Reflections in submanifolds
type: Item
instance of: scholarly article
title: Reflections in submanifolds
zbMATH Open document ID: 0656.53055
DOI: 10.1007/BF00147801
author: Philippe Tondeur
author: Lieven Vanhecke
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: The main topic of this paper is a study of local geodesic reflections of a Riemannian manifold (M,g) with respect to its submanifold \(P\subset M\). For instance, it is proved that if the reflection is volume-preserving, then the ``mirror'' P is necessarily a minimal submanifold. This result holds in both directions if (M,g) is a space of constant curvature and \(P\subset M\) is two-dimensional. Further, similar characterizations are given for those reflections of (M,g) in P which preserve the mean curvature of all tubes about P. The most interesting problem, namely to determine all minimal submanifolds just as mirrors of reflections with simple geometrical properties remains still open.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4071849
zbMATH Keywords: geodesic reflections
zbMATH Keywords: volume-preserving
zbMATH Keywords: minimal submanifold
zbMATH Keywords: mean curvature
zbMATH Keywords: mirrors of reflections
reviewed by: Oldřich Kowalski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147801
OpenAlex ID: W2094536652
description: scientific article; zbMATH DE number 4071849