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

Montesinos-Amilibia, Angel

doi:10.4171/RMI/530

Revista Matemática Iberoamericana

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Volume 24, Issue 1, 2008, pp. 71–90

DOI: 10.4171/RMI/530

Published online: 2008-04-30

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

Angel Montesinos-Amilibia^[1]

(1) Universitat de València, Burjassot (Valencia), Spain

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.

Keywords: flat, semiumbilical surfaces in $\mathbb{R}^4$, Bianchi permutability, Bäcklund transformation, evolute

Montesinos-Amilibia Angel: Transformations between surfaces in $\mathbb{R}^4$ with flat normal and/or tangent bundles. Rev. Mat. Iberoam. 24 (2008), 71-90. doi: 10.4171/RMI/530