  In the 1920's Marston Morse developed what is now known as Morse theory
trying to study the topology of the space of closed curves on S^2. We propose
to attack a very similar problem, which 80 years later remains open, about the
topology of the space of closed curves on S^2 which are locally convex (i.e.,
without inflection points). One of the main difficulties is the absence of the
covering homotopy principle for the map sending a non-closed locally convex
curve to the Frenet frame at its endpoint.
  In the present paper we study the spaces of locally convex curves in S^n with
a given initial and final Frenet frames. Using combinatorics of B^{+}_{n+1} =
B_{n+1} \cap SO_{n+1}, where B_{n+1} \subset O_{n+1} is the usual Coxeter-Weyl
group, we show that for any n \ge 2 these spaces fall in at most
$\lceil\frac{n}{2}\rceil+1$ equivalence classes up to homeomorphism. We also
study this classification in the double cover Spin(n+1).
  For $n = 2$ our results complete the classification of the corresponding
spaces into two topologically distinct classes, or three classes in the spin
case.
