Recent zbMATH articles in MSC 57N35https://www.zbmath.org/atom/cc/57N352021-04-16T16:22:00+00:00WerkzeugResults on the homotopy type of the spaces of locally convex curves on $\mathbb{S}^3$.https://www.zbmath.org/1456.570242021-04-16T16:22:00+00:00"Alves, EmÃlia"https://www.zbmath.org/authors/?q=ai:alves.emilia"Saldanha, Nicolau C."https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcaoA smooth curve \(\gamma:[0,1]\to \mathbb{S}^3\) in \(4\)-dimensional Euclidean space \(\mathbb{R}^4\)
with image on the sphere \(\mathbb{S}^3\), is said to be locally convex, if the set of vectors
\({\gamma}(t),{\gamma}'(t), {\gamma}''(t),{\gamma}'''(t)\) is a positive basis in \(\mathbb{R}^4\)
for all \(t\in[0,1]\). By the Gram-Schmidt procedure, we can turn this basis into an orthonormal basis and thereby associate a Frenet frame curve \(\mathcal{F}_{\gamma}: [0,1]\to SO_4\) in the special orthogonal
group \(SO_4\) to a locally convex curve.
For any matrix \(Q\in SO_4\), let \(\mathcal{L}\mathbb{S}^3(Q)\) denote the space of all locally
convex curves \(\gamma:[0,1]\to \mathbb{S}^3\) where \(\mathcal{F}_{\gamma}(0)=I\) (the identity matrix)
and \(\mathcal{F}_{\gamma}(1)=Q\). It was proved by [\textit{N. C. Saldanha} and \textit{B. Shapiro}, J. Singul. 4, 1--22 (2012; Zbl 1292.58002)] that there are at most 3 different homeomorphism types among the path components of \(\mathcal{L}\mathbb{S}^3(Q)\). But otherwise very little seems to be known about these components and their generalizations to higher dimensional spheres \(\mathbb{S}^n\).
In the present paper the authors prove several interesting theorems on the homotopy and homology of these path components, in particular for the case \(Q=-I\). The results are technical and cannot be given in detail.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)