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Girl’s Surface - mklabgr - 09-15-2026
The paper “Topological structure of optimal flows on the Girl’s surface” by Maria Loseva and Alexandr Prishlyak studies dynamical systems on the Girl’s surface, one of the two known one-triple-point immersions of the real projective plane in three-dimensional space, the other being Boy’s surface.
The authors first describe the surface using a cellular decomposition, representing the projective plane as a disk whose opposite boundary points are identified. They then investigate flows—motion generated by vector fields—on this singular surface. In particular, they classify the possible flows having a single fixed point and no separatrices, showing that there are exactly three distinct structures of this type.
The central part of the paper concerns Morse–Smale flows, which are especially well-behaved dynamical systems whose trajectories and fixed points have a stable topological structure. The authors prove that Morse–Smale flows are precisely the structurally stable flows on both Boy’s and Girl’s surfaces.
They then determine all possible optimal Morse–Smale flows on the Girl’s surface, classify the corresponding flows on the immersed projective plane, and describe their isotopy (continuous deformation) classes. The resulting classification provides a systematic way of encoding these flows through separatrix diagrams and topological graphs, connecting the geometry of the unusual Girl’s surface with broader ideas from dynamical systems, topology, and Morse theory.
ARTICLE [PDF]