Abstract
Let \(M_g^2\) be a closed orientable surface of genus \(g \geqslant 2\), endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering \(\Delta\), there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set \(U_g\) on the absolute having the cardinality of the continuum and such that if an arbitrary flow on \(M_g^2\) has a semitrajectory whose covering has asymptotic direction defined by a point from \(U_g\), then this flow is not analytical and has infinitely many stationary points.
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Aranson, S.K., Zhuzhoma, E.V. Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces. Mathematical Notes 68, 695–703 (2000). https://doi.org/10.1023/A:1026696213559
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DOI: https://doi.org/10.1023/A:1026696213559