A note on conformal connections on lightlike hypersurfaces

Degenerate submanifolds of pseudo-Riemannian manifolds are quite difficult to study because there is no prefered connection when the submanifold is not totally geodesic. For the particular case of degenerate totally umbilical hypersurfaces, we show that there are "Weyl" connections adapted to the induced structure on the hypersurface. We begin the study of these with their holonomy

Representations admitting two pairs of supplementary invariant spaces

We examine the lattice generated by two pairs of supplementary vector subspaces of a finite-dimensional vector space by intersection and sum, with the aim of applying the results to the study of representations admitting two pairs of supplementary invariant spaces, or one pair and a reflexive form. We show that such a representation is a direct sum of three canonical sub-representations which we characterize. We then focus on holonomy representations with the same property

Einstein-Weyl structures on lightike hypersurfaces

We study Weyl structures on lightlikes hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambiant Lorentzian space $\mathbb{R}^{n+2}_{1}$ and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves. Finally, we establish necessary and sufficient conditions for a Weyl structure defined by the $1-$form of an almost contact structure given by an additional complex structure in case of an ambiant Kaehler manifold to be closed.Comment: soumi

Totally reducible holonomies of torsion-free affine connections

That announcement gives the structure of totally reducible linear Lie algebras which are the Lie algebra of the holonomy group of (at least) one torsion-free connection. The result uses the (already known) classi cation of the irreducible ones and some previous (unpublished) works by the author giving the classi cation for the pseudo-riemannian totally reducible case. One describes those Lie subalgebras through a general structure theorem involving two constructions and some lists. These constructions give new examples of non irreducible totally reducible holonomy algebras and also recover some irreducible ones which seem missing in the previous classi cation.Comment: 13 page

Canonical Torsion-Free Connections on the Total Space of the Tangent and the Cotangent Bundle

In this paper we define a class of torsion-free connections on the total space of the (co-)tangent bundle over a base-manifold with a connection and for which tangent spaces to the fibers are parallel. Each tangent space to a fiber is flat for these connections and the canonical projection from the (co-)tangent bundle to the base manifold is totally geodesic. In particular cases the connection is metric with signature (n,n) or symplectic and admits a single parallel totally isotropic tangent n-plane

Semi-Riemannian Symmetric spaces

Avertissement : ce texte n'est qu'un extrait d'un (futur) preprint sur le sujet mentionné dans le résumé. Il contient cependant un énoncé précis (voir section 2.2 théorème 2.2.9) qui circule sous frome de notes manusrites et est utilisé par des articles déjà publiés.Jordan Holder decompositions of representations and applications to orthogonal Lie algebras semi-riemannian symmetric spaces and holonomy problems

Remark on an example by R.Schoen concerning the scalar curvature

soumis à Differential Geometry and its ApplicationsGeneralization of an example by R.Schoen of multiple solutions for the Yamabe Problem on Manifolds with positive scalar curvature

Examples of multiple solutions for the Yamabe problem on scalar curvature

preprint 2006In the conformal class of Riemannian metric on a compact connected manifold, there exists at least one metric with constant scalar curvature. In the case with positive scalar curvature, there may be many (non-homothtic) metrics with constant scalar curvature in a conformal class. R. Schoen gave a beautiful example of that phenomenon for a one-parameter family of metrics on $S^1 \times S^{n-1}$. In a preceding paper, we showed that Shoen's construction may be generalized on products $S^1 \times N$ (and other related examples). A unique (ordinary) differential equation, depending only on the dimenson, is the key to that construction. Here we give some more details on the solutions of that equation and their behavior on a one-parameter family