L'équation de rappel

In this thesis we are interested in the following problem : given two differential k–forms g and f, most of the time they will be assumed closed, on what conditions can we pullback g to f by a map φ ? In other words we ask when it is possible to solve the pullback equation φ*(g) = f. This equation originates from differential geometry. This work is divided into five main parts. In Part I we focus on exterior forms. The study of these forms, which is part of the domain of multilinear algebra, will allow us to understand the pullback equation at the algebraic level and will turn out to be a main first step for the study of the problem. Part II deals with the study of differential forms. A special attention will be brought to different versions of Poincaré lemma which will be constantly used all the way through this thesis. In Part III we solve the problem for volume forms k = n. This part, which will use very seldom the machinery of differential geometry, can be read independently of the previous two parts. Part IV sheds light on the case 0 ≤ k ≤ n – 1. Emphasis will be given on the case k = 2. The best results of this present work work will concern 2–forms and volume forms. The cases k = 0, 1, n – 1 are the easiest and will be discussed briefly while for the case 3 ≤ k ≤ n – 2, which will turn much more difficult even already at the algebraic level, we will only obtain results for k-forms with a special structure. Finally, in Part V, we gather numerous useful properties of Hödler spaces. These spaces will be very suitable for the pullback equation ; indeed there are an algebra (contrary to Sobolev spaces in general) and allow to get results with sharp regularity.