By D. Arnal (auth.), M. Cahen, M. De Wilde, L. Lemaire, L. Vanhecke (eds.)

This quantity includes the textual content of the lectures which have been given on the Differential Geometry assembly held at Liege in 1980 and on the Differential Geometry assembly held at Leuven in 1981. the 1st of those conferences used to be extra oriented towards mathematical physics; the second one has a far better flavour of study. The Editors are happy to thank the lectures who contributed scientifically to those conferences. also they are thankful to Professor M. F1ato who has inspired booklet of those contributions within the Mathematical Physics experiences sequence. We additionally thank the F.N.R.S. who supported financially the touch crew in differential geometry. The Universite de Liege and the Katholieke Universiteit Leuven that have given us a hot hospitality have contributed to the luck of those conferences. We show our gratitude. The Editors. M. Caken et al. (6ds.), Differential Geametry and Mathematical Physics, vii. vii Copyright e 1983 via D. Reidel Publishing corporation. Lectures given on the assembly of the Belgian touch crew on Differential Geometry held at Liege, might 2-3,1980 SIMULTANEOUS DEFORMATIONS OF A LIE ALGEBRA AND ITS MODULES D. Arnal college of Dijon advent We reveal the following a few effects that are got via a crew on the college of Dijon. This workforce incorporated Jean-Claude Cortet, Georges Pinczon and myself.

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**Additional info for Differential Geometry and Mathematical Physics: Lectures given at the Meetings of the Belgian Contact Group on Differential Geometry held at Liège, May 2–3, 1980 and at Leuven, February 6–8, 1981**

**Example text**

1) we know therefore that x ~ Tx is an application of the space of motions U into the vector space of eulerian distributions of E4 (fig. V). 2) U being a function taken to he equal to zero in the past and equal to one in the future (fig. IV). 1) allows one to show that I does not depend on the choice of u, by making some assumptions on the behaviour of T at infinity (14? One can thus calculate I by making u jump from 0 to I in a small neighborhood of a space-like surface; the fact that the result is independent of the choice of this surface expresses the "preserved quantity" character of I.

Characterized in general relativity by its potentiaZs guv; the space of all motions is always a symplectic manifold U, whose structure depends on the field. Now we choose a compact K of space-time E4 (see fig. II) wherein we perturb the guv . The new space of motions u' is still a symplectic manifold, which can be connected to U by the technique of diffusion; this technique will be described in the case of a spinless particle, whose motion is characterized by the world line; if this line does not meet K, it characterizes a motion equally in U as in u'.

Gutt : "Invariance des equations de Maxwell", Bull. Soc. Math. de Belgique, Volume en hommage a J. Geheniau (to appear). (2) M. Cahen, S. Gutt : "Maxwell's equations in Segal's model: solutions and their invariance". Lett. in Math. Phys. 4 (1980). 29 ON THE CLASSIFICATION OF MANIFOLDS FOLIATED BY THE ACTION OF A NILPOTENT LIE GROUP G. Hector University of Lille A homogeneous space of a Lie group G is a manifold admitting a transitive differentiable action of G; these manifolds were the subject of many studies.