Manifold submetries from compact homogeneous spaces

TL;DR

The paper extends the algebraicity of manifold submetries from spheres to compact normal homogeneous spaces by developing a robust link between spectral data and geometric foliations. It defines Laplacian algebras as Laplace–Beltrami-preserved subalgebras of eigenfunctions and proves a 1-1 correspondence with manifold submetries that have basic mean curvature, via a construction that uses finite separating sets and a Hilbert-space submersion. A central technical advance is a formula for mean curvature derived from focal data, showing that on compact normal homogeneous spaces the mean curvature of fibers is basic, which enables the algebraic description. These results unify and generalize prior spherical cases, with implications for Riemannian submersions from Lie groups with bi-invariant metrics and connections to invariant theory.

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.

Theorems & Definitions (55)

• Theorem 1
• Theorem 2
• Corollary 3
• Lemma 5
• proof
• Definition 6
• Definition 7
• Lemma 8
• proof
• Lemma 9