On the Gauss map of null scrolls
The authors provide a geometric characterization for a class of non-degenerate ruled surfaces immersed in the 3-dimensional Minkowski space \(\mathbb{R}^3_1\), the so-called null scrolls, in terms of their Gauss maps. After reviewing the terminology and fundamental properties for a null scross \(M\) in \(\mathbb{R}^3_1\) and calculating the Laplacian \(\Delta\), the Gaussian curvature \(K\) and the mean curvature \(H\), they prove the following main result of the article: Theorem. Let \(M\) be a null scroll along a framed null curve with proper frame field. Then, the Gauss map \(\xi\) of \(M\) satisfies \(\Delta\xi= A\xi\), with \(A\in \text{gl}(3, \mathbb{R})\), if and only if the mean curvarature \(H\) is constant. In this case, \(A\) is always equal to a scalar matrix. This result is also used to state a corollary concerning the characterization of \(B\)-scrolls, previously considered in [\textit{L. K. Graves}, Trans. Am. Math. Soc. 252, 367-392 (1979; Zbl 0415.53041)].
scientific article; zbMATH DE number 1383412