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On conformally flat Lorentz parabolic manifolds

Yoshinobu Kamishima (2014)

Open Mathematics

We introduce conformally flat Fefferman-Lorentz manifold of parabolic type as a special class of Lorentz parabolic manifolds. It is a smooth (2n+2)-manifold locally modeled on (Û(n+1, 1), S 2n+1,1). As the terminology suggests, when a Fefferman-Lorentz manifold M is conformally flat, M is a Fefferman-Lorentz manifold of parabolic type. We shall discuss which compact manifolds occur as a conformally flat Fefferman-Lorentz manifold of parabolic type.

On rank one symmetric space

Inkang Kim (2004/2005)

Séminaire de théorie spectrale et géométrie

In this paper we survey some recent results on rank one symmetric space.

On the asymptotic geometry of gravitational instantons

Vincent Minerbe (2010)

Annales scientifiques de l'École Normale Supérieure

We investigate the geometry at infinity of the so-called “gravitational instantons”, i.e. asymptotically flat hyperkähler four-manifolds, in relation with their volume growth. In particular, we prove that gravitational instantons with cubic volume growth are ALF, namely asymptotic to a circle fibration over a Euclidean three-space, with fibers of asymptotically constant length.

On the holonomy of Lorentzian metrics

Charles Boubel (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Indecomposable Lorentzian holonomy algebras, except 𝔰𝔬 ( n , 1 ) and { 0 } , are not semi-simple; they possibly belong to four families of algebras. All four families are realized as families of holonomy algebras: we describe the corresponding set of germs of metrics in each case.

Super Wilson Loops and Holonomy on Supermanifolds

Josua Groeger (2014)

Communications in Mathematics

The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects occurring with auxiliary Graßmann generators coming from S -points. A key feature of our model is a supergeometric parallel transport, which allows for a natural notion of holonomy on...

Sur l'holonomie des variétés pseudo-riemanniennes de signature (2,2+n).

A. Ikemakhen (1999)

Publicacions Matemàtiques

In this paper, we determine a class of possible restricted holonomy groups for a non-irreducible indecomposable pseudoriemannian manifold with signature (2,2 + n). In particular, we deduce that which associated to symmetric spaces; and give some examples of such spaces. Finally, we construct some examples of metrics whose restricted holonomy groups are not closed.

Tangent Lie algebras to the holonomy group of a Finsler manifold

Zoltán Muzsnay, Péter T. Nagy (2011)

Communications in Mathematics

Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal holonomy algebra by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to...

The Srní lectures on non-integrable geometries with torsion

Ilka Agricola (2006)

Archivum Mathematicum

This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of...

Transitive conformal holonomy groups

Jesse Alt (2012)

Open Mathematics

For (M, [g]) a conformal manifold of signature (p, q) and dimension at least three, the conformal holonomy group Hol(M, [g]) ⊂ O(p + 1, q + 1) is an invariant induced by the canonical Cartan geometry of (M, [g]). We give a description of all possible connected conformal holonomy groups which act transitively on the Möbius sphere S p,q, the homogeneous model space for conformal structures of signature (p, q). The main part of this description is a list of all such groups which also act irreducibly...

Weakly irreducible subgroups of Sp ( 1 , n + 1 )

Natalia I. Bezvitnaya (2008)

Archivum Mathematicum

Connected weakly irreducible not irreducible subgroups of Sp ( 1 , n + 1 ) ⊂ SO ( 4 , 4 n + 4 ) that satisfy a certain additional condition are classified. This will be used to classify connected holonomy groups of pseudo-hyper-Kählerian manifolds of index 4.

Weitzenböck Formula for SL(q)-foliations

Adam Bartoszek, Jerzy Kalina, Antoni Pierzchalski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

A Weitzenböck formula for SL(q)-foliations is derived. Its linear part is a relative trace of the relative curvature operator acting on vector valued forms.

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