Comparing Riemannian foliations with transversally symmetric foliations
In this paper Riemannian foliations are compared with transversally homogeneous foliations, where the model transverse structure is of the type of an irreducible compact symmetric space G/K. The datum needed for this comparison is a connection in the normal bundle, having similar properties as the canonical connection in the case of a transversally symmetric foliation. This similarity is most conveniently formulated in terms of the corresponding Cartan connections. For the symmetric model case the curvature of the Cartan connection vanishes. An almost transversally symmetric foliation is one where this curvature is small in an appropriate Sobolev norm. In the spirit of Rauch's comparison theorem, and more specifically, the comparison theorem of Min-Oo and Ruh, one wishes to conclude that this assumption already implies the existence of a transversally symmetric structure of type G/K. The main result of this paper is that it is true for tense Riemannian foliations with small mean curvature. Here the tenseness means that the mean curvature form of the Riemannian foliation is a basic 1-form. A weaker form of this result was announced earlier, where the mean curvature form was assumed to vanish. The idea of the proof is to construct on the bundle P of normal frames of \({\mathcal F}^ a \)Cartan connection \({\bar \omega}\) with vanishing curvature. This yields a developing map \(\Phi\) : \~P\(\to G\) on the universal covering \~P, equivariant with respect to a homomorphism \(\pi _ 1(P)\to \Gamma \subset G\) (holonomy of \({\bar \omega}\)). It induces in turn a map \(\phi\) : \~M\(\to G/K\), possibly after an averaging process. This map defines the transversally symmetric structure of type G/K for the foliation \({\mathcal F}\) as desired, via its lift to the universal covering \~M. The construction of a Cartan connection with vanishing curvature makes use of the de Rham-Hodge theory for the basic cohomology recently developed [see \textit{F. Kamber} and \textit{Ph. Tondeur}, Math. Ann. 277, 415-431 (1987)].
scientific article; zbMATH DE number 3992349