Topological structure of optimal flows on the Girl's surface

TL;DR

This paper classifies and analyzes the topological and dynamical structures of flows, especially Morse-Smale flows, on the Girl's surface, an immersion of the projective plane with specific singularities.

Contribution

It provides a complete classification of optimal Morse-Smale flows on the Girl's surface and describes their topological and structural stability properties.

Findings

Unique cellular structure of the Girl's surface identified.

Three types of flows with one fixed point characterized.

Morse-Smale flows are structurally stable on these surfaces.

Abstract

We investigate the topological structure of flows on the Girl's surfaces which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of the selfintersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a 2-disc, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the 1-skeleton of the surface. Second, we described three structures of flows with one fixed point and no separatrix on the Girl's surface and proved that there are no other such flows. Third, we proved that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we have found all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we have obtained a classification of Morse-Smale flows on the projective plane, that immersed on the Girl's surface. And finally, we described the components of linear connectivity of sets of these flows