Ehresmann connection for the canonical foliation on a manifold over a local algebra

Abstract

For a canonical foliation on a manifold MA over a local algebra, the double-struck A-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle of double-struck A-jets, is defined. In the case of a complete manifold MA, the double-struck A-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that the double-struck A-affine horizontal distribution on MA exists if and only if the Atiyah class of a certain foliated principal bundle vanishes. ©1996 Plenum Publishing Corporation.