接触Anosov流的次黎曼几何
On the subriemannian geometry of contact Anosov flows
摘要：
我们研究次黎曼几何和双曲动力系统之间的某些自然联系。特别是，我们研究动态定义的水平分布分裂成两个可积的，并问：如何是一个subriemannian测地线之间的投影到可积和共享的能量？我们表明，如果水平分布是一种特殊类型的接触Anosov流在三维的强稳定和强不稳定的分布的总和，那么对于任何足够短的亚黎曼测地线连接点的Anosov流的同一轨道上，能量的测地线之间共享\emph{equally}其投影到稳定和不稳定的束。证明依赖于测地线方程和谐振子方程之间的联系，以及由雅可比椭圆函数得到的显式解。利用不同的思想，我们证明了一个类似的结果在高维的常负曲率的封闭黎曼流形的测地流。
英文摘要：
We investigate certain natural connections between subriemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a subriemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough subriemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared \emph{equally} between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.
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