Counterexamples to the Seifert conjecture

Abstract

Since H. Seifert proved in 1950 the existence of a periodic orbit for a vector field on the 3-dimensional sphere $S^3$ which forms small angles with the fibers of the Hopf fibration, several examples of aperiodic vector fields on $S^3$ have been produced as well as results showing that in some situations a compact orbit must exist. This paper surveys presently known types of vector fields without periodic orbits on $S^3$ and on other manifolds.