Topological structure of simple Hamiltonian flows on the projective plane and the Klein bottle
DOI:
https://doi.org/10.17721/1812-5409.2025/2.8Keywords:
topological equivalence, Reeb graph, classification, non-orientable surfaceAbstract
For every non-orientable surface, there exists an oriented double cover. We consider flows whose lift to the double cover are Hamiltonian flows with a Hamiltonian being a Morse function. By analogy with oriented manifolds, a flow is simple if there are no saddle connections between different saddles within it.
To describe the topological properties of functions on two-dimensional manifolds, an invariant known as the Reeb graph is used. In the case of simple Morse functions on closed oriented two-dimensional manifolds, this is a complete topological invariant for stratified equivalence and topological equivalence of Hamiltonian flows. However, in our case, on a non-orientable manifold, there is no function that generates a Hamiltonian flow. Therefore, the concept of the Reeb graph requires redefinition.
To investigate the topological properties of simple Hamiltonian flows on non-orientable surfaces, we describe the structure of all possible atoms, which are analogous to the Bolsinov-Fomenko atoms for Morse functions and Hamiltonian flows on oriented surfaces. In particular, there are five types of such atoms on non-orientable surfaces. We construct a complete topological invariant, which is the Reeb graph with marked vertices and half-edges incident to the vertices, describe its properties, and prove a criterion for the topological equivalence of flows. For the projective plane and the Klein bottle, we prove theorems about the existence of flows with a given Reeb graph. For cyclic Reeb graphs on the Klein bottle, we describe an algorithm for their enumeration. For all other types of Reeb graphs on the projective plane and the Klein bottle, we obtain recursive formulas for determining their numbers. The results provide calculations of the number of topological structures of flows with no more than eight saddles. The obtained results open up the possibility of studying the structures of Hamiltonian flows of greater complexity on compact non-orientable surfaces.
Pages of the article in the issue: 60 - 72
Language of the article: English
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