Abstract. This paper is devoted to the study of geometric structures modeled on homogeneous spaces G/P, where G is a real or complex semisimple Lie group and P ⊂ G is a parabolic subgroup. We use methods from differential geometry and very elementary finite–dimensional representation theory to construct sequences of invariant differential operators for such geometries, both in the smooth and the holomorphic

.91> X;/\Omega Y ] = ' /\Omega [X; Y ] + ' LX/\Omega Y \Gamma L Y ' /\Omega X + (\Gamma1) k (d' i X/\Omega Y + i Y ' d/\Omega X) ; where X and Y are vector fields, ' is a k-form, and / is an l-form. It is a bilinear differential operator of bidegree (1; 1). The Frolicher-Nijenhuis bracket is natural in the same way as the Lie bracket for vector fields: if f : M ! N is smooth and K i 2\Omega k i (M ;

In this paper we propose a new equivalence relation for dynamical and control systems called bisimulation. As the name implies this definition is inspired by the fundamental notion of bisimulation introduced by R. Milner for labeled transition systems. It is however, more subtle than its namesake in concurrency theory, mainly due to the fact that here, one deals with relations on manifolds. We further show that the bisimulation relations for dynamical and control systems defined in this paper are captured by the notion of abstract bisimulation of Joyal, Nielsen and Winskel (JNW). This result not only shows that our equivalence notion is on the right track, but also confirms that the abstract bisimulation of JNW is general enough to capture equivalence notions in the domain of continuous systems. We believe that the unification of the bisimulation relation for labeled transition systems and dynamical systems under the umbrella of abstract bisimulation, as achieved in this work, is a first step towards a unified approach to modeling of and reasoning about the dynamics of discrete and continuous structures in computer science and control theory.

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauß-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.

this article a general theory of Cartan connections is developed and some applications are indicated. The starting idea is to consider a Cartan connection as a deformation of a local Lie group structure on the manifold, i.e. a 1-form

1. Liouville 1-forms on fiber bundles and the lifting of vector fields..... 3 2. The canonical Poisson structure on T ∗ G............... 5 3. Generalizing momentum mappings.................. 11

The almost Hermitian symmetric structures include several important geometries, e.g. the conformal, projective, quaternionic or almost Grassmannian ones. The conformal case is known best and several efficient techniques have been worked out in the last 90 years. The present note provides links of the development presented in [CSS1, CSS2, CSS3, CSS4] to several other approaches and it suggests extensions of some techniques to all geometries in question.

. In this short note we use the concept of the principal prolongation of principal fiber bundles to develop an alternative procedure for the construction of prolongations of a class of G-structures of the first order. The motivation comes from the so called almost Hermitian structures which can be defined either as standard first order structures, or higher order structures, but if they do not admit a connection without torsion, the classical constructions fail in general. AMS Classification: 58 A 20, 53 A 55 In the sequel, we shall always work with smooth finite dimensional manifolds and Lie groups. First we shall recall the standard procedure of the principal prolongation, due to [Libermann, 71] and [Kol'ar, 71], the details can be found in [Kol'ar, Michor, Slov'ak, 93, sections 15, 16]. This generalization of the concept of frame bundles will enable us to construct prolongations of the first order structures which are principal fiber bundles over the original base manifold with nic...

Skew critical problems occur in continuous and discrete nonholonomic Lagrangian systems. They are analogues of constrained optimization problems, where the objective is differentiated in directions given by an apriori distribution, instead of tangent directions to the constraint. We show semiglobal existence and uniqueness for nondegenerate skew critical problems, and show that the solutions of two skew critical problems have the same contact as the problems themselves. Also, we develop some infrastructure that is necessary to compute with contact order geometrically, directly on manifolds. 1