Classification of two-dimensional cyclic recurrent surfaces in the Euclidean space
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 NEWLINE\[NEWLINE\left(\bar\nabla _Zb\right)(X,Y)=\mu(Z)b(X,Y)+\mu(X)b(Y,Z)+\mu(Y)b(Z,X) NEWLINE\]NEWLINE 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 \(V^2\).NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].
scientific article; zbMATH DE number 6568565