Isotropic submanifolds with pointwise planar normal sections
type: Item
instance of: scholarly article
title: Isotropic submanifolds with pointwise planar normal sections
zbMATH Open document ID: 0591.53049
DOI: 10.1007/BF01227835
author: Guo-Qiang Wang
author: C. S. Houh
published in: Journal of Geometry
publication date: 1986-01-01T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional submanifold of a Euclidean \(m\)-space \(E^ m\). For a unit tangent vector t at a point \(p\) in \(M\), the vector \(t\) and the normal space of \(M\) at \(p\) determine an \((m-n+1)\)-dimensional vector space \(E(p,t)\) in \(E^ m\). The intersection of \(M\) and \(E(p,t)\) gives rise to a curve \(\sigma(s)\) in a neighborhood of \(p\), called the normal section at \(p\) in the direction \(t\). \(A\) submanifold \(M\) is said to have pointwise planar normal sections if each normal section \(\sigma\) at \(p\) satisfies \(\sigma'\wedge \sigma'' \wedge \sigma'''=0\) at \(p\) for each \(p\) in \(M\). The authors classify isotropic submanifolds in \(E^ m\) with pointwise planar normal sections and show that such submanifolds are either open portions of linear subspaces or open portions of a compact rank one symmetric space imbedded in \(E^ m\) by its first standard imbedding.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 3949239
zbMATH Keywords: normal section
zbMATH Keywords: planar normal sections
zbMATH Keywords: isotropic submanifolds
zbMATH Keywords: symmetric space
reviewed by: Bang-Yen Chen
MaRDI profile type: Publication
cites work: Differential geometry of submanifolds with planar normal sections
cites work: Submanifolds with geodesic normal sections
cites work: Manifolds with planar geodesics
cites work: Isotropic and Kähler Immersions
full work available at URL: https://doi.org/10.1007/bf01227835
OpenAlex ID: W2027517223
description: scientific article; zbMATH DE number 3949239