 
Summary: Divergence of C1
vector fields and nontrivial minimal sets
on 2manifolds
Konstantin Athanassopoulos
Department of Mathematics, University of Crete, GR71409 Iraklion, Greece
email : athanako@math.uoc.gr
Abstract
We prove a BendixsonDulac type criterion for the nonexistence of nontrivial
compact minimal sets of C1 vector fields on orientable 2manifolds. As a corollary we
get that the divergence with respect to any volume 2form of such a vector field must
vanish at some point of any nontrivial compact minimal set. We also prove that all the
nontrivial compact minimal sets of a C1 vector field on an orientable 2manifold are
contained in the vanishing set of any inverse integrating factor. From this we get that
if a C1 vector field on an orientable 2manifold has a nontrivial compact minimal set,
then an infinitesimal symmetry is inessential on the minimal set.
MSC: 34C40, 37C10, 37E35, 37B20.
Keywords: C1 vector field, divergence, inverse integrating factor, infinitesimal symmetry,
nontrivial compact minimal set.
1. Introduction
A classical problem in the qualitative theory of 2dimensional ordinary differential
