Equifocal submanifolds with non-flat section and topological Tits buildings

TL;DR

This paper proves the non-existence of certain equifocal submanifolds with non-flat sections in symmetric spaces of compact type, using topological and geometric methods involving Tits buildings and root systems.

Contribution

It provides a new proof of a non-existence theorem for equifocal submanifolds with non-flat sections, introducing the notion of slice topology and applying topological Tits building theory.

Findings

No equifocal submanifold with non-flat section exists in certain symmetric spaces.

The universal cover of the slice space is homothetic to a sphere.

Symmetric spaces of rank greater than one cannot admit such submanifolds.

Abstract

From the Lytchak's result for polar foliations on an irreducible simply connected symmetric space $G/K$ of compact type and rank greater than one, we can derive that there exists no equifocal submanifold with non-flat section whose codimension is greater than two in the symmetric space $G/K$. In the first-half part of this paper, we give a new proof of this non-existence theorem. The recipi of our new proof is as follows. Suppose that there exists an equifocal submanifold $M$ with non-flat section whose codimension is greater than two in an irreducible symmertric space $G/K$ of compact type and rank greater than one. We introduce the notion of a slice topology of $G/K$ associated to $M$. We consider the universal covering $\pi:\widehat{G/K}\to G/K$ of the slice topological space $G/K$ and give $\widehat{G/K}$ the manifold structure and the Riemannian metric such that $\pi$ is a Riemannian submersion onto the symmetric space $G/K$. First we show that a simplicial decomposition of the Riemannian manifold $\widehat{G/K}$ gives an irreducible topological Tits building of spherical type and rank greater than two. By applying Burns-Spatzier's theorem to this topological Tits building, we show that the Riemannian manifold $\widehat{G/K}$ is homothetic to the unit sphere. Furthermore, from this fact, we show that $G/K$ is isometric to a sphere, a complex projective space or a quaternionic projective space. This contradicts that $G/K$ is of rank greater than one. This is the recipi of our proof. In the second-half part, we estimate the codimension of $M$ from above by using the multiplicities of the roots of the root system of $G/K$. As its result, we can show that there exists no equifocal submanifold with non-flat section in some irreducible simply connected symmetric spaces of compact type.