We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

This paper is devoted to the investigation of Lie algebras of local infinitesimal CR automorphisms. Such algebras are naturally associated to germs of homogeneous CR manifolds. The authors introduce a corresponding abstract notion of CR algebra. A CR algebra is a pair $(L,L_1)$(L,L1), consisting of a real Lie algebra $L$L and a subalgebra $L_1$L1 of the complexification $\bold C\otimes_{\bold R} L$C⊗RL, such that the factor space $L/L\cap L_1$L/L∩L1 is finite-dimensional.\ud The authors investigate some formal properties of CR algebras and construct some "fibrations'' (i.e., $L$L-equivariant submersions) of such algebras. They introduce three new notions of nondegeneracy of CR algebras---strict, weak and ideal nondegeneracy. These three concepts are weaker than those used previously by some other authors. The authors intend to extend the application of the É. Cartan method of investigating the equivalence of CR structures to some larger classes of CR manifolds.\ud One of the main ideas of this paper is a decomposition of arbitrary CR algebras into three "parts'': totally real, totally complex and weakly nondegenerate CR algebras (Theorems 5.3 and 5.4). There are some results about these three special classes of CR algebras. Some results about prolongations for transitive CR algebras are also obtained, in particular about maximality of parabolic CR algebras with respect to transitive prolongations

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We consider the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients. In order to avoid the problem of the formal coherence of the initial data, which leads to the notion of formally non-characteristic introduced by Andreotti-Nacinovich in the 80's, we allow formal solutions (in the sense of Whitney) of the given system on any closed subset as initial data. The problem is then to find classical smooth solutions of the system, whose restrictions in the sense of Whitney are the given initial data. This gives a unifying point of view encompassing several different problems, ranging from questions of smoothness of the solutions, to the classical Cauchy problem, to comparison of formal to actual solutions, to Hartog's type phenomena