A general approach to almost structures in geometry

Abstract

Differential Geometry studies various kinds of geometric structures on ``nice'' spaces (manifolds). Such structures appear naturally, e.g. when trying to measure, or even make sense of, distances, areas, etc., or when formalising physical theories. For instance, to talk about distances on such nice spaces, one needs to look at the class of Riemannian structures; or, for the mathematical structure that supports Hamilton's formulation of classical mechanics, one looks at the class of symplectic (or even Poisson) structures. For each kind of geometric structures, one is interested in studying the entire class of them - let us call it S. While the main interest is on such ``S-structures'', one can also talk about ``almost S-structures'', which arise naturally as their ``shadows''. When saying ``shadow'', one should have in mind an ``over-simplified version'' - which gives a rough idea/first order approximation of the actual structure, but without going into intricate details. A fundamental question, known as the integrability problem, is to understand when such a ``shadow'' is actually real, i.e. when an almost S-structure arises from an S-structure. There are various frameworks to make precise sense of S-structures but, somehow, each framework has its own limitations: there are always more examples of geometric structures than the frameworks can accommodate. The situation is even more problematic when looking for a general theory allowing to handle also ``almost S-structures'': the existing literature is always restricted to the so-called ``transitive case'' (when symmetries allow one to pass from one point to another). This thesis presents a general framework for studying almost structures and for proving integrability results, which is not restricted to the ``transitive case''. The standard approach to (almost) geometric structures, at least in the transitive case, is via the theory of G-structures, for an appropriate group of symmetries $G$. Here is an important remark at the foundation of this thesis: in the non-transitive case, one has to allow for local symmetries as well. Accordingly, one should concentrate not only on groups G, but allow also more general, point-dependent, version of groups: groupoids and pseudogroups Γ. The key concept of our approach to almost structures is that of a ``principal Pfaffian bundle''. Roughly speaking, it consists of a principal action by a groupoid G on a space P, together with two differential forms, one on P and one on G, which are compatible in an appropriate sense. To make this precise, we draw techniques and inspiration from Poisson geometry and the formal theory of PDEs.