Locally Homogeneous Four-Dimensional Manifolds of Signature (2,2)

Chaichi, M.; Zaeim, Amirhesam

doi:10.1007/s11040-013-9135-0

Cited by 7 publications

(5 citation statements)

References 18 publications

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“…Finally, not only locally symmetric examples, but all conformally flat (simply connected, complete) pseudo-Riemannian manifolds satisfying the weaker condition R(X, Y ) • Q = 0 have been completely described in [7]. Sorting out the conformally flat Ricci-parallel (hence, locally symmetric) examples in the classification given in the Main theorem of [7] and in [4], [6], we get the following. PROPOSITION 4.1.…”

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confidence: 94%

“…We first consider the Ricci-parallel examples. Note that if a four-dimensional homogeneous pseudo-Riemannian manifold (M, g) is Ricci-parallel, then its Ricci operator Q is necessarily degenerate [4], [6]. Moreover, it is well known that a Ricci-parallel conformally flat manifold is locally symmetric.…”

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confidence: 99%

“…If Q is diagonalizable, we may refer to Theorem 2.2 for the complete classification of Ricci-parallel examples in any dimension greater than or equal to three. On the other hand, Ricci-parallel homogeneous pseudo-Riemannian four-manifolds have been investigated in [4] for the Lorentzian case and in [6] for the neutral signature case. Finally, not only locally symmetric examples, but all conformally flat (simply connected, complete) pseudo-Riemannian manifolds satisfying the weaker condition R(X, Y ) • Q = 0 have been completely described in [7].…”

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confidence: 99%

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Conformally flat homogeneous pseudo-Riemannian four-manifolds

Calvaruso¹,

Zaeim²

2014

Tohoku Math. J. (2)

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We obtain a complete classification of four-dimensional conformally flat homogeneous pseudo-Riemannian manifolds. Introduction.Conformally flat manifolds are a classical field of investigation in pseudo-Riemannian geometry. In this framework, it is a natural problem to classify conformally flat homogeneous pseudo-Riemannian manifolds. A conformally flat (locally) homogeneous Riemannian manifold is (locally) symmetric [17]. Hence, it admits as univeral covering either a space form R n , S n (k),In pseudo-Riemannian settings, the problem of classifying conformally flat homogeneous manifolds is more complicated and interesting. Three-dimensional examples were classified independently in [8] and [3], showing the existence of non-symmetric examples. Using the general results introduced in [8], the same authors contributed in [9] to solve the classification problem for Lorentzian manifolds of any dimension, under some assumptions on the structure of the eigenvalues of the Ricci operator of such a manifold. Up to our knowledge, no classification results have been obtained yet for metrics of different signatures, except for the cases with a diagonalizable Ricci operator [8].In the present paper we shall provide a complete classification of four-dimensional conformally flat homogeneous pseudo-Riemannian manifolds. A fundamental step for this classification will be to understand which forms (Segre types) of the Ricci operator, and under which restrictions, may exist for conformally flat pseudo-Riemannian manifolds. As we shall see, nondegenerate forms of the Ricci operator can only occur when a conformally flat homogeneous pseudo-Riemannian four-manifold is (locally) isometric to some Lie group, while the possible degenerate forms are also realized by some homogeneous spaces with nontrivial isotropy, for which we can use Komrakov's classification [10] to deduce all possible examples.Because of the results obtained in [9], we shall focus mainly on the case of a pseudo-Riemannian metric of neutral signature. However, we shall also explain how our results apply to the Lorentzian case.The paper is organized in the following way. In Section 2, we report some basic information on four-dimensional conformally flat pseudo-Riemannian manifolds and describe

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Conformally flat homogeneous pseudo-Riemannian four-manifolds

Calvaruso¹,

Zaeim²

2014

Tohoku Math. J. (2)

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“…Although homogeneous manifolds have been previously examined in the context of Riemannian manifolds (e.g., [25]), recent research has primarily focused on pseudo-Riemannian geometry. Three-dimensional homogeneous Lorentzian manifolds were studied in [3], while the four-dimensional case due to the signature of the invariant metric was considered in [8], [15]. Special homogeneous pseudo-Riemannian manifolds were also studied in several cases.…”

Section: Introductionmentioning

confidence: 99%

On conformal geometry of four-dimensional generalized symmetric spaces

Zaeim,

AryaNejad,

Gheitasi

2023

Revista De La Unión Matemática Argentina

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“…Homogeneous spaces are the subject of many interesting research projects in the pseudo-Riemannian framework. Four-dimensional homogeneous Lorentzian and neutral signature manifolds were studied in [3] and [5] respectively and Lorentzian Lie groups with complete classification of Einstein and Ricci parallel examples were considered in dimension four in [4].…”

Section: Introductionmentioning

confidence: 99%

On Conformally flat homogeneous Walker four-manifolds

Chaichi,

Zaeim,

Keshavarzi

2015

Preprint

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Locally Homogeneous Four-Dimensional Manifolds of Signature (2,2)

Cited by 7 publications

References 18 publications

Conformally flat homogeneous pseudo-Riemannian four-manifolds

Conformally flat homogeneous pseudo-Riemannian four-manifolds

On conformal geometry of four-dimensional generalized symmetric spaces

On Conformally flat homogeneous Walker four-manifolds

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