Clifford algebras and new singular Riemannian foliations in spheres

Abstract

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are nonhomogeneous. This generalises the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Münzner.A singular Riemannian foliation on a Riemannian manifold M is, roughly speaking, a partition of M into connected complete submanifold, not necessarily of the same dimension, that locally stay at a constant distance from each other. Singular Riemannian foliations on round spheres provide local models of general singular Riemannian foliations around a point.An example of singular Riemannian foliation on round spheres is given by the decomposition into the orbits of an isometric group action, and such a foliation is called homogeneous.A different family of singular Riemannian foliations on spheres is induced by isoparametric hypersurfaces. A hypersurfaces of S n is called isoparametric if it has constant principal curvatures. Isoparametric hypersurfaces were first studied by Cartan who classified those with g ≤ 3 distinct principal curvatures, and a lot of progress has been made (cf. for example the surveys [Cec08,Tho00]), even though the complete classification is still an important open problem. Every isoparametric hypersurface partitions the sphere into parallel hypersurfaces, which are isoparametric as well, and this partition is a special example of a singular Riemannian foliation. For a long time all the known codimension 1 singular Riemannian foliations from isoparametric hypersurfaces appeared to be orbits of some isometric group action on S n , so much so that Cartan asked [Car39] whether every isoparametric hypersurface arised in this way. The question was answered in the negative by Ozeki and Takeuchi [OT75,OT76], who found infinite families of non homogeneous isoparametric foliations with 4 distinct principal curvatures defined in terms of representations Mathematics Subject Classification: 53C12, 57R30

A slice theorem for singular Riemannian foliations, with applications

“…They include polar representations with generalized Weyl group of the form Z/2 × D 4 or Z/2 × D 6 , where D k denotes the dihedral group with 2k elements. Other such F 0 include the quaternionic Hopf foliation of R 8 , the (inhomogeneous) octonionic Hopf foliation of R 16 , and, more generally, the Clifford foliation associated to any Clifford system C m,k with m ≡ 0 (mod 4) and k odd, see [Rad14]. A different class of non-examples are the level sets in round spheres of some homogeneous polynomial F such that ∇F 2 (x) is a power of x 2 .…”

“…In fact, there is a construction that associates to each infinitesimal foliation an infinite family of (higher dimensional) inhomogeneous infinitesimal foliations. These are called composed foliations, they are related to Clifford systems, and include classical examples such as the octonionic Hopf fibration of R 16 , and the isoparametric foliations in [FKM81] (see [Rad14,GR15]). …”

Mean Curvature Flow of Singular Riemannian Foliations

“…This includes a new class of foliations, neither homogeneous nor polar, constructed using Clifford algebras; cf. [14]. By contrast, (irreducible) isoparametric foliations in spheres either have cohomogeneity one, or arise from a polar representation [17].…”

“…Like in the case of orbits of isometric actions in compact manifolds (cf. [13]), it is possible to prove that for generalized isoparametric foliations the mean curvature of a singular leaf is tangent to the stratum that contains it; see [15,Prop. 2.10] for the case of singular Riemannian foliations in spheres.…”

Sobre folheações Finslerianas singulares

“…Nesta seção, usando o resultado de Radeschi [23], serão construídos in nitos exemplos de folheações Randers singulares não-homogêneas.…”