Differential Geometry studies various kinds of geometric structures on “nice” spaces (manifolds). Such structures arise 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 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 that allow one to make precise sense of S-structures but 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 that allows one to also handle “almost S-structures”: the existing literature is always restricted to the so-called “transitive case”. This thesis presents a general framework for studying almost structures and for proving integrability results, which is not restricted to the “transitive case”.