Remove the paracompactness requirement for connection-induced local additions

Determine whether the paracompactness requirement on the base Banach manifold can be removed from the construction of a local addition on a Banach vector bundle equipped with linear connections on the bundle and its base tangent bundle.

Background

The paper constructs local additions on Banach vector bundles by using the spray of a linear connection. The argument invokes a result requiring paracompactness and therefore assumes that the base Banach manifold is paracompact, in addition to assuming the existence of suitable linear connections.

The authors explicitly state that they do not know how to avoid this assumption. Thus, an unresolved issue is whether the local-addition construction, or the underlying result used to obtain it, remains valid without paracompactness.