Variational Invariants of Riemannian Manifolds

Abstract

Abstract. This paper treats higher-dimensional analogues to the minimum geodesic distance in a compact Riemannian manifold M with finite fundamental group. These invariants are based on the concept of homotopy distance in M. This defines a parametrized variational problem which is approached by globalizing the Morse theory of the spaces of paths between two points of M to the space of all paths in M. We develop machinery that we apply to calculate the invariants for numerous examples. In particular, we shall observe that knowledge of the invariants for the standard spheres determines the question of the existence of elements of Hopf invariant one.