Isometric immersions of \({{\mathbb R}^2}\) into \({{\mathbb R}^4}\) and perturbation of Hopf tori. (English) Zbl 1209.53009
The authors construct a large family of new flat tori in \(\mathbb R^4\), which are with flat normal bundle and regular Gauss map. These tori are not contained in any affine 3-sphere and cannot be expressed as the product of two curves.
To do that they first apply the representation formula of Sect. 3 of this paper to obtain a procedure of unfolding a Hopf torus in \(S^3\), so that one gets a flat surface in \(\mathbb R^4\) (possibly with singular points) with flat normal bundle that does not lie in any affine 3-sphere in \(\mathbb R^4\). Then they show that for a certain family of Hopf tori this procedure generates flat tori in \(\mathbb R^4\).
To do that they first apply the representation formula of Sect. 3 of this paper to obtain a procedure of unfolding a Hopf torus in \(S^3\), so that one gets a flat surface in \(\mathbb R^4\) (possibly with singular points) with flat normal bundle that does not lie in any affine 3-sphere in \(\mathbb R^4\). Then they show that for a certain family of Hopf tori this procedure generates flat tori in \(\mathbb R^4\).
Reviewer: Stefka Hineva (Sofia)
MSC:
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
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