Hyperpolar homogeneous foliations on symmetric spaces of noncompact type

Abstract

A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M . A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F , and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with José Carlos Díaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in [1], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SL r+1 (R)/SO r+1 .2000 Mathematics Subject Classification. Primary 53C12, 53C35; Secondary 53S20, 22E25.

Polar actions on symmetric spaces of higher rank

“…Note that Theorem 1.1 does not generalize directly to non-compact symmetric spaces. In fact, there are counterexamples of polar actions with non-flat sections on non-compact symmetric spaces of higher rank; see [1,Proposition 4.2]. However, an analogous statement as in Theorem 1.1 still holds for actions on non-compact irreducible spaces if one requires the action to be given by a reductive algebraic subgroup of the isometry group.…”

Submanifold geometry in symmetric spaces of noncompact type

“…A nice survey that includes a detailed description of the space SL n (R)/SO n can be found in [6]. In this section we mainly follow [6] and [64].…”

“…In these papers one can also find the only complete classifications known so far on symmetric spaces of higher rank, namely on SL 3 /SO 3 , SL 3 (C)/SU 3 , G 2 2 /SO 4 , G C 2 /G 2 and SO 0 2,n /SO 2 SO n , n ≥ 3. In the more general setting of hyperpolar actions, the only known result is the classification of hyperpolar actions with no singular orbits on any symmetric space of noncompact type, up to orbit equivalence, due to Berndt, Díaz-Ramos and Tamaru [10]. In other words, this result describes all hyperpolar homogeneous regular Riemannian foliations on symmetric spaces of noncompact type.…”

Canonical Extension of Submanifolds and Foliations in Noncompact Symmetric Spaces

“…Recently, Kollross and Lytchak [15] concluded that polar actions on irreducible symmetric spaces of compact type and rank at least two are hyperpolar. This result is false in the noncompact setting, as shown in [3,Proposition 4.2]: the group V ×N Φ , where V is any subgroup of A Φ and Φ is a non-empty subset of simple roots, acts polarly, but not hyperpolarly, on the corresponding noncompact symmetric space.…”

“…When such boundary component is a real hyperbolic plane RH 2 , the extended examples are known; see [3,Section 6]. However, if the boundary component has at least dimension three, then it admits many nontrivial examples of minimal and CMC hypersurfaces.…”