Universally polar cohomogeneity two Riemannian manifolds of constant negative curvature

In this paper, we suppose that M is a Riemannian manifold of constant negative curvature under the action of a Lie subgroup G of IsoðMÞ such that the maximum of the dimension of the orbits is dim M 2. Then, we study topological properties of M under some conditions. Subjects: Geometry; Topology; Pure Mathematics


Introduction
An isometric action of a Lie group on a Riemannian manifold M is called polar if there exists a connected closed submanifold ∑ of M which intersects the orbits orthogonally and meets every orbit. Such a submanifold ∑ is called a section of the group action. In the special case where the section is flat in the induced metric, the action is called hyperpolar. The classification problem of polar actions was initiated by Dadok (1985), who classified polar actions by compact Lie groups on Euclidean spaces. In Heintze, Palais, Terng, and Thorbergsson (1995), the authors mentioned the interest of classifying (hyper-)polar actions on symmetric spaces of compact type. Kollross (2007) classified polar actions on compact symmetric spaces with simple isometry group and rank greater than one. Berndt (2011) presented a survey about polar actions on Riemannian symmetric spaces, ABOUT THE AUTHORS Reza Mirzaie is professor and researcher of mathematics in Imam Khomeini international University, Qazvin, Iran. He is interested in differential geometry and its applications to physics.
Mojtaba Heidari is a PhD student. The present paper is a portion of his thesis under the supervision of Dr. Reza Mirzaie.

PUBLIC INTEREST STATEMENT
In differential geometry, a Riemannian manifold is said to be homogeneous if the geometric structure is similar for all points. For example, sphere is homogeneous, because there is no difference between geometric properties of the sphere in any two different points. But, ellipsoid with non-equal semi-axes is not homogeneous. In homogeneous Riemannian manifolds, a connected lie group acts transitively such that each action is isometric (preserves distance). For an example consider the group of all clockwise rotations around the origin of the sphere. A generalization of homogeneous Riemannian manifolds is cohomogeneity K Riemannian manifolds, where transitivity of the action is reduced to another weaker condition called cohomogeneity of the action. In special case, when K = 0, the manifold will be homogeneous.
The present paper is devoted to study topological properties of some cohomogeneity two Riemannian manifolds.
with emphasis on the noncompact case. This classification showed that these actions are in fact all hyperpolar. J. C. Diaz-Ramos and A. Kollross obtain a classification of polar actions with a fixed point on symmetric spaces (Diaz-Ramos & Kollross, 2011).
Let M n be a connected and complete Riemannian manifold of dimension n, and let G be a closed and connected subgroup of the Lie group of all isometries of M. If x 2 M then GðxÞ ¼ gx : g 2 G f gis the orbit containing x. The cohomogeneity of the action of G on M is defined by CohðG; MÞ ¼ n À maxfdim GðxÞ : x 2 Mg. If CohðG; MÞ ¼ m then M is called cohomogeneity m Riemannian manifold. If CohðG; MÞ ¼ 0 then M is called homogeneous Riemannian manifold. Kobayashi (1962) proved that a homogeneous Riemannian manifold M of negative curvature is simply connected. Recently, Riemannian manifolds of cohomogeneity one have been studied from different points of view. Alekseevsky and Alekseevsky (1993) gave a description of such manifolds in terms of Lie subgroup G. Podesta and Spiro (1996) got interesting results about Riemannian manifold of negative curvature and of cohomogeneity one. Among other results, they proved that, if M, dimðMÞ ! 3, is a Riemannian G-manifold of negative curvature and CohðG; MÞ ¼ 1, then either M is diffeomorphic to R k Â T r , r þ k ¼ dimðMÞ, or π 1 ðMÞ ¼ Z and the principal orbits are covered by S nÀ2 Â R, n ¼ dimðMÞ.
In this paper, Mirzaie (2011a) studied topological properties and G-orbits of a flat Riemannian G-manifold M of cohomogeneity two, and in Mirzaie (2009) he characterized a Riemannian G-manifold of negative curvature and of cohomogeneity two from topological view point, under the condition that M G Þϕ.
In this paper, in combination of the concept of polarity and cohomogeneity, we study topological properties of a Riemannian G-manifold of constant negative curvature and of cohomogeneity two.

Preliminaries
In the following, we mention some facts needed for the proof of our theorems. (2) If x 2 M and e x 2 e M such that κðe xÞ ¼ x then κð e Gðe xÞÞ ¼ GðxÞ.
( Kobayashi, 1962). A homogeneous Riemannian manifold of negative curvature is simply connected.
Theorem 2.3 (Podesta & Spiro, 1996). If M is a complete and connected cohomogeneity one Riemannian manifold of negative curvature, then either M is simply connected or π 1 ðMÞ ¼ Z p , p ! 1.
Theorem 2.4 (Mirzaie & Kashani, 2002). Let M be a flat non-simply connected cohomogeneity one Riemannian manifold under the action of Lie group G & IsoðMÞ.
(a) If there is a singular orbit, then π 1 ðMÞ ¼ Z p .
(b) If there is no singular orbit and M=G ¼ R then M is diffeomorphic to R r Â T t for some nonnegative integers r; t; r þ t ¼ dimM. Definition 2.6. If G 1 ; G 2 & IsoðMÞ then we say that G 1 and G 2 are orbit equivalent if for each x 2 M, G 1 ðxÞ ¼ G 2 ðxÞ.
If M is a complete and simply connected Riemannian manifold of nonpositive curvature then the geodesics γ 1 and γ 2 in M are called asymptotic provided there exists a number c > 0 such that dðγ 1 ðtÞ; γ 2 ðtÞÞ c for all t ! 0. The asymptotic relation is an equivalence relation on the set of all geodesic in M, the equivalence classes are called asymptotic classes. If γ is a geodesic in M, then we denote by ½γ the asymptotic class of geodesics containing γ. The following set is by definition the infinity of M: For any x 2 M and ½γ 2 Mð/Þ, there exists a unique geodesic γ x 2 ½γ such that x 2 γ x and there is a unique hypersurface S x , which contains x and is perpendicular to all elements of ½γ. The hypersurface S x is called the horosphere determined by x and ½γ.
Consider the Lorentzian space R n;1 ð¼ R nþ1 Þ with a non-degenerate scaler product h; i given by It is well known that any simply connected Riemannian manifold of constant negative curvature c < 0, is isometric to the hyperbolic space of curvature c defined by It is well known that each horosphere in H n ðcÞ is isometric to R nÀ1 .
Theorem 2.7 (Di Scala & Olmos, 2001). Let G be a connected Lie subgroup of the isometries of hyperbolic space H n . Then, one of the following assertions is true: (i) G has a fixed point.
(ii) G has a unique nontrivial totally geodesic orbit.
(iii) All orbits are included in horospheres centered at the same point at infinity. -Ramos & Sanchez et al., 2013). Let G act polarly on H n . Then the action of G is orbit equivalent to:
(b) The action of N Â K, where N is the nilpotent part of the Iwasawa decomposition of SOð1; mÞ, m 2 2; :::; n f g , and K is a compact group acting polarly on R nÀm .
Remark 2.9 (see (Eberlein & O'Neil, 1973), pp. 57, 58). Let e M be a complete and simply connected Riemannian manifold of strictly negative curvature, and let S be a horosphere in e M determined by asymptotic class of geodesics ½γ. The function f : e M ! R; f ðpÞ ¼ lim t!1 dðp; γðtÞÞ À t, is called a Bussmann function.
For each point p 2 e M there is a point η S ðpÞ is S, which is the unique point in S of minimal distance from p, and the following map is a homeomorphism: ϕ : e M ! S Â R; ϕðpÞ ¼ ðη S ðpÞ; f ðpÞÞ: Do Carmo, 1992). If M is nonsimply connected Riemannian manifold of negative curvature and there is a geodesic in e M such that ΔðγÞ ¼ γ then Δ is isomorphic to ðZ; þÞ.
Definition 2.11. We say that a non-simply connected Riemannian G-manifold M is universally polar, when the covering group e G of G acts polarly on its universal Riemannian covering manifold e M.
Theorem 2.12 (Heidari & Mirzaie, 2018). Let M n , n ! 3, be a nonsimply connected Riemannian G-manifold of constant negative curvature and of cohomogeneity two. Then either M is universally polar or it is diffeomorphic to S 1 Â R nÀ1 or B 2 Â R nÀ2 (B 2 is the Moebius band). Proof: Since M n is a Riemannian manifold of constant negative curvature, then H n is its universal covering manifold. We denote the covering maps by k : H n ! M n . Since e M e G ¼ ϕ, by Theorem 2.8, one of the following is true:

Results
(i) e G has a unique nontrivial totally geodesic orbit.
(ii) All e G-orbits of H n are included in horospheres centered at the same point at infinity.
We consider each case separately.
(i) We denote the unique totally geodesic orbit of the action of e G on H n by Q. By the fact that Δ maps e G-orbits of H n onto e G-orbits and since Q is the unique totally geodesic orbit, then ΔðQÞ ¼ Q. If dimQ ¼ 1, then Q is a geodesic in H n . Thus by Fact 2.12, Δ ¼ Z and we get part (a). If dim Q > 1, put W ¼ kðQÞ. Since Q is a the unique totally geodesic e G-orbit in H n , then W is unique totally geodesic G-orbit in M. Thus W is homogeneous and of constant negative curvature and by Theorem 2.2, W is simply connected. Therefore, Δ is trivial and M is simply connected which is contradiction.
(ii) In this case without loss of generality, we suppose that all e G-orbits of H n are included in horospheres determined by an asymptotic geodesics of class ½γ. If S is a horosphere determined by ½γ then e GðSÞ ¼ S. The homeomorphism ϕ mentioned in Remark 2.11, induces a homeomorphism It is well known that each horospere in H n is isometric to R nÀ1 . Thus S Δ is a flat Riemannian G-manifold of cohomogeneity one, and we get part (b).
Lemma 3.2. Let M n be a closed, connected and non-simply connected Riemannian manifold of constant negative curvature and of cohomogeneity two under the action of a connected Lie group G & IsoðMÞ such that e G is orbit equivalent to SOð1; mÞ Â K, where K & IsoðR nÀm Þ is compact and m 2 0; :::; n À 1 f g . Then one of the following is true: (a) M is homeomorphic to R r Â T l for some non-negative integers r; t; r þ t ¼ dimM (T l is a l-torus).
(b) There is a positive integer p such that π 1 ðMÞ ¼ Z p .
(c) M is homeomorphic to M 1 Â R, where M 1 is a flat cohomogeneity one G-submanifold of M without singular orbit and M 1 G ¼ S 1 .
Proof: Since SOð1; mÞ acts transitively on H m , then e M SOð1;mÞÂK ¼ ϕ. Thus, by Fact 3.1, we get that π 1 ðMÞ ¼ Z p or M is homeomorphic to M 1 Â R, where M 1 is a flat cohomogeneity one G-submanifold of M. In the second case, we can consider the following cases.
Case 1. There is a singular orbit for the G-action on M 1 .
Case 2. There is no singular orbit for the G-action on M 1 .
Case 2. In this case, if M 1 G ¼ R then by Theorem 2.4 (b), we get part (a) of the lemma, otherwise M is homeomorphic to M 1 Â R, where M 1 is a flat cohomogeneity one G-submanifold of M without singular orbit and M1 G ¼ S 1 then we get part (c) of the lemma.