Recent zbMATH articles in MSC 57N35 https://www.zbmath.org/atom/cc/57N35 2021-04-16T16:22:00+00:00 Werkzeug Results on the homotopy type of the spaces of locally convex curves on $\mathbb{S}^3$. https://www.zbmath.org/1456.57024 2021-04-16T16:22:00+00:00 "Alves, Emília" https://www.zbmath.org/authors/?q=ai:alves.emilia "Saldanha, Nicolau C." https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcao A 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)