Almost splitting maps, transformation theorems and smooth fibration theorems

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 ([16]) and Cheeger-Jiang-Naber ([14]). More precisely, we show that a transformation theorem of Cheeger-Jiang-Naber (see Proposition 7.7 in [14]) holds for possibly collapsed manifolds. Some other applications of the transformation theorems are given in this paper.