Surfaces in \(\mathbb{R}^ 3\) with prescribed curvature
type: Item
instance of: scholarly article
title: Surfaces in \(\mathbb{R}^ 3\) with prescribed curvature
zbMATH Open document ID: 0842.53004
DOI: 10.1016/0393-0440(94)00054-9
published in: Journal of Geometry and Physics
publication date: 1996-07-28T00:00:00Z
review text: The author studies surfaces in 3-dimensional Euclidean and 3-dimensional Minkowski space. His main result is a relation of the derivatives of the surface immersion, the Gauß curvature, and the derivative of the Gauß map. To me, this relation seems to be a re-expression of the classical fact that the Gauß curvature is the determinant of the Weingarten map. He then writes down integrability conditions for the surface in terms of the Gauß curvature and the Gauß map. He notes that under certain specializations these integrability conditions are equivalent to the Ernst equation in general relativity which governs stationary, axisymmetric spacetimes whose Killing vector fields commute.
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 83C15
zbMATH DE Number: 839167
zbMATH Keywords: 3-dimensional Minkowski space
zbMATH Keywords: Gauß curvature
zbMATH Keywords: Gauß map
zbMATH Keywords: integrability conditions
zbMATH Keywords: Ernst equation
author: J. Tafel
reviewed by: Marcus Kriele
MaRDI profile type: Publication
full work available at URL: 0393-0440(94)00054-9
OpenAlex ID: W2020506752
cites work: Q4296901
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cites work: The Large Scale Structure of Space-Time
cites work: On some integrable cases in surface theory
cites work: Bianchi surfaces in \(\mathbb{R}^ 3\) and deformation of hyperelliptic curves
cites work: Q3896182
cites work: Integrable Hamiltonian systems and interactions through quadratic constraints
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description: scientific article; zbMATH DE number 839167