The homotopy type of spaces of locally convex curves in the sphere

Saldanha, Nicolau C

doi:10.2140/gt.2015.19.1155

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The homotopy type of spaces of locally convex curves in the sphere

Nicolau C Saldanha

Geometry & Topology 19 (2015) 1155–1203

DOI: 10.2140/gt.2015.19.1155

Abstract

A smooth curve $γ : [0, 1] \to S^{2}$ is locally convex if its geodesic curvature is positive at every point. J A Little showed that the space of all locally convex curves $γ$ with $γ (0) = γ (1) = e_{1}$ and $γ^{'} (0) = γ^{'} (1) = e_{2}$ has three connected components $ℒ_{- 1, c}$ , $ℒ_{+ 1}$ , $ℒ_{- 1, n}$ . The space $ℒ_{- 1, c}$ is known to be contractible. We prove that $ℒ_{+ 1}$ and $ℒ_{- 1, n}$ are homotopy equivalent to $(Ω S^{3}) \lor S^{2} \lor S^{6} \lor S^{10} \lor \dots$ and $(Ω S^{3}) \lor S^{4} \lor S^{8} \lor S^{12} \lor \dots$ , respectively. As a corollary, we deduce the homotopy type of the components of the space $Free (S^{1}, S^{2})$ of free curves $γ : S^{1} \to S^{2}$ (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces $Free ([0, 1], S^{2})$ with fixed initial and final frames.

Keywords

convex curves, topology in infinite dimension, periodic solutions of linear ODEs

Mathematical Subject Classification 2010

Primary: 53C42, 57N65

Secondary: 34B05

References

Publication

Received: 8 August 2012

Revised: 29 August 2013

Accepted: 9 April 2014

Published: 21 May 2015

Proposed: Yasha Eliashberg

Seconded: Dmitri Burago, Leonid Polterovich

Authors

Nicolau C Saldanha
Departamento de Matemática PUC-Rio Rua Marquês de São Vicente, 225 Edifício Cardeal Leme, sala 862 - Gávea 22451-900 Rio de Janeiro Brazil
http://www.nicolausaldanha.com

Volume 19, issue 3 (2015)