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

V. V. Shurygin

Matematičeskie zametki, Tome 59 (1996) no. 2, pp. 303-310

Résumé

For a canonical foliation on a manifold $M^{\mathbb A}$ over a local algebra, the $\mathbb A$-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle of $\mathbb A$-jets, is defined. In the case of a complete manifold $M^{\mathbb A}$, the $\mathbb A$-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental–Hebda. It is shown that the $\mathbb A$-affine horizontal distribution on $M^{\mathbb A}$ exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.