Cartan submersions and Cartan foliations
type: Item
instance of: scholarly article
title: Cartan submersions and Cartan foliations
zbMATH Open document ID: 0608.53027
author: Robert A. Blumenthal
published in: Illinois Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: \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.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 3984044
zbMATH Keywords: Cartan connections
zbMATH Keywords: submersions
zbMATH Keywords: fibrations
zbMATH Keywords: Cartan foliations
zbMATH Keywords: Reinhart stability theorem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3984044