Monge surfaces and planar geodesic foliations

TL;DR

This paper studies Monge surfaces, characterized by planar geodesic foliations, revealing their global properties, classifications, and providing methods to construct examples like tori and Klein bottles.

Contribution

It introduces PGF surfaces as a broader class, characterizes compact orientable PGF surfaces as tori, and offers constructions for various Monge surfaces.

Findings

Only compact orientable PGF surfaces are tori

All such tori are globally Monge surfaces with a simple directrix characterization

Many examples of Monge tori and Klein bottles are constructed

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.