Abstract
A smooth curve 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 is known to be contractible. We prove that and are homotopy equivalent to and , respectively. As a corollary, we deduce the homotopy type of the components of the space of free curves (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces with fixed initial and final frames.
Citation
Nicolau C Saldanha. "The homotopy type of spaces of locally convex curves in the sphere." Geom. Topol. 19 (3) 1155 - 1203, 2015. https://doi.org/10.2140/gt.2015.19.1155
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