表面上有封闭端的测地线叠层和莫尔斯指数;库普卡-斯梅尔度规

Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds

A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.

曲面上测地流的同宿轨

Homoclinics for geodesic flows of surfaces

该研究在动力系统与微分几何领域取得进展，证明了闭曲面上Kupka-Smale黎曼度量的测地流中，所有双曲闭测地线都具有同宿轨道。