manuscripta mathematica

, Volume 160, Issue 3–4, pp 315–325 | Cite as

Flat approximations of hypersurfaces along curves

  • Irina Markina
  • Matteo RaffaelliEmail author


Given a smooth curve \(\gamma \) in some m-dimensional surface M in \(\mathbb {R}^{m+1}\), we study existence and uniqueness of a flat surface H having the same field of normal vectors as M along \(\gamma \), which we call a flat approximation of M along \(\gamma \). In particular, the well-known characterisation of flat surfaces as torses (ruled surfaces with tangent plane stable along the rulings) allows us to give an explicit parametric construction of such approximation.

Mathematics Subject Classification

53A07 (primary) 53B20 (secondary) 


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We thank the referee for the careful reading and valuable suggestions. We also thank J. Bohr and S. Markvorsen for useful comments on earlier versions of the manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.DTU ComputeTechnical University of DenmarkKongens LyngbyDenmark

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