Submanifolds with geodesic normal sections
type: Item
instance of: scholarly article
title: Submanifolds with geodesic normal sections
zbMATH Open document ID: 0536.53053
DOI: 10.1007/BF01450703
author: Bang-Yen Chen
author: Paul Verheyen
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163942
review text: Let m be a submanifold of a Euclidean m-space \(E^ m\). For each point \(p\in M\) and a unit tangent vector t at p, the normal space of M at p and t determine an affine subspace E(p,t) which intersects M along a curve in a neighborhood of p, called the normal section at p in the direction t. Since M is Riemannian with respect to the induced metric, there is another canonical curve on M, the geodesic, through p with t as its tangent at p. In this paper, the authors ask and study the following geometric question: ''which are the submanifolds of \(E^ m\) whose normal sections are geodesics?'' The authors give many examples of submanifolds of \(E^ m\) with geodesic normal sections which include all ''standard immersions'' of strongly harmonic spaces. They prove that every submanifold with geodesic normal sections is in fact constant isotropic. Furthermore, two characterization theorems together with several other fundamental results for such submanifolds are obtained. Moreover, they also obtain some classification theorems for surfaces with geodesic normal sections. As a consequence, a new characterization of the Veronese surface in terms of geodesic normal sections is derived. [Remark. Submanifolds with ''simple'' geodesics are also studied very recently by \textit{D. Ferus} and \textit{S. Schirrmacher} [Math. Ann. 260, 57-62 (1982; Zbl 0474.53004)] and \textit{K. Sakamoto\&(Math. Ann. 261, 63-80 (1982; Zbl 0487.53054).] }
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 3851964
zbMATH Keywords: normal sections
zbMATH Keywords: geodesics
zbMATH Keywords: strongly harmonic spaces
zbMATH Keywords: Veronese surface
MaRDI profile type: Publication
cites work: Q4169382
cites work: Q5678153
cites work: Minimal Surfaces with Constant Gauss Curvature
cites work: Q3945159
cites work: Differential geometry of submanifolds with planar normal sections
cites work: Q3951172
cites work: Minimal immersions of spheres into spheres
cites work: Isometric immersions of manifolds with plane geodesics into Euclidean space
cites work: Q3679831
cites work: Manifolds with planar geodesics
cites work: On a certain minimal immersion of a Riemannian manifold into a sphere
cites work: Isotropic and Kähler Immersions
cites work: Planar geodesic immersions
OpenAlex ID: W1997191097
description: scientific article; zbMATH DE number 3851964