Topological structure of optimal flows on the Girl's surface

Abstract

We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. 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 $2$-disk, 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 describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows.

Gradient vector fields of codimension one on the 2-sphere with at most ten singular points

“…In [18,21,22,23,40,1,29,32,45,49,30,31,38,28], the classifications of flows on closed 2-manifolds and [5,19,32,29,25,37,38] on manifolds with the boundary were obtained. Topological properties of Morse-Smale vector fields on 3-manifolds was investigated in [46,48,36,24,49,26,27,11,4,3].…”