Straight Lines in Three-Dimensional Space and the Ultrahyperbolic Equation

Abstract

The straight lines three-dimensional vector space realize the shortest distance for various metrics. This property is reformulated in terms of the inverse problem of the calculus of variations and closely related to the ultrahyperbolic equation with four independent variables. The interrelation is useful in both directions. For instans, polynomial solutions of the ultrahyperbolic equation provide all polynomial metrics with extremals the straight lines and conversely, a slight generalization of the Hilbert metrics leads to rather nontrivial (multi-valued of focusing) solutions of the ultrahyperbolic equation. In general, the article clarifies some well-known achievements concerning the 4th Hilbert Problem.