Integrability of geodesic flows and isospectrality of Riemannian manifolds
Integrability of geodesic flows and isospectrality of Riemannian manifolds
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测地流的可积性和黎曼流形的等谱性
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
D. Schueth
中科院分区：
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作者：
D. Schueth
我们构造了一对紧凑的八维两步黎曼尼尔曼流形 M 和 M'，它们对于函数上的拉普拉斯算子是等谱的，并且使得 M 具有刘维尔意义上的完全可积测地流，而 M' 则没有。此外，对于这两个流形，我们分析了由具有通用速度场的闭合测地线的两个最大连续族给出的单位切丛的子流形的结构。这些子流形的结构反映了上述（不可）可积性质。另一方面，它们的维数大于 M 中的拉格朗日环面的维数，表明存在简并性，这可能解释了波不变量不区分可积系统和不可积系统的事实。最后，我们证明对于M，由闭测地线组成的不变八维环面在单位切丛中是稠密的，并且M和M'都满足所谓的净交假说。
We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds M and M′ which are isospectral for the Laplace operator on functions and such that M has completely integrable geodesic flow in the sense of Liouville, while M′ has not. Moreover, for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by two maximal continuous families of closed geodesics with generic velocity fields. The structure of these submanifolds turns out to reflect the above (non)integrability properties. On the other hand, their dimension is larger than that of the Lagrangian tori in M, indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here. Finally, we show that for M, the invariant eight-dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle, and that both M and M′ satisfy the so-called Clean Intersection Hypothesis.