Conformal Combescure correspondence of \(m\)-surfaces in the Euclidean space \(E^n\) (Q2760844)

A mapping between two \(m\)-dimensional surfaces in Euclidean spaces is called a Combescure correspondence if it maps tangent planes into tangent planes and there are normal fields to these surfaces such that the mapping preserves the curvature lines with respect to these fields. It is proved that if \(m \geq 3\), the mapping is conformal and sectional curvatures of the surfaces do not vanish then such a correspondence is either a homothety or each of the surfaces fibers into \((m-1)\)-dimensional surfaces such that these surfaces contain \((m-1)\) curvature lines with respect to the normal fields and the restriction of the mapping on every fiber is a homothety.