereira-Saez, A. B. Rodriguez-Raposo, M. J. Souto-Salorio, A. D. Tarrio-Tobar
Summary: In this paper, we analyze the characteristic polynomial of an ellipsoid and another quadric in the context of contact detection. We derive a necessary and sufficient condition for an efficient contact detection method, known as the smallness condition, based on the size and shape of the quadrics. This condition can be directly checked from the parameters of the quadrics. Under this assumption, contact can be detected using the expressions and coefficients of the characteristic polynomial, and the relative positions can be classified based on the sign of the coefficients. As an application, we present a method to detect contact between a small ellipsoid and a combination of quadrics.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Mathematics, Applied
Esteban Calvino-Louzao, Maria Ferreiro-Subrido, Eduardo Garcia-Rio, Ramon Vazquez-Lorenzo
Summary: We demonstrate that a four-dimensional homogeneous manifold with a half-harmonic Weyl curvature tensor is either symmetric or homothetic, and it can be isomorphic to either the only nonsymmetric anti-self-dual homogeneous manifold or the 3-symmetric space.
FORUM MATHEMATICUM
(2022)
Article
Mathematics, Applied
M. Brozos-Vazquez, S. Caeiro-Oliveira, E. Garcia-Rio
Summary: This study investigates curvature homogeneous three-manifolds modeled on a symmetric space, which exhibit critical characteristics for some quadratic curvature functional. If the Ricci operator is diagonalizable, the critical metrics are 1-curvature homogeneous Brinkmann waves and are critical for a specific functional. Otherwise, the critical metrics are modeled on Cahen-Wallach symmetric spaces and they are Kundt spacetimes critical for all quadratic curvature functionals.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
E. Garcia-Rio, R. Marino-Villar, M. E. Vazquez-Abal, R. Vazquez-Lorenzo
Summary: This study focuses on solitons for the two-loop renormalization group flow in four-dimensional gauge field theory, providing a classification of algebraic steady four-dimensional solitons.
JOURNAL OF FIXED POINT THEORY AND APPLICATIONS
(2023)
Article
Mathematics
E. Calvino-Louzao, M. Ferreiro-Subrido, E. Garcia-Rio, R. Vazquez-Lorenzo
Summary: In this study, we identify all the homogeneous structures on non-symmetric three-dimensional Riemannian Lie groups and prove that a non-symmetric three-dimensional Riemannian Lie group has a non-canonical homogeneous structure if and only if its isometry group has dimension four.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2023)
Proceedings Paper
Mathematics
E. Calvino-Louzao, E. Garcia-Rio, I Gutierrez-Rodriguez, R. Vazquez-Lorenzo
Summary: This paper reviews generalizations of Einstein metrics, focusing on the four-dimensional homogeneous case to present classification results and new examples.
GEOMETRY, LIE THEORY AND APPLICATIONS
(2022)
