  In this paper, we introduce a notion, called generalized Reifenberg
condition, under which we prove a smooth fibration theorem for collapsed
manifolds with Ricci curvature bounded below, which gives a unified proof of
smooth fibration theorems in many previous works (including the ones proved by
Fukaya and Yamaguchi respectively). A key tool in the proof of this fibration
theorem is the transformation technique for almost splitting maps, which
originates from Cheeger-Naber (\cite{CN}) and Cheeger-Jiang-Naber
(\cite{CJN21}). More precisely, we show that a transformation theorem of
Cheeger-Jiang-Naber (see Proposition 7.7 in \cite{CJN21}) holds for possibly
collapsed manifolds. Some other applications of the transformation theorems are
given in this paper.
