# Laplacian algebras, manifold submetries and the Inverse Invariant Theory Problem

## TL;DR

This paper establishes a correspondence between manifold submetries of the sphere and maximal Laplacian algebras, solving the Inverse Invariant Theory problem for these partitions and related classes.

## Contribution

It introduces a one-to-one correspondence between manifold submetries and maximal Laplacian algebras, advancing the understanding of invariant theory in geometric contexts.

## Key findings

- Established correspondence between manifold submetries and Laplacian algebras
- Solved the Inverse Invariant Theory problem for these partitions
- Extended solutions to orthogonal representations of finite groups and transnormal systems

## Abstract

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.