Introduction and Motivation
The Grothendieck connection is a generalization of the Gauss-Manin connection constructed in a manner analogous to that in which the Ehresmann connection generalizes the Koszul connection. The construction itself must satisfy a requirement of geometric invariance, which may be regarded as the analog of covariance for a wider class of structures including the schemes of algebraic geometry. Thus the connection in a certain sense must live in a natural sheaf on a Grothendieck topology. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection.
Let M be a manifold and π : E → M a surjective submersion, so that E is a manifold fibred over M. Let J1(M,E) be the first-order jet bundle of sections of E. This may be regarded as a bundle over M or a bundle over the total space of E. With the latter interpretation, an Ehresmann connection is a section of the bundle (over E) J1(M,E) → E. The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.
Grothendieck's solution is to consider the diagonal embedding Δ : M → M × M. The sheaf I of ideals of Δ in M × M consists of functions on M × M which vanish along the diagonal. Much of the infinitesimal geometry of M can be realized in terms of I. For instance, Δ* (I/I2) is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood M(2) of Δ in M × M to be the subscheme corresponding to the sheaf of ideals I2. (See below for a coordinate description.)
There are a pair of projections p1, p2 : M × M → M given by projection the respective factors of the Cartesian product, which restrict to give projections p1, p2 : M(2) → M. One may now form the pullback of the fibre space E along one or the other of p1 or p2. In general, there is no canonical way to identify p1*E and p2*E with each other. A Grothendieck connection is a specified isomorphism between these two spaces.
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