We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative $C^*$-algebra of continuous functions on the Riemannian isometry group.

中文翻译：

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紧度量空间量子等距群的存在性和刚性

我们证明了新的度量空间类别的量子等距群的存在：（i）紧凑连接黎曼流形（可能具有边界）的测地线度量（ii）允许均匀分布的概率度量的度量空间。在前一种情况下，第二作者最近的结果也得出量子等距群是经典的，即黎曼等距群上连续函数的交换$C^*$-代数。