Smooth right quasigroup structures on 1-manifolds (Q2902442)

Let \(G\) be the group \(PSL_{2}(\mathbb R)\), let \(H\) be a \(2\)-dimensional subgroup of \(G\) and let \(\sigma: G/H\rightarrow G\) be a smooth section such that \(\sigma(G/H)\) generates \(G\) and \(\sigma(H) = 1 \in G\). Any smooth loop \(L\) homeomorphic to the \(1\)-sphere and having \(G\) as the group generated by the right translations of \(L\) can be realized on the space \(G/H\) of right cosets of \(H\) in \(G\) by the multiplication \(Hx\circ Hy = Hx\sigma (Hy)\). The authors give nice smooth sections \(\sigma\) related to polynomials functions.