2021/01/12 by Naoyuki Koike, Koike, Naoyuki
Mathematics · #53C35 #53C40 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C35 #msc:53C40
paper · pdf · doi:10.48550/arxiv.2101.04331
20 pages
arxiv created 2021/05/04 · arxiv updated 2021/05/05
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 π:\widehatG/K→ G/K of the slice topological space G/K and give \widehatG/K the manifold structure and the Riemannian metric such that π is a Riemannian submersion onto the symmetric space G/K. First we show that a simplicial decomposition of the Riemannian manifold \widehatG/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 \widehatG/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.