Consider a foliate Rn-action on a compact connected foliated manifold (M,F). Let mand r be the codimension of F and the (transverse)rank of (M,F)respectively. Suppose r<m.In this paper we prove that either there ex...Consider a foliate Rn-action on a compact connected foliated manifold (M,F). Let mand r be the codimension of F and the (transverse)rank of (M,F)respectively. Suppose r<m.In this paper we prove that either there exists an orbit of the Rn-action of transverse dimension< (m + r)/2 or F can be arbitrarily approached by foliations with rank≥r+1. Moreover weshow that this kind of orbits exists in the following three cases: if F is Riemannian ;when all itsleaves are closed or if X(M)≠0(then r=0).On the other hand all foliate Rn-action on (S3,F) has a fixed leaf if dimF=1.Our result generalies a well known Lima's theorem about Rn-actions on surfaces.展开更多
文摘Consider a foliate Rn-action on a compact connected foliated manifold (M,F). Let mand r be the codimension of F and the (transverse)rank of (M,F)respectively. Suppose r<m.In this paper we prove that either there exists an orbit of the Rn-action of transverse dimension< (m + r)/2 or F can be arbitrarily approached by foliations with rank≥r+1. Moreover weshow that this kind of orbits exists in the following three cases: if F is Riemannian ;when all itsleaves are closed or if X(M)≠0(then r=0).On the other hand all foliate Rn-action on (S3,F) has a fixed leaf if dimF=1.Our result generalies a well known Lima's theorem about Rn-actions on surfaces.
