Transformations between surfaces in $\mathbb{R}^4$ with flat normal and/or tangent bundles

Abstract

We exhibit several transformations of surfaces $M$ in $\mathbb{R}^4$: a transformation of flat surfaces that gives surfaces with flat normal bundle (semiumbilical surfaces); and its inverse that from a semiumbilical surface obtains a flat surface; then a one-parameter family of transformations $f$ on flat semiumbilical immersed surfaces (FSIS), such that $df(T_pM)$ is totally orthogonal to $T_pM,$ and that give FSIS. This family satisfies a Bianchi type of permutability property.