The bitangent Bianchi transformation of a submanifold \(H^n\) of constant negative curvature of the Euclidean space \(\mathbb R^{2n}\) (Q881244)

Let \(H^ n\) be an \(n\)-dimensional regular pseudospherical submanifold (a submanifold of constant sectional curvature \(-1\)) in the Euclidean space \(\mathbb{R}^ {2n}\), parametrized by the orispheric coordinates \(y_ 1,\dots,y_ n\) such that the induced metric is \(ds^ 2=e^ {2y_ n}(dy_ 1^ 2+\dots +dy_ {n-1}^ 2)+dy_ n^ 2\). The Bianchi bitangent transformation of \(H^ n\) is the submanifold \(\overline{H}{}^ n\) of \({\mathbb R}^ {2n}\) defined by \(\overline{r}(y_ 1,\dots,y_ n)=r(y_ 1,\dots ,y_ n)-{\frac{\partial r}{\partial y_ n}}\), under the condition that \(\frac{\partial r}{\partial y_ n}\) is also tangent to \(\overline{H}{}^ n\) in \(\overline r\). First, the author shows that there exists a field of common normals for \(H^ n\) and \(\overline{H}{}^ n\) in the corresponding points. Next he obtains some conditions under which the submanifold \(\overline{H}{}^ n\) has constant sectional curvature \(-1\).