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Lipschitz Geometry of Real Semialgebraic Surfaces

In: Handbook of Geometry and Topology of Singularities IV

Author

Listed:
  • Lev Birbrair

    (Universidade Federal do Ceará (UFC), Departamento de Matemática
    Jagiellonian University, Institute of Mathematics)

  • Andrei Gabrielov

    (Purdue University, Department of Mathematics)

Abstract

We present here basic results in Lipschitz Geometry of semialgebraic surfaceSemialgebraic surface germs. Although bi-Lipschitz classification problem of surface germs with respect to the inner metric was solved long ago, classification with respect to the outer metric remains an open problem. We review recent results related to the outer and ambient bi-Lipschitz classification of surface germs. In particular, we explain why the outer bi-Lipschitz classification is much harder than the inner classification, and why the ambient Lipschitz GeometryLipschitzGeometry of surface germs is very different from their outer Lipschitz Geometry. In particular, we show that the ambient Lipschitz Geometry of surface germs includes all of the Knot Theory.

Suggested Citation

  • Lev Birbrair & Andrei Gabrielov, 2023. "Lipschitz Geometry of Real Semialgebraic Surfaces," Springer Books, in: José Luis Cisneros-Molina & Lê Dũng Tráng & José Seade (ed.), Handbook of Geometry and Topology of Singularities IV, chapter 0, pages 449-462, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-31925-9_8
    DOI: 10.1007/978-3-031-31925-9_8
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