Applied geodesic foliations
This paper contains applications to many problems of geometry, of results on (non existence of) geodesic foliations. Some results on geodesic foliations belong to (\textit{A. Zeghib}: ``Feuilletages géodésiques des variétés localement symétriques'', Lyon: ENS, Preprint 1992), but most of the applications can be derived from a result of this paper itself. That is a classification of geodesic ``partially hyperbolic'' foliations, with complete leaves, on finite volume (not necessarily complete) Riemannian manifolds. Here partially hyperbolic means that the curvature tensor applied to three vectors, for which one (at least) is tangent to the foliation, has the same form as in the hyperbolic space of curvature \(-1\). A typical application is the following. Let \(V^ n\) and \(W^{n+q}\) be two hyperbolic manifolds (i.e. of curvature \(-1\)), and \(f:V\to W\), a \(C^ 3\) isometric immersion. Suppose \(q<n\), and \(V\) of finite volume and complete; then \(f(V)\) is a geodesic submanifold of \(W\).
scientific article; zbMATH DE number 562485
scientific article