Some basic theorems on flat G chains

For a complete normed Abelian group G, we show that a flat G chain (respectively, normal flat G chain) of codimension 0 in a finite dimensional Banach space corresponds to a G   valued functions that is HmHm summable (respectively, of bounded variation). We also prove a constancy theorem for an m dimensional flat G chain T in a metric space assuming supp T is contained in the closure of a connected m dimensional Lipschitz submanifold M   with M∩supp∂T=∅M∩supp∂T=∅ and Hm(M)=Hm(ClosM)<∞.