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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
The homotopy type of spaces of locally convex curves in the sphere

Nicolau C Saldanha

Geometry & Topology 19 (2015) 1155–1203
Abstract

A smooth curve γ: [0,1] S2 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) = e1 and γ(0) = γ(1) = e2 has three connected components 1,c, +1, 1,n. The space 1,c is known to be contractible. We prove that +1 and 1,n are homotopy equivalent to (ΩS3) S2 S6 S10 and (ΩS3) S4 S8 S12 , respectively. As a corollary, we deduce the homotopy type of the components of the space Free(S1, S2) of free curves γ: S1 S2 (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces Free([0,1], S2) 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