Dynamics and global geometry of manifolds without conjugate points

Abstract. Manifolds with no conjugate points are natural generalizations
of manifolds with nonpositive sectional curvatures. They have in
common the fact that geodesics are global minimizers, a variational property
of geodesics that is quite special. The restriction on the sign of the
sectional curvatures of the manifold leads to a deep knowledge about the
topology and the global geometry of the manifold, like the characterization
of higher rank, nonpositively curved spaces as symmetric spaces. However,
if we drop the assumptions concerning the local geometry of the manifold
the study of geodesics becomes much harder. The purpose of this survey
is to give an overview of the classical theory of manifolds without conjugate
points where no assumptions are made on the sign of the sectional
curvatures, since the famous work of Morse about minimizing geodesics
of surfaces and the works of Hopf about tori without conjugate points.
We shall show important classical and recent applications of many tools of
Riemannian geometry, topological dynamics, geometric group theory and
topology to study the geodesic
ow of manifolds without conjugate points
and its connections with the global geometry of the manifold. Such applications
roughly show that manifolds without conjugate points are in many
respects close to manifolds with nonpositive curvature from the topological
point of view.