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Minimal Geodesics on Manifolds with Discontinuous Metrics

Authors:	Giambo, Roberto Giannoni, Fabio

Affiliation:	Dipartimento di Matematica e Informatica, Universitá di Camerino Via Madonna delle Carceri, 62032 Camerino, Italy, roberto.giambo{at}unicam.it Dipartimento di Matematica e Informatica, Universitá di Camerino Via Madonna delle Carceri, 62032 Camerino, Italy, fabio.giannoni{at}unicam.it

Abstract:	The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface ${Sigma}$ disconnecting M. Denote by ${Omega}$ ₁ and ${Omega}$ ₂ the open subsets of M such that M ${Sigma}$ = ${Omega}$ ₁ ${cup}$ ${Omega}$ ₂. Assume that and are endowed with metrics $<$ ·, · $>$ ₍₁₎ and $<$ ·,· $>$ ₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.

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