Reductions, Resolutions and the Copolarity of Isometric Group Actions

Abstract

We present some results on reductions and the copolarity of isometric group actions, which we obtained in our thesis [Mag08]. We also describe a resolution construction for isometric actions with respect to a reduction and give examples.for 2 ≤ k ≤ n − 1 and a minimal section is given by R k 2 , which is embedded into R kn as block matrices with nonzero entries in the upper (k × k)-block only.Example 2.4. Consider the following action of T 2 × S(U(1) × U(2)) on SU(3). The first factor acts by matrix multiplication from the left and the second factor by matrix multiplication from the right by the inverted matrix. The copolarity in this case is equal to 1 and a minimal section is given by SO(3) ⊂ SU(3).The following three lemmas are frequently used throughout the paper. Lemma 2.6 and 2.7 are [GOT04, Lemma 5.1 and Lemma 5.2]. Originally, the second of these was stated for orthogonal representations only, but its proof also works in the general case.Lemma 2.5. Let (G, M) be an isometric action and suppose that M is connected and finite dimensional. If p ∈ M reg , then exp p (ν p (G • p)) intersects every G-orbit.Lemma 2.6. Let (G, M) be an isometric action and let q ∈ M be arbitrary. For v ∈ ν q (G • q) the following assertions are equivalent:(i) v is G q -regular.(ii) There exists ε > 0 such that exp q (tv) is G-regular for 0 < t < ε.(iii) exp q (t 0 v) is G-regular for some t 0 > 0.Lemma 2.7. Let Σ be a fat section of (G, M). For all q ∈ Σ there is a G q -regular v ∈ T q Σ ∩ ν q (G • q). Furthermore, v can be chosen such that p = exp q v is G-regular and arbitrarily close to q.

Core reduction for singular Riemannian foliations in positive curvature

“…Let J(s) be the Jacobi field along α(s) determined by J(0) = 0, and J (0) = w . By Lemma 4.1 in [11] we have that for all s ∈ I, J(s) ∈ ν α(s) (M, N ). We have by [9, Chapter IX, Thm 3.1]:…”

“…It remains to prove that V q satisfies condition (C) of Definition 2.1. Fix v ∈ V q such that L v is a regular leaf of the infinitesimal foliation F q , and observe that, as in [11], the property (C) is equivalent to ν v (ν q (M, L q ), V q ) ⊂ T v L v . Since the infinitesimal foliation is invariant under homotheties, we may assume that v is small enough, so that p = exp q (v) is contained in a tubular neighborhood given by the slice theorem in [12].…”

“…The core in the example above (the entire original manifold) is an example of a 1-section. In [11], Magata referred to k-sections of group actions as fat sections and proved that they are a form of reduction in the sense above (see Thm 3.1 [11]).…”