Holonomy Classification of Lorentz-Kähler Manifolds

Galaev, Anton S.

doi:10.1007/s12220-018-0027-1

Cited by 6 publications

(2 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Leistner provided a complete classification for Lorentzian manifolds [15]. Galaev classified holonomy groups of Lorentz-Kähler, i.e., those with Kähler metrics of index 2, and Einstein pseudo-Riemannian manifolds [11,12]. Aside from these classifications some partial results are known (cf.…”

Section: Introductionmentioning

confidence: 99%

Local Type II metrics with holonomy in $\mathrm{G}_2^*$

Volkhausen¹

2019

Preprint

View full text Add to dashboard Cite

A list of possible holonomy groups with indecomposable holonomy representation contained in the exceptional, non-compact Lie group G * 2 was provided by Fino and Kath. The classification is due to the corresponding holonomy algebras and divided into Type I, II and III, depending on the dimension of the socle being 1, 2 or 3, respectively. It was also shown by Fino and Kath that all algebras of Type I, and by the author that all of Type III are indeed realizable as holonomy algebras by metrics with signature (4,3). This article proves that this is also true for all Type II algebras. Thus, there exists a realization by a metric for all indecomposable holonomy groups contained in G * 2 .

show abstract

Section: Introductionmentioning

confidence: 99%

Local Type II metrics with holonomy in $\mathrm{G}_2^*$

Volkhausen¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…While the holonomy of irreducible pseudo-Riemannian manifolds is also classified by Berger's list, little is known about the holonomy of the indecomposable ones, especially about those with index greater than 2. Holonomy groups of Lorentzian manifolds were classified by Leistner [18,11], while Galaev classified those of pseudo-Kählerian manifolds of index 2 [12] and Einstein not Ricci-flat pseudo-Riemannian manifolds [13]. Beside these classifications only some results associated with certain special geometries are known (cf.…”

Section: Introductionmentioning

confidence: 99%

Local Type III metrics with holonomy in $\mathrm{G}_2^*$

Volkhausen

2018

Preprint

View full text Add to dashboard Cite

Fino and Kath determined all possible holonomy groups of sevendimensional pseudo-Riemannian manifolds contained in the exceptional, noncompact, simple Lie group G * 2 via the corresponding Lie algebras. They are distinguished by the dimension of their maximal semi-simple subrepresentation on the tangent space, the socle. An algebra is called of Type I, II or III if the socle has dimension 1, 2 or 3 respectively. This article proves that each possible holonomy group of Type III can indeed be realized by a metric of signature (4, 3). For this purpose, metrics are explicitly constructed, using Cartan's methods of exterior differential systems, such that the holonomy of the manifold has the desired properties.

show abstract