Minimal Geodesics on Manifolds with Discontinuous Metrics | Zendy

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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.

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