Manifold submetries from compact homogeneous spaces

TL;DR

This paper demonstrates that manifold submetries on compact normal homogeneous spaces are algebraic in nature and establishes a correspondence between algebraic function algebras and submetries, revealing their intrinsic algebraic structure.

Contribution

It introduces the algebraic characterization of manifold submetries on compact normal homogeneous spaces and links them to algebraic functions preserved by the Laplace--Beltrami operator.

Findings

Singular Riemannian foliations on these spaces are algebraic.

A one-to-one correspondence exists between algebraic function algebras and manifold submetries.

The mean curvature vector field of fibers is basic and related to the base.

Abstract

We show that singular Riemannian foliations, or, more generally, manifold submetries, defined on a compact normal homogeneous space, have algebraic nature. Moreover, in this case there exists a one-to-one correspondence between algebras of algebraic functions preserved by the Laplace--Beltrami operator, and manifold submetries. A key intermediate result is that, for any manifold submetry on a compact normal homogeneous space, the vector field given by the mean curvature of the fibers is basic, in the sense that it is related to a vector field in the base.