Parametric versions of Hilbert’s fourth problem

Autor:	V. K. Oganian, R. V. Ambartzumian
Rok vydání:	1998
Předmět:	Pure mathematics Geodesic Differential equation General Mathematics Mathematical analysis Hilbert's fourth problem Orientation (vector space) Riemann hypothesis symbols.namesake Harmonic function Euclidean geometry symbols Mathematics::Metric Geometry Point (geometry) Mathematics
Zdroj:	Israel Journal of Mathematics. 103:41-65
ISSN:	1565-8511 0021-2172
DOI:	10.1007/bf02762267
Popis:	LetH be the class of sufficiently smooth metrics defined on the Euclidean plane for which the geodesics are the usual Euclidean liens. The general problem is to describe all metrics fromH which at each point possess the length indicatrix from a prescribed parametric class of convex figures. As a tool, a differential equation is proposed derived from the “triangular deficit principle” formulated in an earlier paper of R. V. Ambartzumian. The paper demonstrates that for the case where the length indicatrix is segmental this equation leads to a complete solution. Also, there is a partial result stating that in the case of Riemann metrics the orientation of the ellipsi should necessarily be a harmonic function.
Databáze:	OpenAIRE
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