Cohomogeneity Two Nonsemisimple Isometric Actions

Abstract

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 \(\mathbb {R}^{k} \times T^{s}\), \(k+s=\text {dim} M\)), and a theorem by S. Kobayashi [11] states that a homogeneous Riemannian manifold of negative curvature is simply connected. Therefore, it is diffeomorphic to \(\mathbb {R}^{n}\), \(n =\) 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\((M) \ge 3\), then either M is diffeomorphic to \(\mathbb {R}^{k} \times T^{s}\), \(k+s=\text {dim} M\) or \(\pi _{1}(M)=Z\) and the principal orbits are covered by \(S^{n-2} \times \mathbb {R}\), \(n=\)dimM, and \(\frac{M}{G}\) is homeomorphic to one of the spaces R and \([0, \infty )\). 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 \(M=M_{1} \times \cdots \times M_{k}\) of Riemannian manifolds such that \(M_{i}\), \(1\le i\le k\), has negative curvature. In the paper [15], we studied the orbits of cohomogeneity two G-manifolds of this kind, under the condition that \(M^{G} \ne \emptyset \). In the present paper, we replace the condition \(M^{G}\ne \emptyset \) 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 \(\mathbb {R}\), or \(\pi _{1}(M)=Z^{p}\), \(p\ge 1\). 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 G-manifold 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 \(M^{G}=\{ x \in M: G(x)=x \}\). 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 \(\phi \) of a Hadamard manifold M is called elliptic if it has fixed point. \(\phi \) is called hyperbolic (parabolic) if the function \(d^{2}_{\phi }:M \rightarrow \mathbb {R}\) defined by \(d^{2}_{\phi }(x)=d^{2}(x,\phi (x))\) has minimum point (has no minimum).

An isometry \(\phi \) is called axial if there is a geodesic \(\gamma \) translated by \(\phi \) ( there exists a positive constant c such that \(\phi (\gamma (t))=\gamma (t+c)\)). \(\gamma \) is called the axis of \(\phi \).

If \(\gamma \) is a geodesic in the Hadamard manifold M, we denote by \([\gamma ]\) the collection of all geodesics which are asymptotic to \(\gamma \). The collection of all asymptotic classes of the geodesics of M is denoted by \(M(\infty )\) and is called the ideal boundary of M (see [8] for details). In fact, we can imagine \(M\bigcup M(\infty )\) as a manifold with boundary such that M is its interior and \(M(\infty )\) is its boundary.

Remark 2.1

If \(M_{1}\) and \(M_{2}\) are Hadamard manifolds then their product \(M=M_{1} \times M_{2}\) is also a Hadamard manifold. Consider unit speed geodesics \(\gamma _{1}\) and \(\gamma _{2}\) in \(M_{1}\) and \(M_{2}\) and put \(\gamma (t)=(\gamma _{1}(\frac{\sqrt{2}}{2}t), \gamma _{2}(\frac{\sqrt{2}}{2}t))\). Then, \(\gamma \) is a unit speed geodesic in M. Let \(\eta _{1}\) and \(\eta _{2}\) be other unit speed geodesics in \(M_{1}\) and \(M_{2}\) and \(\eta \) be a geodesic in M, defined in the similar way as \(\gamma \). If \([\gamma _{1}]=[\eta _{1}]\) and \([\gamma _{2}]=[\eta _{2}]\), then it is easy to show that \([\gamma ]=[\eta ]\). Consider the map \(\rho : M_{1}(\infty ) \times M_{2}(\infty ) \rightarrow M(\infty )\), defined by \(\rho ([\gamma _{1}],[\gamma _{2}])=[\gamma ]\). Clearly, \(\rho \) is a one-to-one and well-defined map. Then, without loss of precision, we can denote \([\gamma ]\) by \(([\gamma _{1}],[\gamma _{2}])\). In this way, \(M_{1}(\infty ) \times M_{2}(\infty )\) can be considered as a subset of \(M(\infty )\). We call a point \(z \in M(\infty )\) a regular point at infinity, if z belongs to \( M_{1}(\infty ) \times M_{2}(\infty )\) ( that is \(z=(z_{1}, z_{2}), z_{1} \in M_{1}(\infty ), z_{2} \in M_{2}(\infty )\)). 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 \(x, y \in M(\infty )\) there exists a geodesic \(\gamma \) joining x to y.

Lemma 2.3

Let \(\phi \) be a parabolic isometry on a Hadamard manifold M.

  1. (1)

    If z is a fixed point at infinity for \(\phi \), then \(\phi \) leaves each horosphere S centered at z invariant (see [8] for definition of the horosphere).

  2. (2)

    If M has strictly negative curvature, then there is a unique fixed point \( z \in M(\infty )\) for \(\phi \).

Proof

(1) is a direct consequence of [6, Lemma 3]. If M has strictly negative curvature then by [4, Lemma 9.10], M satisfies Axiom 1. Thus, by [8, Theorem 6.5], we get (2). \(\square \)

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. (1)

    \(M^{H} \ne \emptyset \).

  2. (2)

    H translates a unique geodesic.

  3. (3)

    There is a unique point in \(M(\infty )\) fixed by H.

Proof

By [6, Theorem 5], one of the following statements is true:

  1. (1)

    \(M^{H} \ne \emptyset \).

  2. (2)

    H translates a geodesic.

  3. (3)

    There is a point in \(M(\infty )\) fixed by H.

Uniqueness of the geodesic in (2) comes from [4, Proposition 4.2]. If (1) and (2) are not true, then all elements of H are parabolic and uniqueness of the fixed point in (3) comes from Lemma 2.3(2). \(\square \)

Lemma 2.5

Let \(M=M_{1} \times M_{2} \times \cdots \times M_{m}\) such that for each i, \(M_{i}\) is a Hadamard manifold of negative curvature, and suppose that \(\phi \in Iso(M)\) can be decomposed as the product \(\phi =\phi _{1} \times \phi _{2} \times \cdots \times \phi _ {m}\) of non-elliptic isometries \(\phi _{i} \in Iso(M_{i})\). If there is a geodesic \(\beta =(\beta _{1},\ldots ,\beta _{m})\) such that \(\phi (\beta )=\beta \), then \(\beta \) is unique.

Proof

Let \(\gamma =(\gamma _{1},\ldots , \gamma _{m})\) be another geodesic in M such that \(\phi (\gamma )=\gamma \). Then, for each i, \(\phi _{i}(\gamma _{i})=\gamma _{i}\) and \(\phi _{i}(\beta _{i})=\beta _{i}\). We get from the uniqueness of the geodesic left invariant by non-elliptic isometry \(\phi _{i}\) [4, Proposition 4.2] that \(\gamma _{i}=\beta _{i}\). Thus, \(\beta =\gamma \). \(\square \)

Remark 2.6

Let G be a connected solvable Lie subgroup of the isometries of \(M=M_{1}\times M_{2}\times \cdots \times M_{m} \) such that for each i, \( M_{i}\) is simply connected with strictly negative curvature. Since G is connected then each \(g \in G\) can be decomposed as \(g=g_{1}\times \cdots \times g_{m}\), \(g_{i} \in Iso(M_{i})\) (see [10, Vol. 1, p. 240]). For each \(i \in \{1,\ldots ,m\}\), let \(P_{i}:G \rightarrow Iso(M_{i})\) be the map defined by \(P_{i}(g_{1},\cdots g_{i},\ldots ,g_{n})=g_{i}\). Put \(G_{i}=P_{i}(G)\). Clearly, \(G_{i}\) is a closed and connected solvable subgroup of the isometries of \(M_{i}\), and we have:

$$\begin{aligned} G\subset G_{1} \times \cdots \times G_{m} \ \ \ (*) \end{aligned}$$

Lemma 2.7

Under the assumptions of Remark 2.6, there is a unique subset \(\Omega \) of M with the property that \(G(\Omega )=\Omega \) and \(\Omega \) is one of the following sets or a product of them.

  1. (1)

    \(M^{G}\) or product of the fixed point set of some elements of \(\{ G_{1},\ldots ,G_{m}\}\).

  2. (2)

    Image of a geodesic.

  3. (3)

    A regular point at infinity of the product of some elements of the set \(\{ M_{1},\ldots ,M_{m}\}\).

Proof

Since \(M_{i}\) has strictly negative curvature, then by Remark 2.4, one of the following statements is true:

  1. (I)

    For all i, \(M_{i}^{G_{i}} \ne \emptyset \).

  2. (II)

    For each i, there is a unique geodesic \(\gamma _{i}\) in \(M_{i}\) such that \(G_{i}(\gamma _{i})=\gamma _{i}\).

  3. (III)

    There is a unique point \(\zeta _{i} \in M_{i}(\infty )\) for each i, such that \(G_{i}(\zeta _{i})=\zeta _{i}\).

  4. (IV)

    \(\{ G_{1}, \ldots , G_{m}\} \) is a collection of the groups with properties (I) or (II) or (III).

In the case (I), \((M_{1} \times \cdots \times M_{m})^{(G_{1} \times \cdots \times G_{m})} \ne \emptyset \), then \(M^{G} \ne \emptyset \).

In the case (II), \(\gamma =(\gamma _{1},\ldots ,\gamma _{m})\) is a geodesic such that \((G_{1} \times \cdots \times G_{m})(\gamma )=\gamma \).

Thus, \(G(\gamma )=\gamma \) and by Lemma 2.5, \(\gamma \) is unique. In a similar way if the case (III) is true, \(\zeta =(\zeta _{1},\ldots ,\zeta _{m})\) will be the unique regular point in \(M(\infty )\) fixed by G. Now, it is easy to show that if (IV) is true, then there is a unique set \(\Omega \), which is a product of the sets similar to (I), (II), (III), such that \(G(\Omega )=\Omega \). \(\square \)

Results

Remark 3.1

[5]. If M is a complete and connected Riemannian manifold and G is a connected subgroup of Iso(M), and if \(\widetilde{M}\) is the universal Riemannian covering manifold of M with the covering map \(\kappa : \widetilde{M} \rightarrow M\), then there is a connected covering \(\widetilde{G}\) of G with the covering map \(\pi : \widetilde{G} \rightarrow G\), such that \(\widetilde{G}\) acts isometrically on \(\widetilde{M}\) and

  1. (1)

    Each deck transformation \(\delta \) of the covering \(\kappa : \widetilde{M} \rightarrow M\) maps \(\widetilde{G}\)-orbits onto \(\widetilde{G}\)-orbits.

  2. (2)

    If \(x \in M\) and \(\widetilde{x}\in \widetilde{M}\) such that \(\kappa (\widetilde{x})=x\) then \(\kappa (\widetilde{G}(\widetilde{x}))=G(x)\).

  3. (3)

    \(\widetilde{M}^{\widetilde{G}}= \kappa ^{-1}( M^{G}).\)

  4. (4)

    If G is non-semisimple then \(\widetilde{G}\) is non-semisimple.

  5. (5)

    Deck transformation group, which we denote it by \(\Delta \), centralizes \(\widetilde{G}\) (i.e., for each \(\delta \in \Delta \) and \(\widetilde{g}\in \widetilde{G} , \delta \widetilde{g} = \widetilde{g}\delta \)).

Remark 3.2

Let \(\widetilde{M}\) be a Hadamard manifold and S be a horosphere in \(\widetilde{M}\) related to the asymptotic class of geodesics \([\gamma ]\) (i.e., all elements of \([\gamma ]\) intersect S perpendicularly). The function \(f: \widetilde{M} \rightarrow \mathbb {R}\), \( f(p)=lim_{t \rightarrow \infty } d(p, \gamma (t))-t\), is called a Bussmann function. For each point \( p \in \widetilde{M}\), there is a point \(\eta _{_{S}}(p)\) in S which is the unique point of S nearest p, and the following map is a homeomorphism ([8, pp. 47, 58]):

$$\begin{aligned} \phi : \widetilde{M} \rightarrow S \times \mathbb {R}, \ \ \phi (p)=( \eta _{_{S}}(p), f(p)). \end{aligned}$$

Theorem 3.3

(see [15, Theorem 3.5]). Let \(M^{n+2}\) be a nonsimply connected Riemannian manifold which is of cohomogeneity two under the action of a closed and connected subgroup G of isometries, and suppose that M can be decomposed as a product of Riemannian manifolds of negative curvature and \(M^{G}\ne \emptyset \). Then,

  1. (a)

    M is diffeomorphic to \(S^{1} \times \mathbb {R}^{n+1}\) or \(B \times \mathbb {R}^{n}\), where B is the Moebius band.

  2. (b)

    Each principal orbit is diffeomorphic to \(S^{n}.\)

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:

  1. (a)

    M is a parabolic manifold homeomorphic to \(\frac{S}{\pi _{1}(M)}\times \mathbb {R}\). Where, S is a horosphere in the universal Riemannian covering of M, and \(\frac{S}{\pi _{1}(M)}\) is a cohomogeneity one G-manifold.

  2. (b)

    \(\pi _{1}(M)=Z^{p}\) for some positive integer p, and all orbits are diffeomorphic to \(\mathbb {R}^{n-p}\times T^{p}.\)

  3. (c)

    M is diffeomorphic to \(S^{1} \times \mathbb {R}^{n+1}\) or \(B \times \mathbb {R}^{n}\), where B is the Moebius band, and each principal orbit is diffeomorphic to \(S^{n}\).

  4. (d)

    dimM=3 and M has negative curvature. \(\pi _{1}(M)=Z\) and each orbit is diffeomorphic to \(S^{1}\).

Proof

Following Remark 3.1, let \(\widetilde{M}\) be the universal Riemannian covering manifold of M with the deck transformation group \(\Delta \) and let \(\widetilde{G}\) be the corresponding connected covering of G which acts isometrically and by cohomogeneity two on \( \widetilde{M} \). Since by assumptions of the theorem, M can be decomposed as the product of Riemannian manifolds of strictly negative curvature, then \(\widetilde{M}\) can be decomposed as \(\widetilde{M}=M_{1} \times M_{2} \times \cdots \times M_{m}\) such that for all i, \(M_{i}\) has strictly negative curvature, and each \(\delta \in \Delta \) decomposes as \(\delta =\delta _{1} \times \delta _{2} \times \cdots \times \delta _{m}\), \(\delta _{i} \in Iso(M_{i})\). If \(M^{G}\ne \emptyset \), then we get part (c) of the theorem from Theorem 3.3. Thus, in the rest of the proof we suppose that \(M^{G}= \emptyset \) which implies \(\widetilde{M}^{\widetilde{G}} = \emptyset \). Then, according to the assumptions of the theorem, all orbits must be regular and of dimension n. Since G is non-semisimple, \(\widetilde{G}\) is non-semisimple. Let H be a connected solvable normal subgroup of \(\widetilde{G}\). By Lemma 2.7, there exists a unique subset \(\Omega \) of \(\widetilde{M}\) such that \(H(\Omega )=\Omega \) and \(\Omega \) is one of the following sets or product of them.

A: a totally geodesic submanifold of \(\widetilde{M}\) or a totally geodesic submanifold of the product of some elements of the set \(\{M_{1}, \ldots , M_{m}\}\) (because, fixed point sets of connected subgroups of isometries are totally geodesic submanifolds).

\(\gamma \): image of a geodesic of \(\widetilde{M}\) or a geodesic in the product of some elements of the set \(\{M_{1}, \ldots , M_{m}\}\) .

\(\zeta \): a regular point at infinity of \(\widetilde{M}\) or a regular point at infinity for product of some elements of the set \(\{M_{1}, \ldots , M_{m}\}\).

That is

$$\begin{aligned} \Omega \in \{ \gamma , \zeta , A, A \times \gamma , A \times \zeta , \zeta \times \gamma , A \times \zeta \times \gamma \} \ \ \ \ \ (*) \end{aligned}$$

Since H is normal in \(\widetilde{G}\), for each \(g \in \widetilde{G}\) we have:

$$\begin{aligned} g^{-1}Hg=H \Rightarrow g^{-1}Hg(\Omega )=H(\Omega ) \Rightarrow g^{-1}Hg(\Omega )=\Omega \Rightarrow H(g\Omega )=g\Omega . \end{aligned}$$

Now, from the uniqueness of \(\Omega \) with the property \(H(\Omega )=\Omega \), we get that \(g\Omega =\Omega \), then

$$\begin{aligned} \widetilde{G}(\Omega )=\Omega \ \ \ \ \ (**) \end{aligned}$$

Since the elements of \(\Delta \) commute with the elements of \(\widetilde{G}\), for each \(\delta \in \Delta \) we have \(\widetilde{G}( \delta \Omega )= \delta \widetilde{G}(\Omega )= \delta \Omega \). Uniqueness of \(\Omega \) implies that \(\delta (\Omega )=\Omega \). Thus,

$$\begin{aligned} \Delta (\Omega )=\Omega \ \ \ \ (***) \end{aligned}$$

We consider now the seven cases of \(\Omega \) in \((*)\).

Case 1.\(\Omega = \gamma \).

\(\widetilde{G}(\gamma )=\gamma \) and By \((**)\), \(\Delta (\gamma )= \gamma \). Since \(\widetilde{M}^{\widetilde{G}} = \emptyset \), then \(\gamma \) is a \(\widetilde{G}\)-orbit. Since the cohomogeneity is two and all orbits are of dimension n, then \(n=1\). Since by assumptions of the theorem, for all i, dim\(M_{i}\ge 3\), then \(\widetilde{M}=M_{1}\). Thus, M has strictly negative curvature. All \(\widetilde{G}\)-orbits in \(\widetilde{M}\) are diffeomorphic to \(\gamma \), then they are diffeomorphic to R. By Lemma 3.5 in chapter 12 of [7], \(\Delta = Z\). Thus, each G-orbit in M will be diffeomorphic to \(\frac{R}{\Delta }= \frac{R}{Z}=S^{1}\). This is part (d) of the theorem.

Case 2.\(\Omega =\zeta \).

Let \([\gamma ]\) be the asymptotic class of the geodesics related to \(\zeta \). We have \(\widetilde{G}(\zeta )=\zeta \) and by \((**)\), \(\Delta (\zeta )=\zeta \)(i.e, \(\widetilde{G}([\gamma ])=[\gamma ]\) and \(\Delta ([\gamma ])=[\gamma ]\)). First, suppose that there is an axial element \(\delta \in \Delta \) and let \(\lambda \) be the unique geodesic such that \(\delta \lambda =\lambda \) ( uniqueness of \(\lambda \) comes from Lemma 2.5). If \(g\in \widetilde{G}\) then \(\delta (g\lambda )=g\delta \lambda =g\lambda \). Thus, we get from the uniqueness of \(\lambda \) that \(g\lambda =\lambda \). Then, \(\lambda \) is a \(\widetilde{G}\)-orbit, and we get part (d) of the theorem as like as Case 1.

Now, suppose that all elements of \(\Delta \) are non-axial. Non-identity elements of \(\Delta \) are without fixed points, then they must be parabolic and M will be a parabolic manifold. By Lemma 2.3, for each \(\delta \in \Delta \) and each horosphere S related to the asymptotic class \([\gamma ]\), \( \delta S = S\). Fix a horosphere S related to \([\gamma ]\). Put \(M_{1}=\frac{S}{\Delta }\) and let \(\eta _{s}\) and f be the maps defined in Remark 3.2. The homeomorphism \(\phi :\widetilde{M}\rightarrow S\times \mathbb {R}\) mentioned in Remark 3.2, induces a homeomorphism \(\phi _{1}:\frac{\widetilde{M}}{\Delta }=M \rightarrow \frac{S}{\Delta }\times \mathbb {R}\), such that \(\phi _{1}(x)=(\kappa \eta _{s}(\widetilde{x}),f(\widetilde{x})), \widetilde{x}\in \kappa ^{-1}(x)\).

If there is a \(g \in \widetilde{G}\) which is axial and \(\lambda \) is its unique axis, then we get from the fact that the elements of \(\Delta \) and g commute that for all \(\delta \in \Delta \), \(\delta (\lambda )\) is also an axis for g. Since the axis is unique then \(\delta (\lambda )=\lambda \). Now, if \(g' \in \widetilde{G}\), then again we get from the uniqueness of the axis \(\lambda \) for \(\delta \) that \(g' \lambda =\lambda \), thus \(\widetilde{G}(\lambda )=\lambda \), and we get part (d) of the theorem as like as case 1.

Now, Suppose that all elements of \(\widetilde{G}\) are non-axial. If for some \(g \in \widetilde{G}\) and \(x \in \widetilde{M}\), \(gx=x\), then for the geodesic \(\lambda \) in \([\gamma ]\) which passes from x, we have \(g\lambda =\lambda \), and g must be axial which is contradiction. Thus, we can assume that the elements of \(\widetilde{G}\) are parabolic and by Lemma 2.3, \(\widetilde{G}(S)=S\). Thus, S is a cohomogeneity one \(\widetilde{G}\)-manifold and \(\frac{S}{\Delta }\) is a cohomogeneity one G-manifold. This is part (a) of the theorem.

Case 3.\(\Omega =A\).

Similar to the previous cases, we have \(\widetilde{G}(A)=A\) and \(\Delta (A)=A\). Put \(A_{1}=\kappa (A)\). A is a nontrivial totally geodesic submanifold of \(\widetilde{M}\), thus \(A_{1}\) is a totally geodesic submanifold of M. Since \(\Delta (A)=A\), then \(\Delta \) is equal to deck transformation group of \(A_{1}\), and \(\pi _{1}(A_{1})=\Delta =\pi _{1}(M)\). Since all orbits are of dimension n, we have for all \(x \in A\):

$$\begin{aligned} n = \text {dim} \widetilde{G}(x) \le \text {dim} A < \text {dim} \widetilde{M} = n + 2 \Rightarrow \text {dim} A = n orn + 1 \end{aligned}$$

Now, we consider dim \(A = n\), dim \(A = n + 1\), separately.

I) dim\( A = n\).

In this case, A is a \(\widetilde{G}\)-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 \(\delta \in \Delta \), \(d^{2}_{\delta }\) has no minimum point.

(II-2) There is a \(\delta \in \Delta \) such that \(d^{2}_{\delta }\) has minimum point.

(II-1): In this case by [17, Lemma 3.2 ], \(\Delta \) maps each orbit of A onto itself. Thus by a similar way in the proof of [17, Lemma 3.6 ], all orbits of \(A_{1}\) are diffeomorphic to \(T^{p} \times \mathbb {R}^{n-p}\) for some nonnegative integer p, thus we get part (b) of the theorem.

(II-2): By [4, Proposition 4.2], the minimum point set of \(d^{2}_{\delta }\) is equal to the image of all geodesics translated by \(\delta \). But by Lemma 2.5, there is at most one geodesic translated by \(\delta \). Let \(\gamma \) be the unique geodesic such that \(\delta (\gamma )=\gamma \). Since the elements of \(\widetilde{G}\) and \(\Delta \) commute, then we get from the uniqueness of \(\gamma \) that \(\widetilde{G}(\gamma )=\gamma \), and we get part (d) of the theorem in the similar way as Case 1.

Case 4.\(\Omega \in \{ A \times \gamma , A \times \zeta , \zeta \times \gamma , A \times \zeta \times \gamma \}\).

By dimensional reasons, this cases can not occur. We give the proof for \( \Omega = A \times \gamma \), other cases are similar. We can assume (after a possible rearrangement of the indices) that

$$\begin{aligned} \widetilde{M}= & {} M_{1} \times M_{2} \times \ldots M_{k} \times M_{k+1} \times M_{k+2} \times \cdots \times M_{m} \\&A \subset M_{1} \times M_{2} \times \ldots M_{k}, \ \ \ \ \gamma \subset M_{k+1} \times M_{k+2} \times \cdots \times M_{m} \end{aligned}$$

By \((**)\), \(\widetilde{G}(A \times \gamma )=A \times \gamma \). Thus, \(A \times \gamma \) is a union of \(\widetilde{G}\)-orbits. Since A and \(\gamma \) are nontrivial then the codimension of A in \(M_{1} \times M_{2} \times \ldots M_{k}\) is at least 1, and because for all i, dim\(M_{i}\ge 3\), then the codimension of \(\gamma \) in \( M_{k+1} \times M_{k+2} \times \cdots \times M_{m}\) is at least 2. Thus, the codimension of \(A \times \gamma \) in \(\widetilde{M}\) will be at least 3. This is contradiction (because, \(A \times \gamma \) is union of orbits which have codimension two in \(\widetilde{M}\)). \(\square \)

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 \(\widetilde{M}\) to negatively curved manifolds and decomposability of \(\Delta \) ( 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 \(\frac{S}{\Delta }\) with \(\mathbb {R}\). 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 \(S^{1} \times R\) 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 \(\widetilde{M}\) 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 \(R^{k} \times T^{n-k}\), for some positive integer k.

References

  1. 1.

    Alekseevsky A.V., Alekseevsky D.V.: Asystatic G-manifolds. In: Proceedings of the workshop on Diff. Geom. Topol. Ed. R. Caddeo. F. Trierri 1–22 (1993)

  2. 2.

    Alekseevsky, D.V.: Riemannian manifolds of cohomogeneity one. Colloq. Math. Soc. J. Bolyai Diff. Geom. Eger 56, 9–22 (1989)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Ballmann W., Gromov M., Schroeder V.: Manifolds of nonpositive curvatue Brikhauser Boston, (1985)

  4. 4.

    Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bredon, G.E.: Introduction to compact transformation groups. Academic Press, New York, London (1972)

    Google Scholar 

  6. 6.

    Byers, W.: Isometry group of Riemannian manifolds of negative curvature. Proc. Am. Math. Soc. 45, 281–285 (1976)

    Article  Google Scholar 

  7. 7.

    Do Carmo, M.P.: Riemannian geometry. Birkhauser, Boston, Basel, Berlin (1992)

    Google Scholar 

  8. 8.

    Eberlin, P., O’Neil, B.: Visibility manifolds. Pac. J. Math. 46, 45–109 (1973)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Eberlein P.: Geodesic flows in manifolds of nonpositive curvature, http://math.unc.edu/ Faculty/ pbe/AMS_-Summer.pdf

  10. 10.

    Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. I. II. Wiely Interscience, New york (1969)

    Google Scholar 

  11. 11.

    Kobayashi, S.: Homogeneous Riemannian manifolds of negative curvature. Tohoku Math. J. 14, 413–415 (1962)

  12. 12.

    Michor P. W.: Isometric actions of Lie groups and invariants, Lecture course at the university of Vienna (1996/97)

  13. 13.

    Mirzaie, R.: On negatively curved G-manifolds of low cohomogeneity. Hokkaido Math. J. 38, 797–803 (2009)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mirzaie, R.: On Riemannian manifolds of constant negative curvature. J. Korean Math. Soc. 48, 23–21 (2011)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Mirzaie, R., Sorush, H.: Topology of orbits and orbit spaces of some product \(G\)-manifolds. Manuscr. Math. 149, 297–302 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Mirzaie, R.: On homoheneous submanifolds of negatively curved Riemannian manifolds. Publ. Math. Debrecen 82, 267–275 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Mirzaie, R., Kashani, S.M.B.: Topological properties of some cohomogeneity one Riemannian manifolds of nonpositive curvature. Bull. Korean Math. Soc. 37, 587–599 (2000)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Mirzaie, R., Kashani, S.M.B.: On cohomogeneity one flat Riemannian manifolds. Glasgow Math. J. 44, 185–190 (2002)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Mirzaie, R.: On orbits of isometric ations on flat Riemannian amnifolds. Kyushu J. Math. 65, 383–393 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    O’Neill, B.: Semi-Riemannian geomerty with applications to Relativity. Academic Press, New York, Berkeley (1983)

    Google Scholar 

  21. 21.

    Podesta, F., Spiro, A.: Some topological propetrties of cohomogeneity one Riemannian manifolds with negative curvature. Ann. Global Anal. Geom. 14, 69–79 (1996)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Wolf, J.A.: Homogeneity and bounded isometries in manifolds of negative curvature. Illinois J. Math. 8, 14–18 (1964)

    MathSciNet  Article  Google Scholar 

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Correspondence to Reza Mirzaie.

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Mirzaie, R., Bakhtiari, M. Cohomogeneity Two Nonsemisimple Isometric Actions. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00400-x

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Keywords

  • Riemannian manifold
  • Lie group
  • Isometry

Mathematics Subject Classification

  • 53C30
  • 57S25