Intuition drawn from quantum mechanics and geometric optics raises the following long-standing question: Can the length spectrum of a closed Riemannian manifold be recovered from its Laplace spectrum? By demonstrating that the Poisson relation is an equality for a generic bi-invariant metric on a compact Lie group, we establish that the length spectrum of a generic bi-invariant metric on a compact Lie group can be recovered from its Laplace spectrum. Furthermore, we exhibit a substantial collection $${\mathscr{G}}$$ G of compact Lie groups—including those that are either tori, simple, simply connected, or products thereof—with the property that for each group $$U \in {\mathscr{G}}$$ U ∈ G the length spectrum of any bi-invariant metric g carried by U is encoded in the Laplace spectrum of g . The preceding statements are special cases of results concerning compact globally symmetric spaces for which the semi-simple part of the universal cover is split-rank. The manifolds considered herein join a short list of families of non-“bumpy” Riemannian manifolds for which the Poisson relation is known to be an equality.

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关于紧李群的泊松关系

从量子力学和几何光学中得出的直觉提出了以下长期存在的问题：能否从拉普拉斯谱中恢复闭合黎曼流形的长度谱？通过证明泊松关系是紧李群上通用双不变度量的等式，我们建立了紧李群上通用双不变度量的长度谱可以从其拉普拉斯谱中恢复。此外，我们展示了大量紧致李群的集合 $${\mathscr{G}}$$ G - 包括那些环面、简单、简单连接或它们的乘积 - 对于每个组 $$U \in {\mathscr{G}}$$ U ∈ G U 携带的任何双不变度量 g 的长度谱被编码在 g 的拉普拉斯谱中。前面的陈述是关于紧凑全局对称空间的结果的特殊情况，其中全域覆盖的半单部分是分裂秩的。此处考虑的流形加入了非“颠簸”黎曼流形家族的简短列表，已知其泊松关系是等式的。