Topics in the differential geometry of supermanifolds

Given a super principal fiber bundle B = B( M, H) $\to p$ M, equipped with a connection ∇, we introduce a “super holonomy” group Φ(u) with reference point u ∈ B on the super principal fiber bundle B. The super holonomy group is shown to be a sub-super Lie group of the structure super Lie group H of the bundle B. We prove that the set of points of B that can be joined to an arbitrary point u of B by horizontal smooth curves is a sub-super principal fiber bundle of B with structure super Lie group Φ(u).; These results enable us to obtain a super version of the Ambrose - Singer holonomy theorem.