Cartan submersions and Cartan foliations
\textit{W. Ambrose} [Ann. Math., II. Ser. 64, 337-363 (1956)] showed that a local isometry defined on a complete Riemannian manifold is a covering, \textit{N. Hicks} [Ill. J. Math. 3, 242-254 (1959; Zbl 0088.381)] proved a similar result for local affine isomorphisms, and the present author has obtained analogous results for local projective and conformal isomorphisms. \textit{R. Hermann} [Proc. Am. Math. Soc. 11, 236-242 (1960; Zbl 0112.137)] showed that a Riemannian submersion defined on a complete Riemannian manifold is a locally trivial fiber bundle and the present author has shown that an affine submersion defined on a complete affinely connected manifold is a fibration. All of the geometric structures occurring in the results quoted above can be treated in a uniform fashion under the rubric of Cartan connections. We consider submersions between manifolds with Cartan connections and we give sufficient conditions for such maps to be fibrations. As an application we consider foliations whose transverse structure is modeled on a Cartan geometry. Among the examples of Cartan foliations are Riemannian, affine, projective, conformal, Lie, and homogeneous foliations. We prove a stability theorem for Cartan foliations which generalizes the Reinhart stability theorem for Riemannian foliations and we obtain a structure theorem for analytic Cartan foliations.
scientific article; zbMATH DE number 3984044