Reduction of Variables for Minimal Submanifolds
Richard S. Palais, Chuu-Lian Terng · Proceedings of the American Mathematical Society · 1986
If $G$ is a compact Lie group and $M$ a Riemannian $G$ manifold, then the orbit map $\prod :M \to M/G$ is a stratified Riemannian submersion and the well-known "cohomogeneity method" pioneered by Hsiang and Lawson [HL] reduces the problem of finding codimension $k$ minimal submanifolds of $M$ to a related problem in $M/G$. We show that this reduction of variables technique depends only on a certain natural Riemannian geometric property of the map $\prod$ which we call $h$-projectability and which is shared by certain other naturally occurring and important classes of Riemannian submersions.