Linearizability of d-webs, d ≥ 4, on two-dimensional manifolds

• Maks A. Akivis ^[3] ; Vladislav V. Goldberg ^[1] ; Valentin V. Lychagin ^[2]
  1. [1] New Jersey Institute of Technology

     New Jersey Institute of Technology

     City of Newark, Estados Unidos

     
  2. [2] The Arctic University of Norway

     The Arctic University of Norway

     Noruega

     
  3. [3] Department of Mathematics, Jerusalem College of Technology–Machon Lev, Israel

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 10, Nº. 4, 2005, págs. 431-451
• Idioma: inglés
• DOI: 10.1007/s00029-004-0362-x
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• Resumen

  • We find d − 2 relative differential invariants for a d-web, d ≥ 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f(x, y) and g4(x, y),..., g d (x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g4, and d − 4 PDEs of the second order with respect to f and g4,..., g d . For d = 4, this result confirms Blaschke’s conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give the Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage.