Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence between such manifold submetries and maximal Laplacian algebras, thus solving the Inverse Invariant Theory problem for this class of partitions. Moreover, a solution to the analogous problem is provided for two smaller classes, namely orthogonal representations of finite groups, and transnormal systems with closed leaves.

中文翻译：

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拉普拉斯代数，流形子矩阵和逆不变理论问题

圆球体的歧管子几何是圆球体的一类分区，它通过奇异群的正交表示来推广奇异黎曼叶面和轨道分解。我们展示了此类流形子矩阵与最大拉普拉斯代数之间的一一对应关系，从而解决了这类分区的逆不变理论问题。此外，为两个较小的类提供了一个类似问题的解决方案，即有限组的正交表示和具有闭合叶的超正规系统。