On hypersurfaces in Peterson correspondence (Q1902779)

Let \(M\) and \(\overline M\) be hypersurfaces in a Euclidean space \(E_n\). A diffeomorphism \(f: M\to \overline M\) is said to be a Peterson correspondence if the tangent hyperplanes of \(M\) and \(\overline M\) at corresponding points are parallel. Then \(dfX= FX\) determines a field \(F\) of linear transformations, whose eigendistributions be denoted by \(\Delta_\alpha\), \(\alpha= 1,\dots, r\). It is proved that if the second fundamental form of \(M\) is positive definite, then 1) these \(\Delta_\alpha\) are foliations, 2) the Peterson correspondence, restricted to the leaves of a \(\Delta_\alpha\) with \(\dim \Delta_\alpha\geq 2\), reduces to a homothety, 3) the induced Levi-Civita connections on the corresponding leaves coincide, as well as the connections, determined by the second fundamental forms.