Chronoprojectlve Cartan Structures on Four-Dimensional Manifolds

As indicated by its denomination Cartan structures have been derived from Cartan's works [1] initiating the projective and conformal geometries. In the fifties a precise description of this notion in a modern mathematical language has been given by using the fibre bundle of second order frames [2, 3]. The starting point can be viewed as a generalization of the Klein's Erlangen program. Indeed Cartan considered various spaces at each point of which an homogeneous space of the same dimension is tangentially associated, with the possibility of connecting these tangent spaces at different neighbouring points of the base space. Moreover these spaces were endowed with a "normal" connection which allows to develop the base space on the tangent homogeneous space along a curve. In a geometrical language the above depicted situation is described by using the notions of Cartan connection and Cartan structure. The classical geometries i.e. the projective [4] and conformal [5] geometries are the standard examples of Cartan structures; they correspond to the case where the bigger concerned Lie group is semisimple and its Lie algebra is jlj-graded. A general study of this case can be found in the literature [6]. On the contrary the geometrical structures considered in this paper, do not enter this scheme. They deal with a group which is not semi-simple and whose Lie algebra is |2|-graded, the so-called chronopro-