Invariant Fermat metrics (Q1596347)

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.