Integrable Riemannian Submersion with Singularities

Abstract

A map of a Riemannian manifold into an euclidian space is said to be transnormal if its restrictions to neighbourhoods of regular level sets are integrable Riemannian submersions. Analytic transnormal maps can be used to describe isoparametric submanifolds in spaces of constant curvature and equifocal submanifolds with flat sections in simply connected symmetric spaces. These submanifolds are also regular leaves of singular Riemannian foliations with sections. We prove that regular level sets of an analytic transnormal map on a real analytic complete Riemannian manifold are equifocal submanifolds and leaves of a singular Riemannian foliation with sections. (2000). 53C12, 53C20. Mathematics Subject Classifications

Singular riemannian foliations with sections, transnormal maps and basic forms

“…The first one generalizes previous results of Carter and West [9], Terng [20] and Heintze et al [12] for isoparametric submanifolds. It can also be viewed as a converse to the main result in [1], and as a global version of one of the results in [2]. The second theorem generalizes a result of Michor for basic forms relative to polar actions [14,15], and will be used to prove Theorem 1.1.…”

“…2 for the definition. These were introduced by Boualem [6], and then by the first author [1,2] as a simultaneous generalization of orbital foliations of polar actions of Lie groups (see e.g., [18]), isoparametric foliations in simply-connected space forms (see e.g., [20]), and foliations by parallel submanifolds of an equifocal submanifold with flat sections in a simply connected compact symmetric space (see e.g., [22]). S.r.f.s.…”

“…It is also worth noting that there is a global converse to the quoted result from [12], namely, the regular leaves of an analytic transnormal map on a complete analytic riemannian manifold are equifocal manifolds and leaves of a s.r.f.s. [1].…”

“…Let ξ(·) be the normal parallel transport of the vector ξ along the curve β. We define ϕ(x) = exp β (1) (ξ(1)). It follows from Theorem 2.7 that exp β(t) (ξ(t)) ∈ L x , and hence ϕ(x) ∈ L x .…”

Equifocality of a singular Riemannian foliation

“…Applying Proposition 3.1 we can find an open set U 0 ⊂ P p , of the plaque P p , an open set U n+1 of P γ p (1) , open sets U i ⊂ P γ p (r i ) of the plaques P γ p (r i ) (for 1 ≤ i ≤ n) and normal foliated vector fields ξ i along U i (for 0 ≤ i ≤ n) with the following properties: 1) For each U i , the normal foliated vector field ξ i is tangent to the geodesics γ x (t), where…”