Evolute surfaces in \(E^ 4\).
type: Item
instance of: scholarly article
title: Evolute surfaces in \(E^ 4\).
zbMATH Open document ID: 1036.53003
DOI: 10.1023/A:1012928405281
author: M. A. Cheshkova
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: Let \(M\) be a smooth 2-surface in Euclidean space \(E^{4}.\) The surface \(\overline{M}\) enveloping the family of normal 2-planes to \(M\) is the evolute of \(M.\) The tangent 2-planes at a point \(p \in M\) and at the corresponding point \(f(p)\in \overline{M}\) are mutually orthogonal, and the vector \(\overrightarrow{p f(p)} = \tau(p),\) \(\tau(p) \in T_{p}^{\bot}M,\) is the normal vector to \(M.\) The following results are proved: Theorem 1. If the evolute \(\overline{M}\) for a smooth 2-surface \(M \subset E^{4}\) exists, then the normal connection of the surface \(M\) is locally flat. Theorem 2. If a 2-surface \(\overline{M} \subset E^{4}\) is the evolute for some 2-surface \(M,\) then the Levi-Civita connection of \(\overline{M}\) is locally flat. Theorem 3. Minimal surfaces have no evolutes.
reviewed by: V. V. Chueshev
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 1927827
zbMATH Keywords: smooth 2-surface in Euclidean space \(E^{4}\)
zbMATH Keywords: evolute
zbMATH Keywords: tangent 2-planes
zbMATH Keywords: normal connection
zbMATH Keywords: Levi-Civita connection
zbMATH Keywords: minimal surfaces
MaRDI profile type: Publication
full work available at URL: a:1012928405281
OpenAlex ID: W2912836503
description: scientific article; zbMATH DE number 1927827