# Topology of the space of locally convex curves on the 3-sphere

@article{Alves2016TopologyOT, title={Topology of the space of locally convex curves on the 3-sphere}, author={Em'ilia Alves}, journal={arXiv: Geometric Topology}, year={2016} }

A (positive) locally convex curve in the 2-sphere is a curve with positive geodesic curvature (i.e., which always turns left). In the 3-sphere, it is a curve with positive torsion. In this work we discussed the topology of spaces of such curves with prescribed initial and final jets. The case of the 2-sphere is understood (Saldanha-2013); the case of n=3 is not yet thoroughly clarified. In order to obtain partial information about the homotopy type of such spaces in the case n=3, we represented…

## 3 Citations

### On the Homology of the Space of Curves Immersed in The Sphere with Curvature Constrained to a Prescribed Interval

- Mathematics
- 2018

While the topology of the space of all smooth immersed curves on the $2$-sphere $\mathbb{S}^2$ that start and end at given points in given directions is well known, it is an open problem to…

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

- Mathematics
- 2017

A curve $\gamma: [0,1] \rightarrow S^n$ of class $C^k$ ($k \geqslant n$) is locally convex if the vectors $\gamma(t), \gamma'(t), \gamma"(t), \cdots, \gamma^{(n)}(t)$ are a positive orthonormal basis…

### Characterization of some convex curves on the 3-sphere

- Mathematics
- 2020

In this paper we provide a characterization for a class of convex curves on the 3-sphere. More precisely, using a theorem that decomposes a locally convex curve on the 3-sphere as a pair of curves on…

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