Summary: Geometriae Dedicata 72: 113, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Rotation Numbers and Isometries
Department of Mathematics, University of Crete, GR-71409 Iraklion, Greece
(Received: 16 May 1997; revised version: 11 November 1997)
Abstract. Using the notion of rotation set for homomorphisms of compact manifolds, we define
the rotation homomorphism of a connected compact orientable Riemannian manifold and apply it
to prove that the dimension of the isometry group of a connected compact orientable Riemannian 3-
manifold without conjugate points is not greater than its first Betti number. In higher dimensions the
same is true under the additional assumption that the fundamental cohomology class of the manifold
is a cup product of integral one-dimensional classes.
Mathematics Subject Classifications (1991): 58F25, 53C20.
Key words: asymptotic cycle, rotation homomorphism, manifold without conjugate points.
A great amount of work in dynamical systems has been oriented towards the prob-
lem of finding conditions which guarantee the existence of periodic orbits for
homeomorphisms and flows. The best known condition concerning orientation pre-