MINIMAL GEODESICS ON MANIFOLDS WITH DISCONTINUOUS METRICS
Roberto Giambò, Fabio Giannoni · Journal of the London Mathematical Society · 2003
The paper describes some qualitative properties of minimizers on a manifold\sM endowed with a discontinuous metric. The discontinuity occurs on a hypersurface Σ disconnecting\sM. Denote by Ω1 and Ω2 the open subsets of M such that\sM\ Σ=Ω1∪Ω2. Assume that Ω¯1 and Ω¯2 are endowed with metrics 〈 ·, · 〉(1) and 〈·,· 〉(2), respectively, such that Ω¯i (i=1, 2) is convex or concave. The existence of a minimizer of the length functional on curves joining two given points of M is proved. The qualitative properties obtained allows the refraction law in a very general situation to be described.