Minimal surfaces of pseudo-Euclidean spaces with geodesic normal sections
type: Item
instance of: scholarly article
title: Minimal surfaces of pseudo-Euclidean spaces with geodesic normal sections
zbMATH Open document ID: 0845.53041
DOI: 10.1016/0926-2245(95)00020-8
author: Young Ho Kim
published in: Differential Geometry and its Applications
publication date: 1996-09-11T00:00:00Z
review text: Let \(M\) be a pseudo-Riemannian surface of index \(r\) which lies in an \(m\)-dimensional pseudo-Euclidean space \(E^m_s\) of index \(s\). We define a normal section of \(M\) at a point \(p\in M\) in a direction \(t\) \((\neq 0)\in T_p M\), the tangent space of \(M\) at \(p\), in the same way as in the Riemannian case: Let \(E(p; t)\) be the affine space generated by \(t\) and the normal space \(T^\perp_p M\) of \(M\) at \(p\). The intersection of \(M\) and \(E(p; t)\) gives rise to a curve in a neighbourhood of \(p\) which is called the normal section of \(M\) at \(p\) in the direction \(t\). An isometric immersion \(i: M\to E^m_s\) is said to be planar geodesic if the image of each geodesic of \(M\) lies in a 2-plane of \(E^m_s\). In this paper, minimal surfaces of pseudo-Euclidean space with geodesic normal sections are studied. We prove that complete connected minimal surfaces in a 5-dimensional pseudo-Euclidean space with geodesic normal sections are totally geodesic or flat quadric.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 897456
zbMATH Keywords: pseudo-Riemannian submanifold
zbMATH Keywords: geodesic normal section
zbMATH Keywords: pseudo-Euclidean space
zbMATH Keywords: minimal surfaces
Wikidata QID: Q115362559
MaRDI profile type: Publication
full work available at URL: 0926-2245(95)00020-8
OpenAlex ID: W2014526493
cites work: Planar geodesic immersions in pseudo-Euclidean space
cites work: Submanifolds with geodesic normal sections
cites work: Reduction of the codimension of an isometric immersion
cites work: Isometric immersions of manifolds with plane geodesics into Euclidean space
cites work: Surfaces in a pseudo-Euclidean space with planar normal sections
cites work: Manifolds with planar geodesics
cites work: Isotropic and Kähler Immersions
cites work: Q3312869
cites work: Planar geodesic immersions
description: scientific article; zbMATH DE number 897456