Contractible manifolds; Homotopy operator; Poincaré lemma; Exact and antiexact forms; Darboux class of one-forms; Canonical forms; Carathéodory's theorem; Exterior differential equations
6.1 Scope of the Chapter
In this section, we shall attempt to investigate certain fundamental properties of exterior differential forms in depth. The most powerful tool that we can employ for this purpose is the homotopy operator. However, this operator can only be defined on manifolds possessing a particular structure. This structure is treated in Sec. 6.2. A manifold is called locally contractible if every open set in its atlas can be smoothly shrunk to one of its interior points. This situation is realised if the homeomorphic ...