Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Results on the homotopy type of the spaces of locally convex curves on
  $S^3$   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 to $R^{n+1}$ for all $t \in
[0,1]$. Given an integer $n \geq 2$ and $Q \in SO_{n+1}$, let $LS^n(Q)$ be the
set of all locally convex curves $\gamma: [0,1] \rightarrow S^n$ 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 $LS^n(Q)$ when $Q$ varies in $SO_{n+1}$ (in particular, at most $3$ for
$n=3$). For any $n \geqslant 2$, 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 $LS^2(Q)$. 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.

26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga