Bøger af L. Vanhecke

This volume contains the text of the lectures which were given at the Differential Geometry Meeting held at Liege in 1980 and at the Differential Geometry Meeting held at Leuven in 1981. The first of these meetings was more orientated toward mathematical physics; the second has a stronger flavour of analysis. The Editors are pleased to thank the lectures who contributed scientifically to these two meetings. They are also grateful to Professor M. F1ato who has encouraged publication of these contributions in the Mathematical Physics Studies Series. We also thank the F.N.R.S. who supported financially the Contact group in differential geometry. The Universite de Liege and the Katholieke Universiteit Leuven which have given us a warm hospitality have contributed to the success of these meetings. We express our gratitude. The Editors. M. Caken et al. (6ds.), Differential Geametry and Mathematical Physics, vii. vii Copyright e 1983 by D. Reidel Publishing Company. Lectures given at the Meeting of the Belgian Contact Group on Differential Geometry held at Liege, May 2-3,1980 SIMULTANEOUS DEFORMATIONS OF A LIE ALGEBRA AND ITS MODULES D. Arnal University of Dijon INTRODUCTION We expose here some results which are obtained by a team at the University of Dijon. This team included Jean-Claude Cortet, Georges Pinczon and myself.

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.