A classical problem in the
calculus of variations may be expressed: Given two points in the same component of aRiemannian manifold M, what is the length of the shortest path connecting them? In
this paper we discuss parametric analogs to this problem. The width of a
homotopy is the supremum of the lengths of the paths traced by the points of X.
Work of Allan Calder and the first author on the topology of Stone-Cech
compactifications led to the study of the question: Given a space X and twohomotopic maps f,g : X → M, what is the width of the shortest homotopy betweenthem?
In this paper we obtain bounds bq(M) that depend only on the dimension q of X
for the answer to this question in case M is a sphere or a projective space (with the
standard metric). We also introduce a related sequence of invariants Bq(M) and
compute these numbers for spheres and projective spaces. Of particular interest is the
fact that b2n−2(Sn) detects elements of Hopf invariant one while B2n−2(Sn) does
not.
