On the Gauss map of null scrolls
type: Item
instance of: scholarly article
title: On the Gauss map of null scrolls
zbMATH Open document ID: 0952.53011
author: Soon Meen Choi
author: U-Hang Ki
author: Young Jin Suh
published in: Tsukuba Journal of Mathematics
publication date: 2001-01-14T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 53A25
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1383412
zbMATH Keywords: third curvature
zbMATH Keywords: ruled surface
zbMATH Keywords: Gauss map
zbMATH Keywords: Minkowski space
zbMATH Keywords: null scroll
zbMATH Keywords: framed null curve
zbMATH Keywords: \(B\)-scrolls
reviewed by: Salvador D. Gigena
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163485
OpenAlex ID: W150536525
DOI: 10.21099/TKBJM/1496163485
description: scientific article; zbMATH DE number 1383412