Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds

Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of order 3; they are not timelike Jordan Osserman. They are k-spacelike higher order Jordan Osserman for $2\le k\le s$; they are k-timelike higher order Jordan Osserman for $s+2\le k\le 2s$, and they are not k timelike higher order Jordan Osserman for $2\le s\le s+1$.Comment: Update bibliography, fix minor misprint

Reflections with respect to submanifolds in contact geometry

summary:We study to what extent some structure-preserving properties of the geodesic reflection with respect to a submanifold of an almost contact manifold influence the geometry of the submanifold and of the ambient space

Reflections With Respect To Submanifolds In Contact Geometry

. We study to what extent some structure-preserving properties of the geodesic reflection with respect to a submanifold of an almost contact manifold influence the geometry of the submanifold and of the ambient space. 1. Introduction Reflections with respect to points and curves and, more generally, with respect to submanifolds in Riemannian manifolds are generalizations of reflections with respect to linear subspaces of a Euclidean space. The reflections with respect to points and curves have been studied by different authors. It turns out that their properties strongly influence the curvature of the manifold and that one can characterize certain classes of manifolds (e.g., locally symmetric spaces and real space forms) by using properties of the reflections with respect to their points or their geodesics. For a survey of results of this type, we refer to [4], [14]. Later, one also started investigating similar problems concerning reflections with respect to submanifolds. As before, ..

Integrable Hamiltonian systems associated to families of curves and their bi-Hamiltonian structure

In this paper we show how there is associated an integrable Hamiltonian system to a certain set of algebraic-geometric data. Roughly speaking these data consist of a family of algebraic curves, parametrized by an affine algebraic variety B, a subalgebra C of O(B) and a polynomial '(x; y) in two variables. The phase space is constructed geometrically from the family of curves and has a natural projection onto B; the regular functions on B lead to an algebra of functions in involution and the level sets of the moment map are symmetric products of algebraic curves. While completely transparant from the geometrical point of view, a slight change of these integrable Hamiltonian systems is needed in order to explicitly realize these integrable Hamiltonian systems. Thus, we associate to the same data another integrable Hamiltonian system and show how they relate to the first one: there is a birational map between them (which is regular in one direction) which is (in the regular direction) a m..