Theorem 4 .
4Consider the following commutative diagram. Then π N (H) : M H → M H /N (H) is a bundle reduction of π : M → M/G and π ′ : M H → M H /(N (H)/H) is a principal bundle reduction of M H → M H /N (H). There is a one-to-one correspondence between principal bundle connections on M H → M H /(N (H)/H) and singular connections on M . Furthermore, the curvature form for the singular connection restricted to M H is equal to the curvature form Ω of the principal connection. Proof. As we have already remarked, every G-orbit in M intersects M H in a unique N (H)-orbit. Therefore for m ∈ M H , π N (H) (m) = π(ι(m)), and then M/G ≃ M H /N (H). Furthermore, since H fixes every point of M H , the free action of N (H)/H on M H has the same orbits as the action of N (H) on M H and therefore the bundle M H → M H /(N (H)/H) is a principal bundle reduction of M H → M H /N (H). To set up the one-to-one correspondence of connections, we start with a connection on the principal bundle π ′ : M H → M H /(N (H)/H). Denote this connection by Γ consisting of horizontal spaces Q m for each m ∈ M H and its connection form by ω : T M H → n/h. We show how to induce from this **data** a connection on M → M/G where the connection **data** will consist of a horizontal distribution invariant with respect to the group action. Given m ′ ∈ M we have m ′ = g · m for some m ∈ M H and g ∈ G.