Reduction of variables for minimal submanifolds

Palais, Richard S.; Terng, Chuu-Lian

doi:10.1090/S0002-9939-1986-0857946-2

Reduction of variables for minimal submanifolds
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by Richard S. Palais and Chuu-Lian Terng PDF

Proc. Amer. Math. Soc. 98 (1986), 480-484 Request permission

Abstract:

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.

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