First Neighbourhood of the Diagonal, and Geometric Distributions

For any manifold, we describe the notion of geometric distribution on it, in terms of its first neighbourhood of the diagonal. In these “combinatorial” terms, we state the Frobenius Integrability Theorem, and use it to give a combinatorial proof of the Ambrose–Singer Theorem on connections in principal bundles. The consideration of the k’th neighbourhood of the diagonal of a manifold M , M(k) ⊆ M ×M , was initiated by Grothendieck to import notions from differential geometry into the realm of algebraic geometry. These notions were reimported into differential geometry by Malgrange [14], Kumpera and Spencer [12], . . . . They utilized the notion of ringed space (a space equipped with a structure sheaf of functions). The only points of (the underlying space of)M(k) are the diagonal points (x, x) with x ∈ M . But it is worthwhile to describe mappings to and fromM(k) as if M(k) consisted of “pairs of k-neighbour points (x, y)” (write x ∼k y for such a pair; such x and y are “points proches” in the terminology of A. Weil). The introduction of topos theoretic methods has put this “synthetic” way of speaking onto a rigourous basis, and we shall freely use it. We shall be only interested in the case k = 1, so we are considering the first neighbourhood of the diagonal, M(1) ⊆M ×M , and we shall write x ∼ y instead of x ∼1 y whenever (x, y) ∈ M ×M belongs to the subspace M(1). The relation ∼ is reflexive and symmetric, but unlike the notion of “neighbour” relation in Non Standard Analysis, it is not transitive. In some previous writings, [6], [7], [8], [10], [11], we have discussed the paraphrasing of several differential–geometric notions in terms of the combinatorics of the neighbour relation ∼. We shall remind the reader about some of them concerning differential forms and connections in Section 5 below.