Abstract: Consider a foliate R"-action on a compact connected foliated manifold(M,F).Let m and 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 R"-action of transverse dimension <(m+r)/2 or F can be arbitrarily approached by foliations with rank ≥r+1.Moreover we show that this kind of orbits exists in the following three cases:if F is Riemannian;when all its leaves are closed or if x(M)≠0(then r=0).On the other hand all foliate R"-action on(S³,F)has a fixed leaf if dimF=1.Our result generalizes a well known Lima's theorem about R"-actions on surfaces. 

Key words:  , transverse action, foliation