Magnetic rigidity of horocycle flows
Magnetic rigidity of horocycle flows
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四环流的磁刚度
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发表时间：
2004
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通讯作者：
G. Paternain
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作者：
G. Paternain
令 M 为具有黎曼度量 g 的闭定向曲面，并令 O 为 2-形式。我们证明，当且仅当 g 具有恒定的高斯曲率、O 是 g 的面积形式的常数倍并且磁流是四周流时，磁流对 (g,O) 的渐近马斯洛夫指数为零，刘维尔作用为零。
星环流的这种表征意味着，如果一对 (g,O) 的磁流与双曲度量 ? 的星环流是 C1 共轭，则存在常数 a > 0，使得 ag 和 ?是等轴测的，a-1O 是 g 的面积形式（直到符号为止）。它还意味着，如果磁流是鬃毛临界且唯一遍历的，那么它一定是星环流。
作为副产品，我们证明了在满足特定技术条件的任意维度的闭合流形上的弱精确磁场的情况下，几乎所有能级都存在闭合磁测地线
Let M be a closed oriented surface endowed with a Riemannian metric g and let O be a 2-form. We show that the magnetic flow of the pair (g,O) has zero asymptotic Maslov index and zero Liouville action if and only if g has constant Gaussian curvature, O is a constant multiple of the area form of g and the magnetic flow is a horocycle flow.
This characterization of horocycle flows implies that if the magnetic flow of a pair (g,O) is C1-conjugate to the horocycle flow of a hyperbolic metric ?, there exists a constant a > 0 such that ag and ? are isometric and a-1O is, up to a sign, the area form of g. It also implies that if a magnetic flow is Mane-critical and uniquely ergodic it must be the horocycle flow.
As a byproduct we show the existence of closed magnetic geodesics for almost all energy levels in the case of weakly exact magnetic fields on closed manifolds of arbitrary dimension satisfying a certain technical condition