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Stratification of Spaces of Locally Convex Curves by Itineraries

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Abstract

Locally convex (or nondegenerate) curves in the sphere \({{\mathbb {S}}}^n\) (or the projective space) have been studied for several reasons, including the study of linear ordinary differential equations of order \(n+1.\) Taking Frenet frames allows us to translate such curves \(\gamma \) into corresponding curves \(\Gamma \) in the flag space, the orthogonal group \({\text {SO}}_{n+1}\) or its double cover \({\text {Spin}}_{n+1}.\) 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 regarding the behavior of fundamental systems of solutions to linear ordinary differential equations. We define the itinerary of a locally convex curve \(\Gamma :[0,1]\rightarrow {\text {Spin}}_{n+1}\) as a (finite) word w in the alphabet \(S_{n+1}{\smallsetminus }\{e\}\) of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of \({\text {Spin}}_{n+1}\) pierced by \(\Gamma (t)\) as t ranges from 0 to 1. We prove that, for each word w,  the subspace of curves of itinerary w is an embedded contractible (globally collared 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, where transversality has the usual meaning. In both cases we also study the adjacency 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. Somewhat surprisingly, 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.

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Acknowledgements

We would like to thank: Emília Alves, Boris Khesin, Ricardo Leite, Carlos Gustavo Moreira, Paul Schweitzer, Boris Shapiro, Michael Shapiro, Carlos Tomei, David Torres, Cong Zhou and Pedro Zülkhe for helpful conversations and the referee for a careful review of the text. We also thank the University of Toronto and the University of Stockholm for the hospitality during our visits. Both authors thank CAPES, CNPq and FAPERJ (Brazil) for financial support. More specifically, the first author benefited from CAPES-PDSE grant 99999.014505/2013-04 during his Ph. D. and also CAPES-PNPD post-doc grant 88882.315311/2019-01.

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Authors and Affiliations

  1. Departamento de Matemática, UFES, Av. Fernando Ferrari 514, Campus de Goiabeiras, Vitória, ES, 29075-910, Brazil

    Victor Goulart

  2. Departamento de Matemática, PUC-Rio, R. Marquês de S. Vicente 255, Rio de Janeiro, RJ, 22451-900, Brazil

    Nicolau C. Saldanha

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  1. Victor Goulart
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  2. Nicolau C. Saldanha
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Correspondence to Victor Goulart.

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Goulart, V., Saldanha, N.C. Stratification of Spaces of Locally Convex Curves by Itineraries. Bull Braz Math Soc, New Series 54, 47 (2023). https://doi.org/10.1007/s00574-023-00362-8

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