Geometric invariant theory and Einstein–Weyl geometry

Kalafat, Mustafa

doi:10.1016/j.exmath.2011.01.002

Cited by 4 publications

(3 citation statements)

References 5 publications

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“…For some recent research on the Weyl geometry from the mathematical point of view, see e.g. [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

A modified variational principle for gravity in the modified Weyl geometry

Yuan¹,

Huang²

2013

Class. Quantum Grav.

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The usual interpretation of Weyl geometry is modified in two senses. First, both the additive Weyl connection and its variation are treated as (1, 2) tensors under the action of Weyl covariant derivative. Second, a modified covariant derivative operator is introduced which still preserves the tensor structure of the theory. With its help, the Riemann tensor in Weyl geometry can be written in a more compact form. We justify this modification in detail from several aspects and obtain some insights along the way. By introducing some new transformation rules for the variation of tensors under the action of Weyl covariant derivative, we find a Weyl version of Palatini identity for Riemann tensor. To derive the energy-momentum tensor and equations of motion for gravity in Weyl geometry, one naturally applies this identity at first, and then converts the variation of additive Weyl connection to those of metric tensor and Weyl gauge field. We also discuss possible connections to the current literature on Weyl-invariant extension of massive gravity and the variational principles in f(R) gravity.

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“…For some recent research on the Weyl geometry from the mathematical point of view, see e.g. [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

A modified variational principle for gravity in the modified Weyl geometry

Yuan¹,

Huang²

2013

Class. Quantum Grav.

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“…Weyl [34] used these geometries in an attempt to unify gravity with electromagnetism -although this approach failed for physical reasons, the resulting geometries are still of importance and there is a vast literature on the subject. See, for example, [1,10,14,15,20,21]; note that the indefinite signature setting is of particular importance [3,11,20,22] as is the complex setting [18,19,23]. The field is a vast one and we only cite a few representative recent examples.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous 4-Dimensional Kähler–Weyl Structures

et al. 2013

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Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kähler-Weyl structure; this structure is locally conformally Kähler if and only if the alternating Ricci tensor ρa vanishes. The tensor ρa takes values in a certain representation space. In this paper, we show that any algebraic possibility Ξ in this representation space can in fact be geometrically realized by a left-invariant Kähler-Weyl structure on a 4-dimensional Lie group in either the pseudo-Hermitian or the para-Hermitian setting. MSC 2010: 53A15, 53C15, 15A72.

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“…The pseudo-Riemannian setting also is important [1,24,32] as are para-complex geometries [11,13]. See also [9,22,30,31] for related results. The literature in the field is vast and we can only give a flavor of it for reasons of brevity.…”

Section: Introductionmentioning

confidence: 99%

(para)-Kähler Weyl Structures

Gilkey

Nikčević

2012

Recent Trends in Lorentzian Geometry

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Abstract. We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kähler Weyl algebraic curvature tensor is in fact Riemannian in dimension m ≥ 6; this yields as a geometric consequence that any (para)-Kähler Weyl geometric structure is trivial for m ≥ 6. By contrast, the 4-dimensional setting is, as always, rather special as it turns out that there are (para)-Kähler Weyl algebraic curvature tensors which are not Riemannian if m = 4. Since every (para)-Kähler Weyl algebraic curvature tensor is geometrically realizable and since every 4-dimensional Hermitian manifold admits a unique (para)-Kähler Weyl structure, there are also non-trivial 4-dimensional Hermitian (para)-Kähler Weyl manifolds. MSC: 53B05, 15A72, 53A15, 53B10, 53C07, 53C25. IntroductionLet ∇ be a torsion free connection on a pseudo-Riemannian manifold (M, g) of even dimension m = 2m ≥ 4. The triple (M, g, ∇) is said to be a Weyl structure if there exists a smooth 1-form φ so that ∇g = −2φ ⊗ g. Such a geometric structure was introduced by Weyl [37] in an attempt to unify gravity with electromagnetism. Although this approach failed for physical reasons, these geometries are still studied for their intrinsic interest [2,10,21,27,28]; they also appear in the mathematical physics literature [12,20,26]. Weyl geometry is relevant to submanifold geometry [25] and to contact geometry [15]. The pseudo-Riemannian setting also is important [1,24,32] as are para-complex geometries [11,13]. See also [9,22,30,31] for related results. The literature in the field is vast and we can only give a flavor of it for reasons of brevity. We shall be primarily interested in the Hermitian setting. However since there are applications to higher signature geometry, we include the pseudo-Hermitian context as well; similarly we treat para-Hermitian geometries as they can be studied with little additional effort. Section 1.1 of the Introduction deals with the real setting. In Theorem 1.1, we recall the basic theorems of geometric realizability for affine, Riemannian, and Weyl curvature models and in Theorem 1.2 provide various characterizations of the notion of a trivial Weyl structure. Section 1.2 treats the (para)-Kähler setting. In Theorem 1.3 we recall geometric realizibility results for (para)-Kähler affine and (para)-Kähler Riemannian curvature models. Theorem 1.4 presents results in the geometric setting for (para)-Kähler Weyl manifolds. Theorem 1.5 is one of the two main results of this paper: every (para)-Kähler curvature model is geometrically realizable. The proof of Theorem 1.5 relies on a curvature decomposition result; the second main result of the paper, Theorem 1.6, discusses the space of (para)-Kähler Weyl algebraic curvature tensors.1.1. Riemannian, Affine, and Weyl geometry. Let (V, ·, · ) be an inner product space of signature (p, q) and dimension m = p+ q; an inner product of signatur...

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Geometric invariant theory and Einstein–Weyl geometry

Cited by 4 publications

References 5 publications

A modified variational principle for gravity in the modified Weyl geometry

A modified variational principle for gravity in the modified Weyl geometry

Homogeneous 4-Dimensional Kähler–Weyl Structures

(para)-Kähler Weyl Structures

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