Monge surfaces and planar geodesic foliations

David Brander*, Jens Gravesen

*Corresponding author for this work

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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
Original languageEnglish
JournalJournal of Geometry
Volume109
Issue number4
Number of pages14
ISSN0047-2468
DOIs
Publication statusPublished - 2018

Keywords

  • Monge surface
  • Planar geodesic
  • Developable surface

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