Cohomogeneity Two Nonsemisimple Isometric Actions

We describe the orbits of a cohomogeneity two Riemannian G-manifold M from topological point of view, under the conditions that G is nonsemisimple and M decomposes as a product of negatively curved Riemannian manifolds.


Introduction
A manifold on which a group G acts is called a G-manifold. In this paper, we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of isometries. Dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds of nonpositive curvature [22] states that a homogeneous Riemannian manifold M of nonpositive curvature is simply connected or it is diffeomorphic to a cylinder over a torus (i.e., it is diffeomorphic to R k × T s , k + s = dimM), and a theorem by S. Kobayashi [11] states that a homogeneous Riemannian manifold of negative curvature is simply connected. Therefore, it is diffeomorphic to R n , n = dimM. There are many interesting theorems about topological properties of cohomogeneity one G-manifolds of nonpositive curvature [2,17,18,21].  The authors of [21] studied cohomogeneity one G-manifolds of negative curvature. Among other results, they proved that if M is a nonsimply connected negatively curved cohomogeneity one Riemannian G-manifold and dim(M) ≥ 3, then either M is diffeomorphic to R k × T s , k +s = dimM or π 1 (M) = Z and the principal orbits are covered by S n−2 × R, n =dimM, and M G is homeomorphic to one of the spaces R and [0, ∞). Also, topological properties of cohomogeneity one Riemannian manifolds of nonpositive curvature has been studied by many authors. But, classification of orbits and orbit spaces of cohomogeneity two Riemannian G-manifolds of nonpositive curvature is an open problem. This article follows previous papers [13][14][15][16], where we proved various results about topological properties of cohomogeneity two Riemannian G-manifolds. One of the main examples of Riemannian manifolds of nonpositive curvature is the product M = M 1 × · · · × M k of Riemannian manifolds such that M i , 1 ≤ i ≤ k, has negative curvature. In the paper [15], we studied the orbits of cohomogeneity two G-manifolds of this kind, under the condition that M G = ∅. In the present paper, we replace the condition M G = ∅ by the condition that G is nonsemisimple and there is no non-principal orbit of positive dimension. Among other results, we show that if M is not simply connected then either it is homeomorphic to the product of a cohomogeneity one G-manifold with R, or π 1 (M) = Z p , p ≥ 1. Our main result is Theorem 3.2. The paper is organized as follows: We recall some definitions and prove some statements about Riemannian manifolds of nonpositive curvature in Preliminaries. In the section of results, first we mention a remark about the relations of the orbits of a Gmanifold and the orbits of its universal covering manifold, which is important in the proof of our theorem. Then, we give our main theorem and its proof.

Preliminaries
In what follows, M is a Riemannian manifold, G is a closed and connected subgroup of the isometries of M, and M G = {x ∈ M : G(x) = x}. All geodesics of M are considered to have unit speed.
A complete connected and simply connected Riemannian manifold of nonpositive curvature is called a Hadamard manifold. An isometry φ of a Hadamard manifold M is called elliptic if it has fixed point. φ is called hyperbolic (parabolic) if the function ) has minimum point (has no minimum). An isometry φ is called axial if there is a geodesic γ translated by φ ( there exists a positive constant c such that φ(γ (t)) = γ (t + c)). γ is called the axis of φ.
If γ is a geodesic in the Hadamard manifold M, we denote by [γ ] the collection of all geodesics which are asymptotic to γ . The collection of all asymptotic classes of the geodesics of M is denoted by M(∞) and is called the ideal boundary of M (see [8] for details). In fact, we can imagine M M(∞) as a manifold with boundary such that M is its interior and M(∞) is its boundary.
. A regular point in the infinity of the product of any finite number of Hadamard manifolds can be defined similarly.

Remark 2.2
A Hadamard manifold M satisfies Axiom 1 (see [8]) if for any distinct points x, y ∈ M(∞) there exists a geodesic γ joining x to y.

Lemma 2.3 Let φ be a parabolic isometry on a Hadamard manifold M.
(1) If z is a fixed point at infinity for φ, then φ leaves each horosphere S centered at z invariant (see [8] for definition of the horosphere).

Lemma 2.4 If H is a closed and connected solvable subgroup of the isometries of a Riemannian manifold M of strictly negative curvature, then one of the following statements is true:
(1) M H = ∅.
We get from the uniqueness of the geodesic left invariant by non-elliptic isometry φ i [4, Proposition 4.2] that γ i = β i . Thus, β = γ .

Remark 2.6
Let G be a connected solvable Lie subgroup of the isometries of M = M 1 × M 2 × · · · × M m such that for each i, M i is simply connected with strictly negative curvature. Since G is connected then each g ∈ G can be decomposed as Clearly, G i is a closed and connected solvable subgroup of the isometries of M i , and we have: Proof Since M i has strictly negative curvature, then by Remark 2.4, one of the following statements is true: . . , G m } is a collection of the groups with properties (I) or (II) or (III).
In the case (I), In the case (II), γ = (γ 1 , . . . , γ m ) is a geodesic such that (G 1 × · · · × G m )(γ ) = γ . Thus, G(γ ) = γ and by Lemma 2.5, γ is unique. In a similar way if the case (III) is true, ζ = (ζ 1 , . . . , ζ m ) will be the unique regular point in M(∞) fixed by G. Now, it is easy to show that if (IV) is true, then there is a unique set , which is a product of the sets similar to (I), (II), (III), such that G( ) = .

Results
Remark 3.1 [5]. (4) If G is non-semisimple then G is non-semisimple. (5) Deck transformation group, which we denote it by , centralizes G (i.e., for each δ ∈ and g ∈ G, δ g = gδ).  As we mentioned in Introduction, classification of orbits of cohomogeneity two Riemannian manifolds of nonpositive curvature is an open problem and seems to be a difficult problem in general case. In the following theorem, we consider an important category of Riemannian manifolds of nonpositive curvature, containing products of negatively curved Riemannian manifolds. In direction of [15], we give a description of the manifold and its orbits under the condition that the acting group is non-semisimple.

Theorem 3.4 Let M n+2 , be a nonsimply connected Riemannian manifold such that it can be decomposed as a product of Riemannian manifolds of strictly negative curvature of dimension bigger than two, and let G be a non-semisimple closed and connected subgroup of the isometries of M. If M is of cohomogeneity two under the action of G, without non-principal orbits of positive dimension, then one of the following statements is true:
(a) M is a parabolic manifold homeomorphic to S π 1 (M) ×R. Where, S is a horosphere in the universal Riemannian covering of M, and S π 1 (M) is a cohomogeneity one G-manifold. That is Since H is normal in G, for each g ∈ G we have: Now, from the uniqueness of with the property H ( ) = , we get that g = , then Since the elements of commute with the elements of G, for each δ ∈ we have G(δ ) = δ G( ) = δ . Uniqueness of implies that δ( ) = . Thus, We consider now the seven cases of in ( * ). ). First, suppose that there is an axial element δ ∈ and let λ be the unique geodesic such that δλ = λ ( uniqueness of λ comes from Lemma 2.5). If g ∈ G then δ(gλ) = gδλ = gλ. Thus, we get from the uniqueness of λ that gλ = λ. Then, λ is a G-orbit, and we get part (d) of the theorem as like as Case 1. Now, suppose that all elements of are non-axial. Non-identity elements of are without fixed points, then they must be parabolic and M will be a parabolic manifold. By Lemma 2.3, for each δ ∈ and each horosphere S related to the asymptotic class [γ ], δS = S. Fix a horosphere S related to [γ ]. Put M 1 = S and let η s and f be the maps defined in Remark 3.2. The homeomorphism φ : M → S × R mentioned in Remark 3.2, induces a homeomorphism φ 1 : If there is a g ∈ G which is axial and λ is its unique axis, then we get from the fact that the elements of and g commute that for all δ ∈ , δ(λ) is also an axis for g. Since the axis is unique then δ(λ) = λ. Now, if g ∈ G, then again we get from the uniqueness of the axis λ for δ that g λ = λ, thus G(λ) = λ, and we get part (d) of the theorem as like as case 1. Now, Suppose that all elements of G are non-axial. If for some g ∈ G and x ∈ M, gx = x, then for the geodesic λ in [γ ] which passes from x, we have gλ = λ, and g must be axial which is contradiction. Thus, we can assume that the elements of G are parabolic and by Lemma 2.3, G(S) = S. Thus, S is a cohomogeneity one G-manifold and S is a cohomogeneity one G-manifold. This is part (a) of the theorem.

Case 3. = A. Similar to the previous cases, we have G(A) = A and (A) = A. Put A 1 = κ(A).
A is a nontrivial totally geodesic submanifold of M, thus A 1 is a totally geodesic submanifold of M. Since (A) = A, then is equal to deck transformation group of A 1 , and π 1 (A 1 ) = = π 1 (M). Since all orbits are of dimension n, we have for all x ∈ A: Now, we consider dim A = n, dim A = n + 1, separately. I) dimA = n. In this case, A is a G-orbit and A 1 must be a G-orbit of nonpositive curvature. Thus, we get part (b) of the theorem ( Because, a homogeneous Riemannian manifold of nonpositive curvature is diffeomorphic to product of a torus and a euclidean space [22]. II) dim A = n + 1. A 1 is a cohomogeneity one G-manifold of non-positive curvature, without singular orbits. Consider the following two cases separately: (II-1) For all δ ∈ , d 2 δ has no minimum point. (II-2) There is a δ ∈ such that d 2 δ has minimum point. δ is equal to the image of all geodesics translated by δ. But by Lemma 2.5, there is at most one geodesic translated by δ. Let γ be the unique geodesic such that δ(γ ) = γ . Since the elements of G and commute, then we get from the uniqueness of γ that G(γ ) = γ , and we get part (d) of the theorem in the similar way as Case 1.
By dimensional reasons, this cases can not occur. We give the proof for = A × γ , other cases are similar. We can assume (after a possible rearrangement of the indices) that Thus, A × γ is a union of G-orbits. Since A and γ are nontrivial then the codimension of A in M 1 × M 2 × . . . M k is at least 1, and because for all i, dimM i ≥ 3, then the codimension of γ in M k+1 × M k+2 × · · · × M m is at least 2. Thus, the codimension of A × γ in M will be at least 3. This is contradiction (because, A × γ is union of orbits which have codimension two in M).

Remark 3.5
In Theorem 3.4, decomposability of M to the product of negatively curved manifolds can be replaced by the weaker condition of decomposability of the universal covering manifold M to negatively curved manifolds and decomposability of ( see [17], definition of UND-manifolds and examples).

Remark 3.6
In case (a) of Theorem 3.4, M is homeomorphic to the product of a cohomogeneity one manifold S with R. By proof of the theorem, in this case, there is no singular orbit. Since the orbit space of cohomogeneity one manifolds with no singular orbit are homeomorphic to S 1 or R, then the orbit space of M under the action of G will be homeomorphic to S 1 × R or R 2 . Study of the orbits in this case reduces to the study of the orbits of cohomogeneity one actions on horospheres. In the special case when M has constant negative curvature (decomposition of M has one factor of constant negative curvature), the horospheres of M are isometric to R n+1 , and from the known results about cohomogeneity one actions on flat Riemannian manifolds, orbits of M are diffeomorphic to R k × T n−k , for some positive integer k.
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