SUSY structures on deformed supermanifolds

We construct a geometric structure on deformed supermanifolds as a certain subalgebra of the vector fields. In the classical limit we obtain a decoupling of the infinitesimal odd and even transformations, whereas in the semiclassical limit the result is a representation of the supersymmetry algebra. In the case of mass preserving structure we describe all high energy corrections to this algebra.Comment: 20 pages. v2 coincides with the version published in Differential Geometry and its Application

Local reflexion spaces

A reflexion space is generalization of a symmetric space introduced by O. Loos. We generalize locally symmetric spaces to local reflexion spaces in the similar way. We investigate, when local reflexion spaces are equivalently given by a locally flat Cartan connection of certain type.Comment: 8 pages, submitted to Archivum Mathematicu

Homogeneous Lorentzian manifolds of a semisimple group

We describe the structure of $d$-dimensional homogeneous Lorentzian $G$-manifolds $M=G/H$ of a semisimple Lie group $G$. Due to a result by N. Kowalsky, it is sufficient to consider the case when the group $G$ acts properly, that is the stabilizer $H$ is compact. Then any homogeneous space $G/\bar H$ with a smaller group $\bar H \subset H$ admits an invariant Lorentzian metric. A homogeneous manifold $G/H$ with a connected compact stabilizer $H$ is called a minimal admissible manifold if it admits an invariant Lorentzian metric, but no homogeneous $G$-manifold $G/\tilde H$ with a larger connected compact stabilizer $\tilde H \supset H$ admits such a metric. We give a description of minimal homogeneous Lorentzian $n$-dimensional $G$-manifolds $M = G/H$ of a simple (compact or noncompact) Lie group $G$. For $n \leq 11$, we obtain a list of all such manifolds $M$ and describe invariant Lorentzian metrics on $M$

Paraquaternionic CR-submanifolds of paraquaternionic Kahler manifolds and semi-Riemannian submersions

In this paper we introduce paraquaternionic CR-submanifolds of almost paraquaternionic hermitian manifolds and state some basic results on their differential geometry. We also study a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of paraquaternionic Kaehler manifolds.Comment: 19 page