Hopf-Rinow定理与奇数维球面上磁测地线的Mañé临界值

半李群上磁测地线的Hopf--Rinow定理与Mañé临界值

The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups

本文研究半李群上的右不变磁系统，定义了弱恰当右不变磁场万有覆叠的Mañé临界值，证明能量高于该值时磁流与Finsler测地流一致且Hopf--Rinow定理成立，将相关结果推广到无限维磁测地流情形。

关于两组分Hunter-Saxton系统的Mañé临界值及无限维磁Hopf-Rinow定理

On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem

本文研究了两组分磁Hunter-Saxton系统的Mañé临界值及其无限维磁Hopf-Rinow定理，通过几何框架分析解的破裂行为并扩展解至全局保守弱解。

Healthy vector spaces and spicy Hopf algebras (with applications to the growth rate of geodesic chords and to intermediate volume growth on manifolds of non-finite type)

We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild assumption on the homology of the based loop space. For the special case of geodesic flows, these lower bounds are: (i) For any Riemannian metric on M, any pair of non-conjugate points p,q in M, and every component C of the space of paths from p to q, the number of geodesics in C of length at most T grows at least like e^{\sqrt T}. (ii) The exponent of the vo…

Quantization of the Geodesic flow on Quaternion Projective Spaces

We study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle of this complex structure is holomorphically trivial by explicitly constructing a nowhere vanishing holomorphic global section. Then we construct a Hilbert space consisting of a certain class of holomorphic functions on the punctured cotangent bundle by the method of pairing polar…

A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$

We consider Reeb flows on the tight $3$-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition then there exist infinitely many periodic trajectories distinguished by their linking numbers with the components of the link. This result admits a natural comparison to the Poincaré-Birkhoff theorem on area-preserving annulus homeomorphisms. An analogous theorem holds on $SO(3)$ and applies to geodesic flows of Finsler metrics on $S^2$.