Results on the homotopy type of the spaces of locally convex curves on $S^3$

Abstract

A curve $\gamma :[0,1]\to S^n$ of class $C^k$ ($k⩾n$) is locally convex if the vectors $\gamma (t),{\gamma }^{\prime }(t),\gamma "(t),\cdots ,{\gamma }^{(n)}(t)$ are a positive orthonormal basis to $R^{n+1}$ for all $t\in [0,1]$. Given an integer $n\ge 2$ and $Q\in S{O}_{n+1}$, let $L{S}^n(Q)$ be the set of all locally convex curves $\gamma :[0,1]\to S^n$ with fixed initial and final Frenet frame $F_{\gamma }(0)=I$ and $F_{\gamma }(1)=Q$. Saldanha and Shapiro proved that there are just finitely many non-homeomorphic spaces among $L{S}^n(Q)$ when $Q$ varies in $S{O}_{n+1}$ (in particular, at most $3$ for $n=3$). For any $n⩾2$, the homotopy type of one of these spaces is well-known, but not of the others. For $n=2$, Saldanha determined the homotopy type of the spaces $L{S}^2(Q)$. The purpose of this work is to study the case $n=3$. We will obtain information on the homotopy type of one of these $2$ other spaces, allowing us to conclude that its connected components are not homeomorphic to the connected components of the known space.

arXiv comment36 pages, 10 figures; based on the PhD Thesis arXiv:1608.04635 of the first author, advised by the second author. Minor changes from previous version. To appear in Annales de l'Institut Fourier

References in corpus (3)

• The homotopy and cohomology of spaces of locally convex curves in the sphere -- II
• Homotopies of Curves on the 2-Sphere with Geodesic Curvature in a Prescribed Interval
• Topology of the space of locally convex curves on the 3-sphere

Cited by in corpus (1)

• Combinatorialization of spaces of nondegenerate spherical curves