Homogeneous pseudo-Riemannian structures of linear type

Batat, Wafaa; Gadea, Pedro M.; Oubiña, José A.

doi:10.1016/j.geomphys.2010.12.006

Cited by 9 publications

(26 citation statements)

References 25 publications

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“…In the pseudo-Kähler case (ǫ = −1), representation of U(p, q) gives [4] the following decomposition into irreducible modules:…”

Section: Homogeneous ǫ-Kähler Structuresmentioning

confidence: 99%

“…The pointwise classification of homogeneous structures in the purely Riemannian case was provided in [20], and in [10], using representation theory, such a description was obtained for all the possible holonomy groups in Berger's list. These techniques have also been used for metrics with signature (see for instance [4]). In many cases (such as Riemannian, Kähler, hyper-Kähler, quaternion Kähler, as well as in the pseudo-Riemannian analogues) these classifications contain a class consisting of sections of a bundle whose rank grows linearly with the dimension of the manifold.…”

Section: Introductionmentioning

confidence: 99%

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Non-degenerate homogeneous $$\epsilon $$ ϵ -Kähler and $$\epsilon $$ ϵ -quaternion Kähler structures of linear type

Luján

Swann

2015

Monatsh Math

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We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and paraquaternion sectional curvature respectively. In addition the corresponding homogeneous models are computed, exhibiting the relation between these kind of structures and the incompleteness of the metric.

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“…In the pseudo-Kähler case (ǫ = −1), representation of U(p, q) gives [4] the following decomposition into irreducible modules:…”

Section: Homogeneous ǫ-Kähler Structuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-degenerate homogeneous $$\epsilon $$ ϵ -Kähler and $$\epsilon $$ ϵ -quaternion Kähler structures of linear type

Luján

Swann

2015

Monatsh Math

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“…It is a straightforward computation to prove (see [3]) Proposition 3.1 A tensor field S on (M, g, J)defined by formula (2) is a homogeneous -Kähler structure if and only if ∇ξ = 0, ∇ζ = 0, ∇R = 0.…”

Section: Degenerate Homogeneous -Kähler Structures Of Linear Typementioning

confidence: 99%

“…A tensor field S satisfying the previous equations is called a homogeneous -quaternion Kähler structure. The classification of such structures was obtained in [3] and [6] , resulting five primitive classes QK 1 , QK 2 , QK 3 , QK 4 , QK 5 . Among them QK 1 , QK 2 , QK 3 have dimension growing linearly with respect to the dimension of M .…”

Section: Proposition 62 [1]mentioning

confidence: 99%

Homogeneous structures of linear type on $\epsilon$-Kähler and $\epsilon$-quaternion Kähler manifolds

López¹,

Luján²

2017

Rev. Mat. Iberoam.

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We analyze degenerate homogeneous structures of linear type in the pseudo-Kähler and para-Kähler cases. The local form and the holonomy of pseudo-Kähler or para-Kähler manifolds admitting such structure are obtained. In addition the associated homogeneous models are studied exhibiting their relation with the incompleteness of the metric. The same questions are tackled in the pseudo-quaternion Kähler and paraquaternion Kähler cases.These results complete the study of homogeneous structures of linear type in pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and paraquaternion Kähler cases

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“…A tensor field S satisfying the previous equations is called a homogeneous pseudo-quaternion Kähler structure. The classification of such structures was obtained in [3], resulting five primitive classes QK 1 , QK 2 , QK 3 , QK 4 , QK 5 . Among them QK 1 , QK 2 , QK 3 have dimension growing linearly with respect to the dimension of M .…”

Section: The Lorentz-kähler Casementioning

confidence: 99%

Strongly degenerate homogeneous pseudo-Kähler structures of linear type and complex plane waves

López

Luján

2013

Journal of Geometry and Physics

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We study the class K2 + K4 of homogeneous pseudo-Kähler structures in the strongly degenerate case. The local form and the holonomy of a pseudo-Kähler manifold admitting such a structure is obtained, leading to a possible complex generalization of homogeneous plane waves. The same question is tackled in the case of pseudo-hyper-Kähler and pseudoquaternion Kähler manifolds.MSC2010: Primary 53C30, Secondary 53C50, 53C55, 53C80. Key words and phrases: homogeneous plane waves, pseudo-hyper-Kähler, pseudo-Kähler, pseudo-quaternion Kähler, reductive homogeneous pseudo-Riemannian spaces. stance), the so-called homogeneous structures of linear type characterize negative constant sectional (holomorphic sectional, quaternionic sectional) curvature (see [9], [17] and [27] for these results as well as indications of other similar results). When the case of metrics with signature is analyzed, the causal nature of the vector fields characterizing a homogeneous structure of linear type gives rise to different scenarios with some physical implications. In [23] and [24] the purely pseudo-Riemannian case with an isotropic structure is studied in full detail: More precisely, it is proved that these spaces have the underlying geometry of a real singular homogeneous plane wave. The aim of this paper is to extend this result to the pseudo-Kähler, pseudo-hyper-Kähler and pseudo-quaternion Kähler settings.The main characterization of this work gives the geometry of pseudo-Kähler manifolds with a so called strongly degenerate homogeneous pseudo-Kähler structure of linear type. In particular, the expression of the metric obtained in the characterization has strong similarities with singular scale-invariant homogeneous plane waves. Furthermore, the manifolds under study and these homogeneous plane waves share some other analogies and features as it is shown in §5. Because of this, since there is not a formal definition of "complex plane wave" (as far as the authors know), pseudo-Kähler manifolds with strongly degenerate linear homogeneous structures seem to be the correct generalization of this special kind of homogeneous plane waves in complex framework, at least in the important particular Kähler case. In addition, the same techniques are successfully applied to a comparison of Cahen-Wallach spaces and one of the possible pseudo-Kähler symmetric spaces of index 2 in the classification given in [19], giving a more general picture of complex plane waves.A relevant fact about these spaces is that they have holonomy group contained in SU (p, q). Consequently, it is natural to study strongly degenerate homogeneous pseudo-hyper-Kähler and pseudo-quaternion Kähler structures of linear type. However, we prove that a manifold admitting any of those structures is necessarily flat, pointing out that the notion of homogeneous plane wave can not be realized in the pseudo-hyper-kähler or pseudo-quaternion Kähler cases in a non-trivial way.The paper is organized as follows. In Section 2 we recall some results concerning homogeneous pseudo-Riemannian s...

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Homogeneous pseudo-Riemannian structures of linear type

Cited by 9 publications

References 25 publications

Non-degenerate homogeneous $$\epsilon $$ ϵ -Kähler and $$\epsilon $$ ϵ -quaternion Kähler structures of linear type

Non-degenerate homogeneous $$\epsilon $$ ϵ -Kähler and $$\epsilon $$ ϵ -quaternion Kähler structures of linear type

Homogeneous structures of linear type on $\epsilon$-Kähler and $\epsilon$-quaternion Kähler manifolds

Strongly degenerate homogeneous pseudo-Kähler structures of linear type and complex plane waves

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