  Locally convex (or nondegenerate) curves in the sphere (or projective space)
have been studied for several reasons, including the study of linear ordinary
differential equations. Taking Frenet frames allows us to translate such curves
into corresponding curves in the flag space, the orthogonal group or its cover
$Spin_{n+1}$. Determining the homotopy type of the space of such closed curves
or, more generally, of spaces of such curves with prescribed initial and final
jets appears to be a hard problem, which has been solved for $n=2$ but
otherwise remains open. This paper is a step towards solving the problem for
larger values of $n$. In the process, we prove a related conjecture of B.
Shapiro and M. Shapiro. We define the itinerary of a locally convex curve
$\Gamma:[0,1]\to Spin_{n+1}$ as a word $w$ in the alphabet of non-trivial
permutations. This word encodes the succession of non-open Bruhat cells of
$Spin_{n+1}$ pierced by $\Gamma$. We prove that, for each word $w$, the
subspace of curves of itinerary $w$ is an embedded contractible (topological)
submanifold of finite codimension, thus defining a stratification of the space
of curves. We show how to obtain explicit (topologically) transversal sections
for each of these submanifolds. We study both a space of curves with minimum
regularity hypotheses, where only topological transversality applies, and
spaces of sufficiently regular curves. In both cases we study the neighboring
relation between strata. This is an important step in the construction of CW
cell complexes mapped into the original space of curves by weak homotopy
equivalences. Our stratification is not as nice as might be desired, lacking
for instance the Whitney property. The differentiability class of the curves
affects some properties of the stratification. The necessary ingredients for
the construction of a dual CW complex are proved.
