Monge surfaces and planar geodesic foliations

Abstract

A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and investigate the global properties of these two naturally defined objects. The only compact orientable PGF surfaces are tori; these are globally Monge surfaces, and they have a simple characterization in terms of the directrix. We show how to produce many examples of Monge tori and Klein bottles, as well as tori that do not have a closed directrix.

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Correspondence to David Brander.

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Brander, D., Gravesen, J. Monge surfaces and planar geodesic foliations. J. Geom. 109, 4 (2018). https://doi.org/10.1007/s00022-018-0413-7

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Keywords

  • Monge surface
  • planar geodesic
  • developable surface

Mathematics Subject Classification

  • Primary 53A05
  • Secondary 53C12
  • 53C22