Manifold submetries from compact homogeneous spaces

2025/12/18 by Lin, Samuel, Mendes, Ricardo A. E., Radeschi, Marco
#13A50 (Secondary) #53C12 (Primary) #53C20 #53C21 #57S15 #58J50 #Analysis of PDEs (math.AP) #Commutative Algebra (math.AC) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2512.16606

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.

Citations

• Smoothness of submetries
• A Weyl's Law for Singular Riemannian Foliations with Applications to Invariant Theory
• Laplacian algebras, manifold submetries and the Inverse Invariant Theory\n Problem
• Totally Geodesic Riemannian Foliations on Compact Lie Groups
• Algebraic nature of singular Riemannian foliations in spheres
• Mean curvature flow of singular Riemannian foliations
• Rigidity theorems for submetries in positive curvature
• Isometries between leaf spaces
• Riemannian submersions from simple, compact Lie groups
• Riemannian manifolds with maximal eigenfunction growth
• Isoparametric submanifolds and a Chevalley-type restriction theorem
• Invariants and fundamental functions

Related

• Smooth foliations on homogeneous compact Kähler manifolds
• Canonical extension of submanifolds and foliations in noncompact\n symmetric spaces
• Orbits of families of vector fields on subcartesian spaces
• On the geometry of the orbits of Killing vector fields
• Leaf Space Isometries of Singular Riemannian Foliations and Their Spectral Properties