We classify the orbits and orbit spaces of cohomogeneity two isometric actions of compact and connected Lie groups on anti-de Sitter spaces. We also give a more simple proof for the previously proven theorem about the similar classifications for cohomogeneity one actions.

1. Introduction

Let Mⁿ be a connected pseudo-Riemannian manifold of dimension n and G be a Lie subgroup of the isometries of M. For each point x ∈ M, the G-orbit containing x is the set {gx : g ∈ G}, which will be denoted by G(x) or x_G. The collection of all orbits, M/G = {G(x) : x ∈ M}, is a topological space with a topology for which the map π : M⟶M/G,

is open and continuous. The dimension of the orbit space is called the cohomogeneity of the action of G on M and is denoted by coh(M, G). It is easy to show that

(1)

When G is connected and coh(M, G) = 0, then M is a homogeneous semi-Riemannian manifold. In this case, M is diffeomorphic to G/G_x, G_x = {g ∈ G : gx = x}. Thus, the study of the geometric properties of M reduces to corresponding problems in algebra. There are many papers and books about the geometry and topology of homogeneous Riemannian and Lorentzian manifolds. G-manifolds mostly have been studied when the cohomogeneity is small, since small cohomogeneity restricts the geometry and topology of M with respect to G. Thus, cohomogeneity one G-manifolds attracted many researchers in the past decade. Two main questions are as follows: Given a semi-Riemannian manifold M, classify the groups G which can act on M with the given cohomogeneity? Given a G-manifold M, characterize the orbit space and the orbits?

If M is a Riemannian cohomogeneity one G-manifold and G is a closed and connected subgroup of the isometries, then the orbit space is homeomorphic to one of the spaces [0, 1], S¹, (0, 1) and [0,1) (see [1, 2]). The orbits of cohomogeneity one Riemannian manifolds are studied in many special cases under the conditions on the dimension or curvature of M (see [1–8]). Cohomogeneity one Lorentzian manifolds are not as well studied as the Riemannian manifolds. They are interesting both from a geometric point of view and their applications in physics. We refer to [9–15] for some results about the cohomogeneity one Lorentzian manifolds. The study of cohomogeneity two Riemannian manifolds goes back to the works performed by Montgomery, Samelson, and Burrell, where they classified the compact and connected groups acting by cohomogeneity two on Euclidean space or the sphere. The orbits related to these actions were also determined by Zippin and Montgomery (see [16–18]). In the direction of a series of papers about the Riemannian and Lorentzian G-manifolds with cohomogeneity two (see [12, 19–22]), we consider in the present article the anti-de Sitter space. Flat l Lorentzian G-manifolds of cohomogeneity two have been studied in [12]. We characterized the orbits and orbit spaces of cohomogeneity two Lorentzian manifolds of constant positive curvature in [13]. The study of the orbits and orbit spaces of cohomogeneity two actions on Lorentzian manifolds of constant negative curvature is an open problem at the moment. We consider a very special case in which the manifold is the anti-de Sitter space and the acting group is compact and connected. Then, we characterize the orbits and the orbit spaces.

2. Results

In what follows, we will use the symbol “X = Y” to denote isometric semi-Riemannian manifolds X, Y, homeomorphic topological spaces X, Y, and diffeomorphic manifolds X, Y.

G will be a connected and closed subgroup of the isometries of a semi-Riemannian manifold M. For each point x ∈ M, we denote the orbit containing x by

(2)

If is a boundary (interior) point of the orbit space M/G, then is called a singular (principal) orbit of M and x is called a singular (principal) point. It is known that the collection of all principal points is an open and dense subset of M. Principal orbits have maximum dimension among the orbits. Each orbit with a dimension less than the dimension of a principal orbit is a singular orbit.

We refer to [23] for more details about the following definitions, symbols, and criteria.

For integer numbers 0 ≤ ν ≤ n, consider the semi-Euclidean space

with the scalar product

(3)

The collection of all linear maps on

preserving the scalar product g is denoted by

. The pseudohyperbolic space in

is defined by

(4)

is diffeomorphic to S^ν × R^n−ν. The isometry group of is equal to O_ν+1(n + 1). is called the anti-de Sitter space.

A model of

: There are many models for

(geometric and algebraic). The following model is useful in geometric arguments (see [23]).

is considered to be a hypersurface of revolution in

(5)

The map

defined by

(6)

is a differentiable covering map, which by using a suitable metric

(the pulled back by κ metric),

, will become the universal semi-Reimannian covering manifold of

usually is denoted by

. Note that

is diffeomorphic to S¹ × Rⁿ⁻¹ and its isometry group is equal to O₂(n + 1) (see [23]).

Definition 1. Let E ⊂ Rⁿ and e be a point in Rⁿ − E. A cone over E with the vertex e is defined as follows:

(7)

A topological space Y is called a cone if it is homeomorphic to a cone Con(E, e).

The following lemma plays a key role in our proofs.

Remark 1. If G is a compact and connected subgroup of O_ν(n), then it is conjugate to a subgroup of O(ν) × O(n − ν).

Proof 1. If we use Lie algebraic tools, the proof comes from a well-known fact that the maximal compact subgroups of O_ν(n) are conjugate. We can show that O(ν) × O(n − ν) is maximal compact in O_ν(n). Thus, G is conjugate to a subgroup of O(ν) × O(n − ν). However, it seems to be useful to give a straight geometric proof. There is a geometric proof for the special case of this theorem in [2] (G is considered to be finite). We use the ideas of [2] and consider the usual scalar and dot products of and Rⁿ as

(8)

Let μ be a Har measure on G and put

(9)

where 〈,〉₂ is a G-invariant inner product on Rⁿ. Since, if g₁ ∈ G, then Gg₁ = G and from the definition of Har measure, we can replace dμ(g) by dμ(gg₁) in the following computations. Thus,

(10)

Let T be a linear map on Rⁿ such that

(11)

where T is symmetric. Let λ₁, …, λ_m be the collection of all different eigenvalues of T and W₁, …, W_m be the corresponding eigenspaces. We have Rⁿ = W₁ + ⋯+W_m. Thus, each vector v of Rⁿ can uniquely be written as v = v₁ + ⋯+v_m, v_i ∈ W_i. We show that for all i, G(W_i) = W_i. Let g ∈ G, v_i ∈ W_i and let i ≠ j. For each vector v, we have

(12)

Since 〈v_i, v〉 = 〈gv_i, gv〉 and

, then we get from (12) that

(13)

For each vector w ∈ W_j, we have w = gv for some v. Thus, by (13), we have

(14)

Now, we get from (14) that

(15)

Since λ_i ≠ λ_j, (15) yields 〈gv_i, w〉 = 0, which by the fact that w and j are arbitrary, we get that gv_i ∈ W_i. Thus, G(W_i) = W_i.

Now, we have

(16)

where 〈,〉₃ is a G-invariant inner product on Rⁿ. Let {e_i,1, …, e_i,n(i)} be a 〈, 〉-orthonormal basis for W_i. Then, the following set is a 〈, 〉-orthonormal basis for

(17)

We show that E is also a 〈,〉₃-orthonormal basis.

Since E is a 〈, 〉-orthonormal basis, then 〈e_i,k, e_i,k〉 = ϵ_i = ±1. Thus, we get from

that

(18)

Since 〈,〉₂ is a inner product, we get from (18) that λ_i and ϵ_i have the same sign, that is,

(19)

Now, if i ≠ j or k ≠ l, then by (16),

. If i = j and k = l, then

(20)

Then, by (19), we get . Therefore, E is a 〈,〉₃-orthonormal basis.

Now, let U be the usual basis of

and A be the matrix of the basis change from U to E. Then,

is the matrix of g relative to the basis E. In other words,

is the matrix of g with respect to the 〈,〉₃-orthonormal basis E, which means that

. Now, we have

(21)

Thus, A⁻¹GA ⊂ O(ν) × O(n − ν).

Remark 2. Let G and K be two groups acting on a manifold M. We say G is (up to diffeomorphism) orbit equivalent to K if there is a diffeomorphism ϕ : M⟶M such that for all x ∈ X, ϕ(G(x)) = K(ϕ(x)). If M is a semi-Riemannian manifold, then the G action is up to isometry orbit equivalent to K if ϕ is an isometry. It is clear that if G is orbit equivalent to K, then the classification of G-orbits and the orbit space M/G is equivalent to the classification of the K-orbits and the orbit space M/K. If G is a subgroup of the isometries of M, then each subgroup K of the isometries which is conjugate to G is (up to isometry) orbit equivalent to G. Since, if K = h⁻¹Gh for some isometry h : M⟶M, then ϕ : M⟶M, ϕ(x) = h⁻¹(x). Then,

(22)

Theorem 1. If G is a compact and connected subgroup of the isometries of , ν > 0, then it is orbit equivalent to a compact and connected subgroup of O(ν + 1) × O(n − ν).

Proof 2. Since the isometry group of is equal to O_ν+1(n + 1), we get the result from Remarks 1 and 2.

Remark 3. If K is a compact and connected subgroup of the isometries of Rⁿ, then K is a subgroup of SO(n). For each point x ∈ Rⁿ, we have K(x) ⊂ Sⁿ⁻¹(r), r = |x|. If coh(Rⁿ, K) = 1, then K(x) = Sⁿ⁻¹(r). Thus, K-action on Rⁿ is orbit equivalent to the action of SO(n) on Rⁿ. It is clear that Rⁿ/K = [0, ∞) (the map is a homeomorphism).

Now, we can mention our results about the cohomogeneity one and cohomogeneity two isometric actions of compact and connected Lie groups on the anti-de Sitter spaces. Part 1 of the theorem has been proved in [15] by Lie algebra tools. Our proof is more simple and ordinary. The second part of the theorem is new.

Theorem 2. Let G be a connected and compact subgroup of the isometries of which acts by cohomogeneity c on .

1.
If c = 1, then each principal orbit is diffeomorphic to S¹ × Sⁿ⁻². The singular orbit is diffeomorphic to S¹. The orbit space is homeomorphic to [0, ∞).
2.
If c = 2, then one of the following is true:
- a.
  Each principal orbit is diffeomorphic to Sⁿ⁻². The singular orbits are zero-dimensional and the union of the singular orbits is diffeomorphic to S¹. The orbit space is homeomorphic to S¹ × [0, ∞).
- b.
  There is a cohomogeneity one isometric action of G on S¹ × Sⁿ⁻² such that each G-orbit of is diffeomorphic to a G-orbit on S¹ × Sⁿ⁻². The orbit space is homeomorphic to [0, ∞) × R or R².

Proof 3. By Theorem 1, G can be considered as a subgroup of O(2) × O(n − 1). We have

(23)

and

(24)

is diffeomorphic to S¹ × Rⁿ⁻¹. Consider an arbitrary point and let o be the origin of Rⁿ⁻¹, then, it is clear that G(x, o) = K₁(x) × {o} ⊂ S¹ × {o}. If x ∈ S¹, then K₁(x) is a compact and connected submanifold of S¹ without boundary, thus one of the following cases is true:

Case 1. K₁(x) = {x}

Case 2. K₁(x) = S¹

Case 1: We have

(25)

Thus, each G-orbit of

is diffeomorphic to a K₂-orbit of Rⁿ⁻¹. Since K₂ is compact, then dim K₂(x) < n − 1 (if dim K₂(x) = n − 1, then connectivity and compactness of K₂(x) lead to a contradiction). Thus, n − dimG(x) = n − dim K₂(x) > 1. This means that

. If c = 2, then the action of K₂ on Rⁿ⁻¹ must be of cohomogeneity one, and by Remark 3, each principal K₂-orbit of Rⁿ⁻¹ is diffeomorphic to Sⁿ⁻² and the unique singular K₂-orbit is a one-point set {∗}. Therefore, each principal G-orbit is in the form

(26)

and each singular G-orbit is in the form

(27)

Thus, each principal G-orbit is diffeomorphic to Sⁿ⁻² and the union of the singular G-orbits is diffeomorphic to S¹. In other words, it is easy to show that the following map is a homeomorphism:

(28)

By Remark 3, we have Rⁿ⁻¹/K₂ = [0, ∞). Thus,

(29)

This is part 2(a) of the theorem.

Case 2: We have

(30)

Clearly, in this case,

. Again, consider the K₂-action on Rⁿ⁻¹. Since K₂ is compact, coh(Rⁿ⁻¹, K₂) ≥ 1 (if coh(Rⁿ⁻¹, K₂) = 0, then K₂(y) = Rⁿ⁻¹, which is in contrast with the compactness of K₂). If c = 1, then coh(Rⁿ⁻¹, K₂) = 1 and by Remark 3, the action of K₂ on Rⁿ⁻¹ is orbit equivalent to the action of SO(n − 1) on Rⁿ⁻¹. Thus, without loss of the generality, we can suppose that K₂ = SO(n − 1). If G(z), z = (x, y), is a principal G-orbit in

, then dim G(z) = n − 1, and clearly, y ≠ o. Thus, K₂(y) is equal to Sⁿ⁻²(r), r = |y|. Now, by (12) and dimensional reasons, we get that

(31)

The singular G-orbit is equal to G(z) for y = o. Thus, G(z) = G(x, o)≃S¹ is the singular orbit. To characterize the orbit space, we consider the following map:

(32)

where ψ is a homeomorphism. Thus,

This is part (12) of the theorem.

If c = 2, then similar to the above argument, K₂ is a (proper) subgroup of SO(n − 1). If z = (x, y), y ≠ o, then G(z) ⊂ S¹ × K₂(y) ⊂ S¹ × Sⁿ⁻²(r), r = |y|. Consider the action of G on S¹ × Sⁿ⁻²(r). Since coh(Hⁿ, G) = 2, then coh(S¹ × Sⁿ⁻²(r), G) = 1. Thus, the structure of the G-orbits on is similar to the structure of the G-orbits of cohomogeneity one G-action on S¹ × Sⁿ⁻². For characterization of the orbit space, we have the following argument.

can be written as

(33)

Since G(x, o) = K₁(x) × {o} = S¹ × {o}, then S¹ × {o}/G is a one-point set which we denote it by {⋆}. Thus,

(34)

For a compact cohomogeneity one G-manifold, the orbit space is homeomorphic to S¹ or [0, 1]. Thus, for all r > 0, S¹ × Sⁿ⁻²(r)/G = [0, 1] or S¹. Let A = [0, 1] or S¹. For each point z = (x, y) ∈ S¹ × Sⁿ⁻²(1) denoted by [z], the G-orbit G(z) in S¹ × Sⁿ⁻²(1)/G (=A). Embed A to R³ and let e be a point in R³ such that e ∉ A. Considering the cone Con(A, e), we have

(35)

where η is a homeomorphism. Thus,

is homeomorphic to Con([0, 1], e) or Con(S¹, e). However, it is easy to show that Cone([0, 1], e) is homeomorphic to [0, ∞) × R and Cone(S¹, e) is homeomorphic to R², which yields

(36)

This is part 2(b) of the theorem.

Remark 4. The more general case of our theorem is the case where the acting group is not necessarily compact. Classification of the orbits and the orbit space for this general case is open yet. There are some results about the structure of the connected subgroups of O₂(n), which can be useful for the first steps in the study of the general case. For instance, Scala and Leistner ([24]) have studied connected subgroups of O₂(n), which can act irreducibly on . They completely classified this kind of group up to conjugacy and proved that G is conjugate to one of the groups SO₀(2, n), SO₀(1, 2), and U(1, n/2), SU(1, n/2), S¹·SO₀(1, n/2) if n is even. Thus, the classification of the orbits and orbit spaces reduces to the case where the action of G is reducible.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

No funding was received for this study.

Open Research

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

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A Note on Anti-De Sitter Spaces as G-Manifolds

Abstract

1. Introduction

2. Results

Conflicts of Interest

Funding

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Data Availability Statement

References

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A Note on Anti-De Sitter Spaces as G-Manifolds

Abstract

1. Introduction

2. Results

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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