Research Article

Flows with constant slope on the Klein bottle

DOI: 10.2989/16073606.2024.2402926
Author(s): Luís Barreira Universidade de Lisboa, Portugal, Jaume Llibre Universitat Autònoma de Barcelona, Spain, Claudia Valls Instituto Superior Técnico, Universidade de Lisboa, Portugal,

Abstract

We consider smooth flows with constant slope on the 2-dimensional (flat) torus and on the (flat) Klein bottle. It is well known that for a flow with constant slope on the torus either all orbits are periodic or all orbits are dense. We obtain a similar classification for the orbits of a flow with constant slope on the Klein bottle: again either all orbits are periodic, although some periods change, or all orbits are dense. This requires considering flows on the unit normal bundle of the Klein bottle.

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