During the last ten years a powerful technique for the study of partial differential equations with regular singularities has developed using the theory of hyperfunctions. The technique has had several important applications in harmonic analysis for symmetric spaces.

0.1. The creators of the Lie theory viewed a Lie group as a group of symmetries of an algebraic or a geometric object; the corresponding Lie algebra, from their point of view, was the set of infinitesimal transformations. Since the group of symmetries of the object is not necessarily finite-dimensional, S. Lie considered not only the problem of classification of subgroups of GL n , but also the problem of classification of infinite-dimensional groups of transformations.

These notes consist of two parts: 1) Selected Topics in Geometry, New York University 1946, Notes by Peter Lax. 2) Lectures on Differential Geometry in the Large, Stanford University 1956, Notes by J. W. Gray. They are reproduced here with no essential change. Heinz Hopf was a mathematician who recognized important mathema­ tical ideas and new mathematical phenomena through special cases. In the simplest background the central idea or the difficulty of a problem usually becomes crystal clear. Doing geometry in this fashion is a joy. Hopf's great insight allows this approach to lead to serious ma­ thematics, for most of the topics in these notes have become the star­ ting-points of important further developments. I will try to mention a few. It is clear from these notes that Hopf laid the emphasis on poly­ hedral differential geometry. Most of the results in smooth differen­ tial geometry have polyhedral counterparts, whose understanding is both important and challenging. Among recent works I wish to mention those of Robert Connelly on rigidity, which is very much in the spirit of these notes (cf. R. Connelly, Conjectures and open questions in ri­ gidity, Proceedings of International Congress of Mathematicians, Hel­ sinki 1978, vol. 1, 407-414 ) • A theory of area and volume of rectilinear'polyhedra based on de­ compositions originated with Bolyai and Gauss.

Dicembre 1974. La casa discografica Numero Uno pubblica il nuovo album di Lucio Battisti. È un disco molto atteso. L’anno precedente Battisti aveva piazzato contemporaneamente al primo e secondo posto delle hit parade ben due 33 giri, un evento raro e difficilmente ripetibile da qualsiasi altro artista. Uno era il mio canto libero, uscito nel novembre del 1972, l’altro il nostro caro angelo, pubblicato appena nove mesi dopo, nel settembre del 1973. Con quegli album Battisti aveva consolidato la sua fa-ma di autore di brani capaci di arrivare al grande pubblico, senza concessioni e cadute di gusto. Il 1973 fu l’anno delle hit a 45 giri Il mio canto libero e La collina dei ciliegi; dal nuovo disco in uscita tutti si aspettavano altre grandi canzoni. Ma Battisti spiazzò tutti, anche lo stesso Mogol..

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.

These notes were written by Camilla Horst on the basis of the lectures I gave during the week of June 22-26, 1981 at the DMV Seminar on Complex Differential Geometry in Dusseldorf. My aim was to make the contents of my survey lecture at the DMV annual meeting in 1980 (published in Jahresberichte, 1981) accessible to beginning research students by providing a little more details. I suggest therefore that the Jahresberichte paper be read as an introduction to these notes.

Whoever you are! How can I but offer you divine leaves . . . ? Walt Whitman The object of study in modern differential geometry is a manifold with a differ­ ential structure, and usually some additional structure as well. Thus, one is given a topological space M and a family of homeomorphisms, called coordinate sys­ tems, between open subsets of the space and open subsets of a real vector space V. It is supposed that where two domains overlap, the images are related by a diffeomorphism, called a coordinate transformation, between open subsets of V. M has associated with it a tangent bundle, which is a vector bundle with fiber V and group the general linear group GL(V). The additional structures that occur include Riemannian metrics, connections, complex structures, foliations, and many more. Frequently there is associated to the structure a reduction of the group of the tangent bundle to some subgroup G of GL(V). It is particularly pleasant if one can choose the coordinate systems so that the Jacobian matrices of the coordinate transformations belong to G. A reduction to G is called a G-structure, which is called integrable (or flat) if the condition on the Jacobians is satisfied. The strength of the integrability hypothesis is well-illustrated by the case of the orthogonal group On. An On-structure is given by the choice of a Riemannian metric, and therefore exists on every smooth manifold.

From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks… The second part, a very short one, consists of hints… The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem…

The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo­ metry during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [l.c.] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, i. e., the corres­ ponding periodic orbit in the geodesic flow is infinitesimally stable.