  This paper stresses the strong link between the existence of partial
holomorphic connections on the normal bundle of a foliation seen as a quotient
of the ambient tangent bundle and the extendability of a foliation to an
infinitesimal neighborhood of a submanifold. We find the obstructions to
extendability and thanks to the theory developed we obtain some new
Khanedani-Lehmann-Suwa type index theorems.
