Premium
Minimal Geodesics on Manifolds with Discontinuous Metrics
Author(s) -
Giambò Roberto,
Giani Fabio
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003952
Subject(s) - hypersurface , geodesic , mathematics , manifold (fluid mechanics) , regular polygon , discontinuity (linguistics) , pure mathematics , metric (unit) , mathematical analysis , combinatorics , geometry , mechanical engineering , operations management , engineering , economics
The paper describes some qualitative properties of minimizers on a manifold\s M endowed with a discontinuous metric. The discontinuity occurs on a hypersurface Σ disconnecting\s M . Denote by Ω 1 and Ω 2 the open subsets of M such that\s M \ Σ=Ω 1 ∪Ω 2 . Assume thatΩ ¯ 1andΩ ¯ 2are endowed with metrics 〈 ·, · 〉 (1) and 〈·,· 〉 (2) , respectively, such thatΩ ¯ 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.