Invariant Fermat metrics
type: Item
instance of: scholarly article
title: Invariant Fermat metrics
zbMATH Open document ID: 0990.53024
published in: Expositiones Mathematicae
publication date: 2002-08-12T00:00:00Z
review text: The author considers a Fermat metric defined as follows: take a vector field \(V\) over an \(n\)-dimensional real manifold \(M\) and a function \(f\) with \(df(V) > 0\) and look for a Riemannian metric \(g\) on an open connected domain \(D \in M\) such that the integral curves of \(V\) are geodesics of \(g\), and that \(V\) is the gradient of \(f\) realtive to \(g\). When such a metric \(g\) exists, it is called a Fermat metric for \((V,f)\). The author points out that: in this case, ``\(f\) must solve on \(D\) the \dots eikonal equation''. This nonlinear equation arises in physics studies, like big wave fronts studies [\textit{E. T. Newman} and \textit{A. Perez}, J. Math. Phys. 40, No. 2, 1093-1102 (1999; Zbl 0947.83006)] and flat space-times [\textit{S. Frittelli, E. T. Newman} and \textit{G. Silva-Ortigoza}, J. Math. Phys. 40, 1041-1056 (1999; Zbl 0946.83012)]. His main result is the following: \((V,f)\) such that \(df(V) > 0\) admits a Fermat metric on a domain \(D\) if and only if it satisfies the pre-eikonal condition, i.e., \(d[df(V)]= 0\). The author provides a geometrical interpretation of the invariance of a Fermat metric and analyses invariant metrics for multiple data.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 78A05
zbMATH DE Number: 1562969
zbMATH Keywords: Fermat metric
zbMATH Keywords: eikonal equation
zbMATH Keywords: geometric optics
author: Philippe P. Delanoë
reviewed by: L. Del Riego
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1562969