2,025 research outputs found
The homotopy type of spaces of locally convex curves in the sphere
A smooth curve \gamma: [0,1] \to \Ss^2 is locally convex if its geodesic
curvature is positive at every point. J. A. Little showed that the space of all
locally convex curves with and
has three connected components ,
, . The space \cL_{-1,c} is known to be contractible. We
prove that \cL_{+1} and \cL_{-1,n} are homotopy equivalent to
(\Omega\Ss^3) \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots and
(\Omega\Ss^3) \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots, respectively.
As a corollary, we deduce the homotopy type of the components of the space
\Free(\Ss^1,\Ss^2) of free curves \gamma: \Ss^1 \to \Ss^2 (i.e., curves
with nonzero geodesic curvature). We also determine the homotopy type of the
spaces \Free([0,1], \Ss^2) with fixed initial and final frames.Comment: 47 pages, 13 figure
Homotopy type of spaces of curves with constrained curvature on flat surfaces
Let be a complete flat surface, such as the Euclidean plane. We determine
the homeomorphism class of the space of all curves on which start and end
at given points in given directions and whose curvatures are constrained to lie
in a given open interval, in terms of all parameters involved. Any connected
component of such a space is either contractible or homotopy equivalent to an
-sphere, and every is realizable. Explicit homotopy equivalences
between the components and the corresponding spheres are constructed.Comment: 39 pages, 13 figures. Differs from previous version by many
improvements of the expositio
- …