Classification of two-dimensional cyclic recurrent surfaces in the Euclidean space. (Russian) Zbl 1366.53002

Bodrenko, I. I.

Труды Всероссийской молодежной щколы-семинара “Анализ, геометрия и топология”, Барнаул, 2–4 октября 2013. Часть 1. Barnaul: Altaĭskiĭ Gosudarstvennyĭ Universitet, Institut Matematiki Imeni S.L. Soboleva So RAN (ISBN 978-5-91556-142-6/pbk). 197-200 (2013).

A submanifold \(F^n\) in the Euclidian space \(E^{n+p}\) is called cyclic recurrent if there exists a 1-form \(\mu\) on \(F^n\) such that the second fundamental form \(b\) satisfies \[ \left(\bar\nabla _Zb\right)(X,Y)=\mu(Z)b(X,Y)+\mu(X)b(Y,Z)+\mu(Y)b(Z,X) \] for all vector fields \(X,Y,Z\) tangent to \(F^n\). For the case \(n=2\) and \(p\geq3\), two classification theorems are proved: \(F^2\) is an open part of the Clifford torus \(S^1\times S^1\subset E^4\subset E^{2+p}\) or an open part of the Veronese surface \(V^2\subset E^5\subseteq E^{2+p}\), provided \(F^2\) is not contained in a 3-dimensional linear subspace of \(E^{2+p}\). If \(\| b(t,t)\|\) is a multiple of \(\| t\|^2\) for all tangent vectors \(t\), \(F^n\) is called isotropic. For such surfaces \(F^2\subset E^{n+p}\) it is shown that \(F^2\) is an open part of the 2-sphere \(S^2\subset E^3\subseteq E^{2+p}\) or an open part of \(V^2\).
For the entire collection see [Zbl 1298.53003].

Reviewer: Hubert Gollek (Berlin)

MSC:

53A07

Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Keywords:

isotropic surface; Clifford torus; Veronese surface; second fundamental form