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Motivated by the geometric theory of differential equations and thevariational approach to the equivalence problem for geometric structures onmanifolds, we consider the problem of equivalence for distributions with fixedsubmanifolds of flags on each fiber. We call them flag structures. Theconstruction of the canonical frames for these structures can be given in thetwo prolongation steps: the first step, based on our previous works, gives thecanonical bundle of moving frames for the fixed submanifolds of flags on eachfiber and the second step consists of the prolongation of the bundle obtainedin the first step. The bundle obtained in the first step is not as a rule aprincipal bundle so that the classical Tanaka prolongation procedure forfiltered structures can not be applied to it. However, under naturalassumptions on submanifolds of flags and on the ambient distribution, thisbundle satisfies a nice weaker property. The main goal of the present paper isto formalize this property, introducing the so-called quasi-principle framebundles, and to generalize the Tanaka prolongation procedure to these bundles.Applications to the equivalence problems for systems of differential equationsof mixed order, bracket generating distributions, sub-Riemannian and moregeneral structures on distributions are given.

Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds