Riemannian metrics obtained from the boundary single-time ODE

Abstract

Our theory of determining a metric by single-time ODE system energy is an advance to those developed by Sharahfutdinov (1994). Section 1 refines the theory of potential curves determined by an ODE system and a Riemannian metric. Section 2 defines the boundary energy of a potential curve and proves that the problem of determining a metric from a single-time flow boundary energy cannot have a unique solution. It continues by linearizing the above-mentioned problem and it is defined the notion of single-ray transform, Udriste, Pitea and Mihaila (2006) and Pitea (2008). All the results in this section are original.