Polar Foliations of Symmetric Spaces

Abstract

We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.2000 Mathematics Subject Classification. 53C20, 51E24.

Lie Groups and Geometric Aspects of Isometric Actions

“…Similar techniques to the ones used by Lytchak [155] were independently employed by Fang, Grove and Thorbergsson [84], who recently proved the following classification result.…”

“…A partial answer was given by Biliotti [40], who proved it under the extra assumption that the symmetric space is Hermitian. The full conjecture was recently settled by Lytchak [155], see Theorem 5.38. Another recent development in the area was the classification of polar manifolds with sec > 0 and cohomogeneity at least two by Fang, Grove and Thorbergsson [84], see Theorem 5.39.…”

“…The following result of Lytchak [155] gives a positive answer to an important conjecture in polar actions on symmetric spaces: Theorem 5.38. Let M be a simply-connected, irreducible symmetric space with sec ≥ 0, and let F be a polar foliation on M of codimension at least 3.…”

The dual foliation of some singular Riemannian foliations

“…Wilking also used the theory of dual foliations to show that the Sharafutdinov projection is smooth. For more applications of dual foliations, the reader is referred to [7,9,10,11,12,16].…”

Clifford and composed foliations

“…In this section we will prove that in fact they are singular Riemannian foliations employing a method due to Lytchak [Lyt14] for a larger class of foliations on spheres that include Clifford foliations. This procedure involves the extension by homotheties of a singular Riemannian foliation F 0 given on S C to a foliation F h 0 of the same type on D C , and then sending the leaves of the latter back to S 2l´1 under the map π C .…”