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

Abstract: A smooth curve $\gamma: [0,1] \to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) = \gamma'(1) = e_2$ has three connected components. Our first aim is to describe the homotopy type of these spaces. One of the connected components is known to be contractible. The two other connected components are homotopically equivalent to $(\Omega S^3) \vee S^2 \vee S^6 \vee S^{10} \vee \cdots$ and $(\Omega S^3) \vee S^4 \vee S^8 \vee S^{12} \vee \cdots$, respectively: we describe the equivalence.

More generally, a smooth curve $\gamma: [0,1] \to S^n$ is locally convex if \[ \det(\gamma(t), \gamma'(t), \ldots, \gamma^{(n)}(t)) > 0 \] for all $t$. A motivation for considering this space comes from linear ordinary differential equations. Again, we would like to know the homotopy type of the space of locally convex curves with prescribed initial and final jets. We present several partial results.

Includes joint work with E. Alves, V. Goulart, B. Shapiro, M. Shapiro, C. Zhou and P. Zuhlke

differential geometry

Audience: researchers in the topic

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Pangolin seminar

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