On the topological nature of Volterra's theorem

Abstract

It is first observed that the original formulation of the Volterra construction for dislocations and disclinations was related to the role that homotopy plays in strain compatibility, whereas the modern discussions are chiefly concerned with how it relates to the holonomy groups of connections that have non-vanishing torsion and curvature. However, the Saint Venant conditions that follow from assuming infinitesimal strain compatibility imply that both torsion and curvature must vanish. The resolution of the confusion is in the fact that when a manifold is multiply connected a flat connection might still have non-trivial discrete holonomy.