j. differential geometry 60 (2002) 355-387

CONFORMAL ACTIONS OF SIMPLE LIE GROUPS

ON COMPACT PSEUDO-RIEMANNIAN MANIFOLDS

URI BADER & AMOS NEVO

Abstract

As is well-known, the real rank of a simple Lie group that acts conformally on a pseudo-Riemannian manifold is bounded by means of the signature of the manifold. We give a precise description of the action whenever the real rank of the group reaches that bound, assuming the action is minimal.

1. Introduction and statement of results

1.1 Introduction

The isometry group of a compact Riemannian manifold must be compact by the Arzela-Ascoli Theorem. This is no longer the case for the group of conformal transformations. Indeed, it is well known that, letting ℍ^{n} denote hyperbolic n-space, the isometry group Iso^{o}(ℍ^{n}) ≃ SO^{o}(n, 1) acts conformally on ∂ℍ^{n} ≃ S^{n−1}. A well known conjecture of A. Lichnerowicz, considered by J. Lelong-Ferrand and M. Obata, asserts that these are the only examples:

Theorem (Lelong-Ferrand [14, 6], Obata [15]). If M is a compact Riemannian manifold and the group of conformal transformations of M, Conf(M), is not compact, then M ≃ S^{n}, endowed with the standard conformal structure.

Supported by BSF.

Received November 15, 2000.

It is natural to inquire whether a similar result can be established for the conformal transformation group of a compact Lorentz, and more generally, pseudo-Riemannian manifold. This problem is one source of motivation for the present study.

Given a general compact pseudo-Riemannian manifold M of signature (p, q), the only constraint we are aware of on the (finite dimensional Lie) group Conf(M) is that the real rank of its semisimple component is bounded by min{p, q} + 1. This fact follows immediately from the following (denoting the real rank of G by rk_{ℝ}(G)):

Theorem (Zimmer, [21]). Assume G is a semisimple Lie group with finite center, acting faithfully on a compact manifold M, and preserving an H-structure, where H is a real-algebraic group. Then rk_{ℝ}(G) ≤ rk_{ℝ}(H).

The algebraic structure present in the case of pseudo-Riemannian manifold is the real algebraic group of conformal linear transformations H = CO(p, q) ≃ SO(p, q) × ℝ_{+}^{∗}, and as is well known rk_{ℝ}(H) = min{p, q} + 1.

Let now G be a semisimple Lie group of higher real-rank that acts on M preserving an H-structure and a volume form. In this case, a number of rigidity phenomena arise, both in terms of the topology of the underlying manifold, as well as the structure of the action itself. These have been studied extensively (see e.g., [22, 7]). We note, however, that the proof of these rigidity results depend crucially on the existence of an invariant volume form (or at least a finite invariant measure) on the manifold. It is an interesting problem to consider the case of conformal transformations, which preserve a non-unimodular structure, and establish whether analogous rigidity phenomena persist in some form. This is our second source of motivation for the present study.

1.2 Statement of results

We first note that each of the groups G = SO^{o}(p, q) admit a conformal action on a compact homogeneous pseudo-Riemannian manifold of signature (p − 1, q − 1). This manifold is the product of the two spheres S^{p−1} ×S^{q−1}, and is a 2-fold cover of the homogeneous projective variety G/Q, where Q is the maximal parabolic subgroup of least codimension in SO^{o}(p, q). The spaces G/Q will be denoted by C^{p−1,q−1}, and will constitute our standard models for conformal actions on compact pseudo-Riemannian manifolds. The pseudo-Riemannian manifold C^{p−1,q−1} is Riemannian if and only if p = 1 (or q = 1), and in that case, we retrieve S^{q−1} (or S^{p−1}), which is the standard model for a conformal action on a compact Riemannian manifold. For a more explicit description of the standard spaces we refer to §2.4.

Our framework will be that of compact manifolds with a bilinear structure, namely endowed with a bilinear form in each tangent space. We emphasize that we do not require anything further from the bilinear forms, and in particular they need not be symmetric or even nondegenerate. This setup is considerably more general than that of G-actions preserving a pseudo-Riemannian structure. We begin by formulating a general lower bound on the dimension of an isotropic subspace in a bilinear manifold.

Theorem 1. Let G be a connected almost simple real Lie group with finite center. Assume G acts conformally (and nontrivially) on a compact manifold with a bilinear structure M. Then there exists some point m ∈ M, where the bilinear form on T_{m}(M) has an isotropic subspace of dimension at least rk_{ℝ}(G) − 1.

As noted already, in §2.4 we will show that the group SO(p, q) acts on compact homogeneous space, preserving a pseudo-Riemannian structure up to a scalar factor. We will also show that SL(3, ℝ) acts on a compact homogeneous space, preserving a symplectic structure up to a scalar factor. Clearly, finite covers of these groups will act similarly on finite covers of these spaces. Furthermore, any outer automorphism of the group gives rise to a twisted action on the same space. The actions obtained by this prescription will be called standard models of conformal actions. We refer to §2.4 for more details and a complete list. In the presence of an upper bound on the dimension of an isotropic subspace in a bilinear manifold, we will show the following

Theorem 2. Let G and M be as in Theorem 1. Assume that the maximum dimension of an isotropic subspace of T_{m}(M) is at most rk_{ℝ}(G) − 1, for all m ∈ M. Then:

• G must be locally isomorphic to either SO(p, q), or SL(3, ℝ).

• There exists a closed G-orbit in M.

• Any closed G-orbit in M is equivariantly and conformally diffeomorphic to a standard model.

Theorem 2 has the following two corollaries. The first considers pseudo-Riemannian manifold admitting a group of conformal transformations with simple subgroup of real rank which is the maximum allowed by Zimmer’s Theorem (or by Theorem 1). It asserts that if the action is minimal, then the group is determined uniquely (up to local isomorphism), and the action, which turns out to be transitive, is also uniquely determined (up to finite covers).

Corollary 3. Assume that G is a connected almost simple Lie group. Assume that G acts conformally and minimally on a pseudo-Riemannian manifold M of signature (p, q), and has real rank min{p, q} +1. Then:

• G is locally isomorphic to SO(p + 1, q + 1).

• M is a homogeneous G-space, and M is conformally and equivariantly diffeomorphic to a finite cover of the standard model C^{p,q} (see §2.4).

Our second corollary considers even dimensional manifold with a bilinear structure where the form at each tangent space is a nondegenerate symplectic form. Such manifolds we call symplectic manifolds (we do not require the 2-form to be closed). Here it turns out that the real rank of a simple subgroup of the conformal group must be strictly smaller than the maximum allowed by Zimmer’s Theorem (or by Theorem 1). The only exception arises from a symplectic structure on ℙ^{2} arising from the cross-product structure on ℝ^{3}, an example that will be discussed further in §2.4.

Corollary 4. Let G be a connected almost simple Lie group of real rank n + 1. Assume that G acts conformally (and nontrivially) on a compact symplectic manifold of dimension 2n. Then G = SL(3, ℝ) or its universal cover, and M is conformally and equivariantly diffeomorphic to ℙ^{2} or its universal cover S^{2}. The action of G on ℙ^{2} is either the one arising from the standard action of G on ℝ^{3} or its twist by the nontrivial outer automorphism of G.

Our proof of Theorem 1 depends on establishing the existence of a continuous nonconstant G-equivariant map from a minimal subset M′ of M to a projective algebraic variety U. The variety we choose to consider is the product of the projective space of bilinear forms on the Lie algebra of G with the Grasmann variety consisting of Lie algebras of stability groups of points of M′. The stability group Q of any point u in the image of M′ stabilizes the form induced on the Lie algebra (via the orbit map). At the same time, Q normalizes the stabilizer of any point in the preimage of u. In addition, for some point u_{0}, Q is co-compact (and algebraic), hence contains a maximal ℝ-split torus, A. One constructs from the action of G on M′ a conformal faithful linear representation of the torus A. The weight spaces associated to a codimension one subtorus which acts isometrically (with respect to the induced form) are used to construct an isotropic subspace of dimension at least rk_{ℝ}(G) − 1 in the tangent space of every point of the pre-image of u_{0}.

The proof of Theorem 2 consists of analyzing further the conformal linear representation of the maximal torus just described. This representation is in fact a subrepresentation of the tensor square of the co-adjoint representation of the torus on the Lie algebra. Inspecting the root systems of real simple groups, one can often construct an isotropic subspace of dimension greater than rk_{ℝ}(G) − 1, contradicting the additional assumption of an upper bound on the dimension of an isotropic subspace. The proof then proceeds by determining explicitly for which Lie algebras the construction yields an isotropic subspace of dimension exactly rk_{ℝ}(G) − 1.

1.3 Remarks and relevant references

As noted already, our work is motivated by a conjecture of Lichnerowicz in conformal Riemannian geometry on the one hand, and by rigidity phenomena associated with simple isometry groups in pseudo-Riemannian geometry on the other hand.

We recall that R. Zimmer [20] showed that the noncompact semisimple component of the isometry group of a compact Lorentz manifold is locally-isomorphic to SL(2, ℝ). This result was generalized by M. Gromov to the case of a pseudo-Riemannian manifold of any signature.

Both of these results are dependent on the fact that the volume form is stabilized by isometries, hence the group preserves a finite measure. This allows the use of variants of the Borel density Theorem (see e.g., [22]).

Another approach, which dispensed with the assumption of finite invariant measure, appeared first in Kowalsky’s thesis [12] (see also [13]). This method allowed Kowalsky to characterize the isometry groups of Lorentz manifolds under the sole assumption of non-properness of the action, rather than the compactness of the manifold. Furthermore, this method yielded some results on conformal actions on Lorentzian manifolds.

One idea introduced by Kowalsky is that the non-properness of the action of a split torus implies the existence of large dimensional isotropic subspace of the tangent space to the orbit at some point of the manifold. To obtain this conclusion she used implicitly a map from the manifold to the space of bilinear forms on the Lie algebra, and considered the dynamics of the action of a split torus in the latter space. Our approach is motivated by these ideas, which we develop systematically in the compact, conformal pseudo-Riemannian case. We have also used some ideas from [21, 17, 16].

We note that isometry groups of compact and noncompact Lorentz manifolds were considered by S. Adams and G. Stuck e.g., in [1, 2] (see also [3]). Isometry groups of Lorentz and more generally pseudo-Riemannian manifolds were considered by A. Zeghib e.g., in [18, 19]. Finally, we note that in the case of symplectic conformal actions, if the 2-form is closed, and the manifold is of dimension ≥ 4, then every conformal transformation is isometric. This result is due to P. Libermann, see [11, p. 27].

1.4 The next chapters

The paper is organized as follows. In Chapter 2 we present the necessary definitions relating to G-actions on manifold with a bilinear structure, and describe the standard models mentioned above. In Chapter 3 we present some preliminary facts on almost simple algebraic groups that will be needed later. In Chapter 4, we establish the existence of the G-equivariant map to the projective variety, and proceed to prove Theorem 1. Chapter 5 is devoted to a streamlined proof of Theorem 2. The technical details relating to analysis of the root systems are relegated to the Appendix. Chapter 6 is devoted to some further remarks on related matters, including the case of isometric actions.

2. Manifolds with bilinear structure

2.1 Lie group actions

Definition 2.1. An action of a Lie group G on a C^{∞}-manifold is a C^{∞}-action if the action map G × M → M is a C^{∞}-map. That is, the induced map G → Diff(M) is a group homomorphism.

For every g ∈ G we have the action map, denoted also by g:

g : M → M, g(m) = gm

and for every m ∈ M we have the orbit map, denoted also by m:

m : G → M, m(g) = gm.

The G-action gives a bundle automorphism of the tangent bundle T M, defined by g(m, V) = (gm, dg_{m}(V)). G also acts by automorphism on the trivial bundle M × 𝔤 by g(m, X) = (gm, Ad(g)X). The orbit map gives a bundle morphism ϕ : M × 𝔤 → T M, defined by ϕ(m, X) = (m, dm_{e}(X)).

Denoting the inner automorphism of G, h ↦ ghg^{−1} by inn(g), the following diagram is commutative:

[figure: M; G; M.; G; g; m; inn(g); gm]

By differentiating at e ∈ G we deduce the commutativity of:

[figure: T_{m}M; 𝔤; T_{gm}M.; 𝔤; dg_{m}; dm_{e}; Ad(g); d(gm)_{e}]

which gives the formula:

∀X ∈ 𝔤 : dg_{m} · dm_{e}(X) = d(gm)_{e}(Ad(g)X). (1)

We conclude that ϕ is a G-equivariant bundle morphism.

Define ψ : M → Gr(𝔤) by ψ(m) = ker(dm_{e}), where Gr(𝔤) = ⋃_{k=0}^{dim(𝔤)} Gr_{k}(𝔤) denotes the space of all linear subspaces of 𝔤. Note the action is locally free if and only if the map ψ takes the constant value {0} ∈ Gr_{0}(𝔤). It follows similarly that ψ is a G-map, where Gr(𝔤) is a G-space via the adjoint action. Note further that the stability group of ψ(m) is the normalizer of the stability group of m.

Denote G_{m} = Stab_{G}(m). There is a canonical representation, induced by the action, π_{m} : G_{m} → GL(T_{m}M) defined by π_{m}(g) = dg_{m}.

2.2 Manifolds with bilinear structure

Definition 2.2. A manifold with a bilinear structure is a couple (M, s), such that M is a C^{∞}-manifold and s : M → T M^{∗} ⊗ T M^{∗} is a C^{∞}section. We call s the structure of (M, s).

When s is clear from the context we will denote (M, s) by M, and s_{m}(X, Y) by ⟨X, Y ⟩_{m}.

The main examples are:

1. (M, s) is called a Riemannian manifold if for all m ∈ M, s_{m} is a positive definite symmetric form on T_{m}M.

2. (M, s) is called a pseudo-Riemannian manifold if M is connected, and ∀m ∈ M, s_{m} is a nondegenerate symmetric form on T_{m}M. The signature σ(s_{m}) is a constant, denoted σ(M).

3. (M, s) is called a symplectic manifold if for all m ∈ M, s_{m} is a nondegenerate anti-symmetric form on T_{m}M (we do not require the form s to be closed).

Recall that given a vector space with a bilinear form (V, B), a subspace W < V is called isotropic if every two vectors in W are perpendicular, that is B|_{W ×W} = 0.

2.3 Lie groups action on manifolds with bilinear structure

Let (M, s) be a manifold with a bilinear structure. Given a diffeomorphism t : M → M we define a new structure on M, ts, by:

(ts)_{m}(X, Y) = s_{t(m)}(dt_{m}(X), dt_{m}(Y)).

Definition 2.3. For a manifold with a bilinear structure (M, s) and a diffeomorphism t, we call t an isometry if ts = s. We call t a conformal map if ts = λs for some scalar valued function λ : M → ℝ_{+}^{∗}.

Definition 2.4. Let G be a Lie group acting on M. If for all g ∈ G, g : M → M is an isometry, we say that G acts isometrically on M. If for all g ∈ G, g : M → M is a conformal map, we say that G acts conformally on M.

Notice that G acts isometrically on M if and only if the section s is a G-map, with respect to the natural action of G on the bundle T^{∗}M ⊗ T^{∗}M. In this case we construct a G-equivariant map r : M → 𝔤^{∗} ⊗ 𝔤^{∗}. The G-bundle-map ϕ : M × 𝔤 → T M gives rise to a G-bundle-map ϕ^{∗} : T^{∗}M ⊗ T^{∗}M → M × (𝔤^{∗} ⊗ 𝔤^{∗}). Denoting r = pr_{2} ∘ ϕ^{∗} ∘ s, where pr_{2} denotes the projection on the second coordinate, we obtain the desire map.

Similarly, if G acts conformally on M then the map s : M → ℙ(T^{∗}M × T^{∗}M) is a G-equivariant map (if s is known to exist then the converse holds also). If for all m ∈ M, dm_{e}(𝔤) = T_{m}(Gm) is not isotropic (under s_{m}) then one can define ϕ^{∗} : ℙ(T^{∗}M ⊗ T^{∗}M) → M × ℙ(𝔤^{∗} ⊗ 𝔤^{∗}), induced by ϕ. This is also a G-equivariant map. Composing with s and projecting on the second coordinate, we obtain the G-equivariant map r : M → ℙ(𝔤^{∗} ⊗ 𝔤^{∗}).

We keep this conclusion for future reference as:

Lemma 2.5. If G acts conformally on the manifold with a bilinear structure M, such that the map r defined above is non-vanishing (that is, r(m) ≠ 0 for all m ∈ M) then r : M → ℙ(𝔤^{∗} ⊗ 𝔤^{∗}) is a well defined C^{∞} G-map.

The following Lemma establishes the fact that in the homogeneous case the bilinear structure is determined uniquely by its value at any point.

Lemma 2.6. Let G be a Lie group. Let M be a G-homogeneous space. Let m_{0} be a point in M, and denote H = Stab_{G}(m_{0}). Assume s_{0} is a bilinear form on T_{m_{0}}M. We then have:

• If the representation π_{m_{0}} : H → GL(T_{m_{0}}M) is orthogonal then there is a unique bilinear structure s on M such that s(m_{0}) = s_{0}, and that G acts isometrically on (M, s).

• If the representation π_{m_{0}} : H → GL(T_{m_{0}}M) is conformal then (up to a conformal equivalence) there is a unique bilinear structure s on M such that s(m_{0}) = s_{0}, and that G acts conformally on (M, s).

2.4 The standard models for conformal actions

Let ℝ^{p,q} be the vector space ℝ^{p+q} endowed with the bilinear form B^{p,q} defined by B^{p,q}(x, y) = ⟨x, y⟩ = ∑_{i=1}^{p} x_{i}y_{i} − ∑_{i=p+1}^{p+q} x_{i}y_{i}. The signature of this form is denoted by (p, q). The group of linear orthogonal transformation of ℝ^{p,q} is denoted by O(p, q), and the group of linear conformal transformation is denoted by CO(p, q).

The additive group ℝ^{p+q} acts on itself, so we have an action of O(p, q) ⋉ ℝ^{p+q} on the homogeneous space O(p, q) ⋉ ℝ^{p,q}/O(p, q) ≃ ℝ^{p+q} (we are identifying the coset O(p, q) with 0 ∈ ℝ^{p+q}). The form s_{0} = B^{p,q} on T_{0}ℝ^{p+q} is O(p, q)-invariant. Using Lemma 2.6 we get a bilinear structure on ℝ^{p,q}. We call ℝ^{p+q} with this bilinear structure 𝔼^{p,q}. 𝔼^{p,q} is a pseudo-Riemannian manifold of signature (p, q). In fact 𝔼^{p,q} is the manifold with a bilinear structure obtained by identifying every tangent space of ℝ^{p+q} with ℝ^{p,q}.

What are the O(p, q) orbits in 𝔼^{p,q}? Identifying 𝔼^{p,q} with ℝ^{p,q} one can define the norm function f(x) = ⟨x, x⟩ on 𝔼^{p,q}. Define F_{t} = {x ∈ 𝔼^{p,q} − {0} | f(x) = t}. The tangent spaces to F_{t} are subspaces of the tangent spaces to 𝔼^{p,q}, so the submanifold F_{t} inherits a bilinear structure from 𝔼^{p,q}. What is the restricted form on F_{t}? Observing that

∀x ∈ 𝔼^{p,q}, ∀y ∈ T_{x}𝔼^{p,q} grad(f)_{x}(y) = ⟨2x, y⟩

we see that T_{x}F_{f(x)} is parallel to {x}^{⊥}, hence both subspaces have the same bilinear form. If t > 0, then F_{t} is a sub-pseudo-Riemannian manifold of signature (p−1, q). If t < 0, then F_{t} is a sub-pseudo-Riemannian manifold of signature (p, q−1). F_{0} is not a pseudo-Riemannian manifold but has a bilinear structure of signature (p − 1, q − 1, 1) (by this notation we mean that the restricted form has a one dimensional radical in every tangent space). Observe that for all x ∈ F_{0}, ℝx − {0} ⊂ F_{0}, and ℝx = rad(T_{x}F_{0}). Passing to projective quotient we get the manifold F_{0}. F_{0} has a natural bilinear structure defined up to a scalar multiple (at any point). Fixing such a structure we obtain our canonical examples of conformal actions, which we denote by C^{p−1,q−1}. Since O(p, q) acts isometrically on F_{0}, its action on C^{p−1,q−1} is conformal.

Standard models for conformal actions on compact pseudo-Riemannian manifolds.

We call the action of SO(p, q) (as well as locally-isomorphic groups with finite center) on C^{p,q} (as well as its finite covers) the standard models of conformal action on a compact pseudo-Riemannian manifold of signature (p, q). In the case where G = SO(4, 4) (and in this case only) there exist three non-equivalent actions of G on C^{3,3} obtained by twisting the standard action by an outer automorphism of G of order three. We consider these additional actions as standard models too.

We note that all the groups SO(q, q) admit a nontrivial outer automorphism, but if q ≠ 4, the twisted action obtained is equivalent to the standard one. This fact will be observed in the course of the proof of Theorem 2.

We now consider the canonical example of a conformal action on a symplectic manifold.

Standard models for conformal actions on compact symplectic manifolds.

Identify the tangent planes to S^{2} with their parallel copies through the origin. For m ∈ S^{2} and x, y ∈ T_{m}S^{2} denote s_{m}(x, y) = ⟨x×y, m⟩. (M, s) is a two-dimensional symplectic manifold. The action of SL(3, ℝ) (or its 2-fold cover group) on S^{2} (or ℙ^{2}) which arises from the natural action on rays in ℝ^{3} is conformal with respect to this symplectic form. In addition there is another action obtained on the same spaces by twisting the above action by the nontrivial outer automorphism of SL(3, ℝ). We call these actions the standard models of conformal action on compact symplectic manifolds.

3. Preliminary results

We list here some simple facts which are well known. We provide short proofs for the sake of completeness.

By a real algebraic group we mean a group of finite index in the group of real points of a complex algebraic group defined over ℝ. For such a group G, we denote by G_{ℂ} the associated connected complex algebraic group.

Recall that a (complex) solvable algebraic group is a semi-direct product of a subtorus and of its unipotent radical. We use the term ℝsplit solvable algebraic group to denote a subgroup of finite index in the set of real points of a solvable algebraic group defined over ℝ, if it has a maximal subtorus which is ℝ-split. Such a connected group, G, admits a composition series G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{s} = {e} consisting of connected closed real algebraic subgroups such that G_{i}/G_{i+1} ≃ (ℝ, +) or (ℝ_{+}^{∗}, ·) for 0 ≤ i < s. The following Lemma is a simple variation on Borel’s fixed point theorem [4, §15.2].

Lemma 3.1. Let G be a connected ℝ-split solvable algebraic group. Let V be an algebraic variety, defined over ℝ, and assume G_{ℂ} acts algebraically on V, and that the action is defined over ℝ. Let U be a G-invariant, (Hausdorff) compact subset of the real points of V. Then there exist a G-fixed point in U.

Proof. We argue by induction on the dimension of G. Assume G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{s} = {e} is a composition series as above, and assume G_{1} has a fixed point in U. We then have an algebraic action of G/G_{1} ≃ (ℝ, +) or (ℝ_{+}^{∗}, ·) on the compact subset U′ consisting of G_{1}- fixed points in U. A minimum dimensional orbit in U′ must be closed in U′, since otherwise its boundary, which consists of smaller dimensional orbits, lies outside U′. But a compact algebraic factor of (ℝ, +) or (ℝ_{+}^{∗}, ·) is a single point. We end up with a G/G_{1}-fixed point in U′, hence a G-fixed point in U. q.e.d.

Recall that an almost-simple group G is of finite index in the set of real-points of a semisimple algebraic group defined over ℝ, i.e., it is a real algebraic group.

Lemma 3.2. Let G be an almost-simple Lie group, and Q < G a cocompact proper algebraic subgroup. Then:

1. Q contains a maximal ℝ-split torus.

2. Q is contained in a proper parabolic subgroup of G.

3. If G is a split form, then Q is of finite index in a parabolic subgroup.

Proof. Let A be a maximal ℝ-split torus. Let Φ be the root system of G with respect to A, and Π the set of positive roots. Let N < G be the connected unipotent subgroup with Lie algebra 𝔫 = ⨁_{α∈Π} 𝔤_{α}. Let N′ < G be the connected unipotent subgroup with Lie algebra 𝔫′ = ⨁_{α∈−Π} 𝔤_{α}. AN is a connected ℝ-split solvable algebraic group which acts algebraically on G/Q. By Lemma 3.1, AN has a fixed point in G/Q, hence AN is contained in a conjugate of Q. We can assume (by picking a conjugate of A in the first place, and if necessary changing the notion of positivity in Φ) that AN < Q. Now:

1. Follows immediately from the discussion above.

2. Q contains N. If it was reductive it would contain N′ too. Indeed, A is a maximal torus of Q, and the root system of A in Q is invariant under multiplication by −1. Furthermore, the multiplicities of α and −α must be the same. The group generated by N and N′ is normal, hence it is G itself. It follows that Q contains a nontrivial unipotent-radical. Q is contained in the normalizer of this radical, and this normalizer itself is contained in a parabolic subgroup (Borel and Tits, [5, §3.1]).

3. If G is a split form then AN is of finite index in a minimal parabolic subgroup. Hence a finite extension of Q is a parabolic subgroup, since it contains a minimal parabolic subgroup.

q.e.d.

Lemma 3.3. Let G be an almost-simple Lie group, H < G a proper subgroup, A ≤ N_{G}(H) an ℝ-split torus. Then the representation induced by the adjoint representation of A on 𝔤/𝔥 is faithful.

Proof. Assuming the representation is not faithful, 𝔥 contains all eigenspaces of some t ∈ A with nonzero eigenvalues. The algebra generates by all such spaces is a nontrivial ideal, hence 𝔥 = 𝔤, contradicting the properness assumption. q.e.d.

It will be significant in our considerations below to have a lower bound for the codimensions of subgroups of a simple group.

Lemma 3.4.

1. Let H be an almost simple complex algebraic group, H′ a proper Zariski closed subgroup. Then codim_{ℂ}(H′) ≥ rk(H).

2. Let G be an almost-simple Lie group, G′ a proper closed subgroup. Then codim_{ℝ}(G′) ≥ rk(G_{ℂ}) ≥ rk_{ℝ}(G).

Proof.

1. Assume H′ < H is a proper algebraic subgroup of minimal codimension. As is well known [4, §5.1], there exists a representation H → GL(V) for some complex vector space V, and there exists a line L < V such that H′ is the stabilizer of L in H. By passing to a quotient space if necessary one can assume there are no H-invariant lines in V. Stating the above in other words, there exist an algebraic action of H on a projective space P = ℙ(V) with no H-fixed points, and there is a point p = [L] ∈ P such that H′ = Stab_{H}(p). In the Zariski closure of the orbit H · p there are no H-fixed points, and no orbits of dimension smaller than codim_{ℂ}(H′) (by minimality of codimension). It follows that the orbit H · p ≃ H/H′ is closed, hence it is a projective variety. Therefore H′ is a parabolic subgroup, hence contains a maximal torus A. Restricting the adjoint representation of H to A, and passing to the representation on the quotient space 𝔥/𝔥′, (which must be faithful by Lemma 3.3), we conclude that rk(H) = dim_{ℂ}(A) ≤ rk(GL(𝔥/𝔥′)) = dim_{ℂ}(𝔥/𝔥′) = codim_{ℂ}(H′).

2. Assume now that G is a real algebraic group which is almost simple. Let G′ < G be a proper subgroup of minimal codimension. Assume also, without loss of generality, that G′ is connected. By the simplicity assumption N_{G}(G′)^{o} = G′, and therefore Ad(G′) is the connected component of Ad(G) ∩ Stab_{GL(𝔤)}(𝔤′). The latter group is the real points of an algebraic group defined over ℝ, so G′ is real-algebraic. By the foregoing result codim_{ℝ}(G′, G) = codim_{ℂ}(G′_{ℂ}, G_{ℂ}) ≥ rk(G_{ℂ}), and of course rk(G_{ℂ}) ≥ rk_{ℝ}(G).

q.e.d.

Finally, we recall the following result (and its proof) from [21].

Lemma 3.5. Let G be a simple Lie group acting on the connected manifold M. Let m ∈ M be a G-fixed point, and assume π_{m} = 1 (see §2.1). Then G acts trivially on M.

Proof. Let K be a maximal compact subgroup in G. Using the compactness of K one can assume M is a Riemannian manifold and K acts on it isometrically. It follows, using the exponential map, that K acts trivially in a neighborhood of n ∈ M if and only if n is fixed and π_{n} = 1. The set of points satisfying both conditions is obviously open and closed. It is not empty by assumption, so the connectivity of M implies the triviality of the action of K. The action of G on M has a nontrivial kernel so it is trivial by simplicity. q.e.d.

4. Proof of Theorem 1

We begin with the following linear version of Theorem 1. Note that the bilinear form we consider below need not be symmetric, nor nondegenerate.

Lemma 4.1. Let (V, B) be a real vector space with a bilinear form. Let O(B), CO(B) be the orthogonal and the conformal groups of B, respectively. Then V has an isotropic subspace U of dimension dim(U) = rk_{ℝ}(O(B)) ≥ rk_{ℝ}(CO(B)) − 1.

Proof. First observe that rk_{ℝ}(O(B)) = rk_{ℝ}(CO(B)) − 1, unless B is the zero form and then obviously, rk_{ℝ}(O(B)) = rk_{ℝ}(CO(B)) = GL(V) = dim(V). Let l = rk_{ℝ}(O(B)). Let A be a maximal ℝ-split torus of O(B), and denote 𝔞 = Lie(A). Then 𝔞 is ℝ-diagonalizable in V. Let α, β be weights of 𝔞, and V_{α}, V_{β} be the corresponding weight spaces. Observe that if α + β ≠ 0 then V_{α} ⊥ V_{β}, and V_{β} ⊥ V_{α}, because for x ∈ V_{α}, y ∈ V_{β} and t ∈ 𝔞 such that (α + β)(t) ≠ 0,

B(x, y) = B(exp(t)x, exp(t)y) = e^{(α+β)(t)}B(x, y).

The representation of 𝔞 on V is faithful, so the weights of this representation span 𝔞^{∗}. Let the weights α_{1}, …, α_{l} be a basis of 𝔞^{∗}. U = ⨁ V_{α_{j}} is isotropic subspace of V of dimension at least l. q.e.d.

Our next Lemma is essential. It ensures the existence of an equivariant map to a projective G-space. This map will be our main tool in the proofs of Theorems 1 and Theorem 2. Recall the definition of the map r : M → 𝔤^{∗} ⊗ 𝔤^{∗}, given at §2.3.

Lemma 4.2. Let G and M be as in Theorem 1. Assume that the maximum dimension of an isotropic subspace of T_{m}(M) is at most rk_{ℝ}(G) − 1, for all m ∈ M. Then the map r : M → 𝔤^{∗} ⊗ 𝔤^{∗} is nonvanishing.

Proof. It follows from Lemma 3.4 that r(m) ≠ 0 whenever m is not a fixed point. Indeed the dimension of the orbit Gm is at least rk_{ℝ}(G), and therefore the tangent space to the orbit cannot be isotropic (by assumption), so r(m) is nonzero.

We will show there are no fixed points. Assume m is a fixed point. G acts conformally, but since every character of G is trivial, its representation π_{m} on T_{m}M is actually orthogonal. It follows that the representation π_{m} is trivial, by our assumption on the maximal dimension of an isotropic subspaces and by Lemma 4.1. Using Lemma 3.5 we get that the action of G on M is trivial, a contradiction. q.e.d.

Proof of Theorem 1. Assume there is no isotropic subspace of T_{m}(M) of dimension rk_{ℝ}(G) − 1, for all m ∈ M. Using Lemma 4.2 r is non-vanishing so the continuous G-map r : M → ℙ(𝔤^{∗} ⊗ 𝔤^{∗}) is well defined (see Lemma 2.5).

We now define another continuous G-map to a different projective space. Let M′ ⊂ M be the set of points with stability group of maximal dimension. M′ is a compact G-invariant subset (see e.g., [17]). Recall the map ψ : M → Gr(𝔤), which assigns to each point in M the Lie algebra of its stability group. ψ is a continuous map on M′ (but not on M) by [17, Lemma 2.1]. From now on we consider the restriction of ψ to M′.

The map r × ψ : M′ → ℙ(𝔤^{∗} ⊗ 𝔤^{∗}) × Gr(𝔤) is a continuous Gmap. Denote the image of M′ by U. G acts algebraically on the space ℙ(𝔤^{∗} ⊗ 𝔤^{∗}) × Gr(𝔤). In particular any minimum dimensional orbit in U is closed (in the relative topology), hence compact. The stabilizer Q of a point in such an orbit is real algebraic and cocompact, hence contains a maximal ℝ-split torus (by Lemma 3.2, part I). We then have a point u ∈ U and a maximal ℝ-split torus A < Q that fixes u.

Let m be a point in the pre-image of u. Denote H = Stab_{G}(m), 𝔥 = Lie(H), and note that H < G since there are no G-fixed points (see the proof Lemma 4.2). By construction, the adjoint action of A on 𝔤 is conformal with respect to the form r(m) since A < Q. 𝔥 is clearly contained in the radical of r(m), and is normalized by Q and hence also by A. Therefore A acts on 𝔤/𝔥 as well, the form r(m) descends to a form on 𝔤/𝔥, and A acts conformally on the latter. Furthermore the representation of A on 𝔤/𝔥 is faithful by Lemma 3.3. By Lemma 4.1, 𝔤/𝔥 contains an isotropic subspace of dimension rk_{ℝ}(G) − 1.

Finally, 𝔤/𝔥 is naturally identified with the tangent space to the orbit T_{m}(Gm), and we obtain a contradiction to the assumption that there is no (rk_{ℝ}(G) − 1)-dimensional isotropic subspace of T_{m}M. q.e.d.

5. Proof of Theorem 2

5.1 Beginning the proof of Theorem 2

Through out this section we will denote l = rk_{ℝ}(G). The proof begins exactly as the proof of Theorem 1, except now our leading assumption is that we do not have any l-dimensional (instead of l − 1) isotropic subspaces of any of the tangent-spaces. We pick-up at the last sentence of the proof, just before using Lemma 4.1, and keep on from there. To summarize: we have a point m ∈ M and a cocompact algebraic subgroup Q < G which both normalizes the stabilizer H of m, and acts conformally on 𝔤 with respect to the bilinear form r(m) via the adjoint representation.

We note that the upper bound on the dimension of an isotropic subspace implies that Q is a proper subgroup of G. Indeed, Q = N_{G}(𝔥)∩ Stab_{G}(r(m)), so in particular if Q = G then 𝔥 is an ideal. Since there are no fixed points and G is simple, 𝔥 = 0, and we can identify T_{m}(Gm) with 𝔤. Furthermore, G acts conformally with respect to the form r(m) on 𝔤. Since G is simple G must preserve the form, hence by Lemma 4.1 there exists an isotropic subspace of dimension l, a contradiction.

Recall that by Lemma 3.2, Q is contained in a proper parabolic subgroup. Fix such a parabolic subgroup and denote it by P. Q also contains a maximal ℝ-split torus, which we denote A. We fix a root system Φ of A in 𝔤 = Lie(G), a set of positive roots Π, and a set of simple roots Δ. We assume from now on that 𝔭 = Lie(P) and all other parabolic subalgebra that occurs are standard (for the construction of the standard parabolic subalgebra see the Appendix).

Let 𝔞 = Lie(A), 𝔥 = Lie(H), 𝔭 = Lie(P) and 𝔮 = Lie(Q). 𝔥 < 𝔮 < 𝔭 < 𝔤 are all semisimple 𝔞-modules, hence admit direct sum complements: 𝔥̆ > 𝔮̆ > 𝔭̆ > 𝔤̆ = 0. 𝔥̆ ≃ 𝔤/𝔥, 𝔤 = 𝔥 ⊕ 𝔥̆, and similarly for 𝔮̆ and 𝔭̆. r(m) restricts to a form on 𝔥̆. 𝔥̆ might be identified with a subspace of T_{m}M as well, and under this identification s(m)|𝔥̆ = r(m)|𝔥̆.

The weights of the natural representation of G on the space of all bilinear forms, 𝔤^{∗} ⊗ 𝔤^{∗}, are simply Φ̂ + Φ̂ (where Φ̂ = Φ ∪ {0}). As an 𝔞-module we have the decomposition

𝔤^{∗} ⊗ 𝔤^{∗} = ⨁_{λ∈Φ̂+Φ̂} V_{λ}

where V_{λ} is the set of bilinear forms, B, on 𝔤 satisfying for all t ∈ 𝔞, and u, v ∈ 𝔤

B(exp(t)u, exp(t)v) = e^{λ(t)}B(u, v).

(Note that V_{λ} is not the weight space of λ, but of −λ, since the action on 𝔤^{∗} involves taking an inverse. Nevertheless, we use the above convention, for convenience.) Keeping in mind that for u ∈ 𝔤_{α} and v ∈ 𝔤_{β} we have

B(exp(t)u, exp(t)v) = e^{(α+β)(t)}B(u, v)

it follows that the forms in V_{λ} are characterized by:

α + β ≠ λ ⇒ 𝔤_{α} ⊥ 𝔤_{β}.

We apply this fact to the form r(m) ∈ 𝔤^{∗} ⊗ 𝔤^{∗}. By our choice of m the form r(m) is A-invariant up to a scalar multiple, hence there is a weight λ ∈ 𝔞^{∗} such that r(m) ∈ V_{λ}.

The proof will now proceed according to the following three steps. In step one we show that 𝔤 is isomorphic to 𝔰𝔬(p, q) or 𝔰𝔩(3, ℝ). This fact is established by showing that only in these Lie algebras the weights, Λ, appearing in the 𝔞-module 𝔭̆ give rise to an isotropic subspace, W, whose dimension is bounded by rk_{ℝ}(𝔤) − 1. We also show that the parabolic subalgebra 𝔭 is uniquely determined (for a given 𝔤). Step one will take most of the effort and will be divided into two parts, one dealing with all root systems except A_{l}, and the other dealing with the case where the root system is of type A_{l}. In step two we show that 𝔭 = 𝔥, so that H = Stab_{G}(m) is cocompact. In step three we show that the compact orbit G/H ⊂ M is conformally equivalent to one of the standard models.

First we note the following facts and we refer to [8, Ch. X, pp. 530-535] for their verification.

Fact 5.1.

• A real simple Lie algebra is determined uniquely by the type of its real root system, together with the multiplicities of the long and the short roots [8, p. 535, Ex. 9].

• If the type of the real root system is B_{l} and the long roots have multiplicity one, then the Lie algebra is isomorphic to 𝔰𝔬(l, l + n), where n is the multiplicity of the short roots [8, Table VI, see also Ex. 8].

• If the type of the real root system is D_{l} there is only one root lengths. If the multiplicity is one, then the Lie algebra is isomorphic to 𝔰𝔬(l, l) [8, Table VI, see also Ex. 8].

• If the type of the real root system is A_{l} there is only one root length. If the multiplicity is one, then the Lie algebra is isomorphic to 𝔰𝔩(l + 1, ℝ) [8, Table VI].

Our strategy is to use the information that the dimension of an isotropic subspace is bounded by l − 1 in order to determine the type of the Lie algebra, together with the multiplicities.

We will now focus our attention on the 𝔞-module 𝔭̆. Denote the restriction of the form from 𝔥̆ to 𝔭̆ by ⟨·, ·⟩. From now on we will consider only this form and only the space 𝔭̆ unless stating otherwise. We suggest that it may be helpful to read the first paragraph of the Appendix for some definitions and notations.

Denote by Λ the set of weights (or roots) of the 𝔞-module 𝔭̆. The existence of λ gives a natural splitting of the set Λ into three parts, Λ = Λ_{1} ∪ Λ_{3} ∪ Λ_{2}, where:

• Λ_{1} = {α ∈ Λ | λ − α ∈ Λ, λ − α ≠ α}. Λ_{1} itself splits into pairs of distinct roots, {α, β} such that α+β = λ. For such a pair, 𝔤_{α} ⊕𝔤_{β} is perpendicular to its complement (in 𝔭̆), and each of 𝔤_{α}, 𝔤_{β} is isotropic.

• Λ_{2} = \{\frac{1}{2}λ\} ∩ Λ. In this case we can not tell a-priori whether the space 𝔤_{\frac{1}{2} λ} is isotropic or not. (Note that \frac{1}{2}λ may fail to be a root.)

• Λ_{3} = {α ∈ Λ | λ − α ∉ Λ}. For such roots, 𝔤_{α} is radical with respect to the form ⟨· , ·⟩ on 𝔭̆.

Notice that any of the above sets might be empty.

Denote by Λ′_{1} a set consisting of one weight of each of the pairs in Λ_{1}. The following subspace of 𝔭̆,

W = ⨁_{α∈Λ′_{1}} 𝔤_{α} ⊕ ⨁_{β∈Λ_{3}} 𝔤_{β}

is an isotropic subspace of dimension at least ⌈\frac{1}{2}(|Λ| − 1)⌉. Indeed, the dimension of the isotropic subspace W is at least

|Λ′_{1}| + |Λ_{3}| ≥ |Λ′_{1}| + \frac{1}{2}|Λ_{3}| = \frac{1}{2}(|Λ| − |Λ_{2}|) ≥ \frac{1}{2}(|Λ| − 1).

On the other hand, by assumption dim(W) ≤ l−1, hence it follows that |Λ| ≤ 2l − 1.

It turns out that the three conditions above, namely the upper bound on the number of weights appearing in the complement of a parabolic subalgebra, the additional structure imposed on Λ by the weight λ, and the upper bound on the dimension of an isotropic subspace, are very restrictive. We now turn to a case by case analysis to determine the possible couples (𝔤, 𝔭) that might occur.

5.2 Beginning Step 1: Root systems other than A_{l}

We will make use of the following lemma, which is an immediate consequence of Lemma A.1 in the Appendix.

Lemma 5.2. Let Φ be a root system of rank l, whose type is not A_{l}. Assume that for some maximal parabolic subalgebra 𝔭 the number of weights in the module 𝔭̆ is less than 2l. Then Φ is of type B_{l}, C_{l} or D_{l}, and 𝔭 is the standard parabolic subalgebra associated with the simple root α_{1} (under the standard ordering of the simple roots, see [8], the table in page 470 and the discussion preceding).

Utilizing this lemma we now analyze the set Λ and determine the multiplicities of the long and the short roots.

B_{l} (l ≥ 2): By Lemma 5.2, Λ = {e_{1}, e_{1} ± e_{i} | 1 < i ≤ l}. We claim that the only possible splitting of Λ obtains for λ = 2e_{1}, and:

Λ_{1} = {e_{1} ± e_{i} | 1 < i ≤ l}

and

Λ_{2} = {e_{1}}, Λ_{3} = ∅.

This claim follows because for any other choice of λ, Λ′_{1} ∪ Λ_{2} will contain at most one weight (indeed, the equation α + β = λ has at most one solution in Λ). Therefore

dim(W) ≥ |Λ′_{1}| + |Λ_{3}| = |Λ| − |Λ′_{1} ∪ Λ_{2}| = 2l − 2

and 2l − 2 ≥ l for l ≥ 2.

Assuming λ = 2e_{1}, in order to have dim(W) ≤ l−1, the dimension of 𝔤_{e_{1}−e_{i}}, 𝔤_{e_{1}+e_{i}} must be exactly one. The latter dimension is the multiplicity of the long roots. Denoting the multiplicity of the short roots by n, Fact 5.1 implies that 𝔤 ≃ 𝔰𝔬(l + n, l). This possibility is realized in the action of SO(l + n, l) on the standard model C^{l+n−1,l−1} (see §2.4).

C_{l} (l ≥ 3): By Lemma 5.2, Λ = {2e_{1}, e_{1} ± e_{i} | 1 < i ≤ l}. One can see, as in the previous case, that λ must be equal to 2e_{1}. On the other hand, λ can not be equal to 2e_{1}, because then 2e_{1} ∈ Λ_{3}, and choosing e_{1} + e_{i} ∈ Λ′_{1} for 1 < i ≤ l, we get dim(W) ≥ l, contradicting our assumptions.

Hence the case that Φ is of type C_{l}, l ≥ 3 can not arise (observe that the case C_{2} is covered by B_{2}).

D_{l} (l ≥ 4): By Lemma 5.2, Λ = {e_{1} ±e_{i} | 1 < i ≤ l}. As in the previous cases it is easy to see that in order to have dim(W) ≤ l − 1 we must have λ = 2e_{1}. In this case, again, the pairs must be {e_{1} + e_{i}, e_{1} − e_{i}}, but in contrast to the B_{l} case, Λ_{2} is empty. Clearly, the dimension of 𝔤_{e_{1}−e_{i}}, 𝔤_{e_{1}+e_{i}} must be exactly one. This latter dimension is the multiplicity of the long (and all) roots. Once again, by Fact 5.1, we get 𝔤 ≃ 𝔰𝔬(l, l). This possibility is realized in the action of SO(l, l) on the standard model C^{l−1,l−1} (see §2.4).

For l = 4 the parabolic subalgebras associated with α_{1}, α_{3} and α_{4} are mapped to one another by an outer automorphism of order three. Hence there exist three distinct homogeneous actions of SO(4, 4) on C^{3,3} (see §2.4). Finally, we note that the case of D_{3} is covered by the A_{3} case discussed below.

5.3 Concluding Step 1: The root system A_{l}

We now turn to the case where Φ is of type A_{l}. We have already seen examples of conformal actions of groups of type A_{l}, for l = 1, 2, 3. Recall that SO(n, 1) is of type A_{1} and it acts conformally on S^{n−1}, SL(3, ℝ) is of type A_{2} and it acts conformally on the symplectic manifold S^{2}, and SO(3, 3) is of type A_{3} and it acts conformally on C^{2,2} (see §2.4). We will show that groups of type A_{l}, l > 3 can not act conformally without having an l-dimensional isotropic subspace. Also in the cases l = 1, 2, 3 we determine what the parabolic subgroup 𝔭 must be. We continue our analysis of the set Λ of weights appearing in 𝔭̆ and the multiplicities of the roots, as in the §5.2.

We first consider the case l = 1.

l = 1: Denoting the roots multiplicity by n, it follows that 𝔤 ≃ 𝔰𝔬(n + 1, 1). 𝔭 must be the only proper standard parabolic subalgebra.

This possibility is realized in the action of SO(n + 1, 1) on the standard model C^{n,0} ≃ S^{n} (see §2.4).

In order to handle the case where l > 1 we will use the following lemma, which follows immediately from Lemma A.2 in the Appendix.

Lemma 5.3. Let Φ be a root system of type A_{l}. Let 𝔭 be a maximal parabolic subalgebra which is different than those associates to the simple roots α_{1} or α_{l}. We then have the following:

• The number of weights in the module 𝔭̆ satisfies |Λ| ≥ l + 1.

• Assume l ≥ 3. If |Λ| = l + 1 then l = 3 and 𝔭 is the maximal parabolic subalgebra associated with α_{2}.

The roots of A_{l} are e_{i} − e_{j} (1 ≤ i, j ≤ l + 1). One can easily see that Λ_{1} contains at most two pairs. The case of two pairs arises for example when λ = e_{1} + e_{2} − e_{3} − e_{4}, and Λ_{1} = {e_{1} − e_{3}, e_{2} − e_{4}, e_{1} − e_{4}, e_{2} − e_{3}}. Then

λ = (e_{1} − e_{3}) + (e_{2} − e_{4}) = (e_{1} − e_{4}) + (e_{2} − e_{3}).

One can see also that Λ_{1} and Λ_{2} can not both be nonempty, because all roots are of the same length (the equation α + β = 2γ does not have a nontrivial solution where α, β and γ are on the circle).

In case Λ_{1} is not empty we have (since Λ_{1} has at most two pairs)

l − 1 ≥ dim(W) ≥ |Λ′_{1}| + |Λ_{3}||Λ| − \frac{1}{2}|Λ_{1}| ≥ |Λ| − 2.

In case Λ_{2} is not empty we have

l − 1 ≥ dim(W) ≥ |Λ_{3}| = |Λ| − |Λ_{2}| ≥ |Λ| − 1.

Either way we get |Λ| ≤ l + 1. By Lemma 5.3, this inequality can occur in only three cases:

l = 3 and 𝔭 is the maximal parabolic subalgebra associated to α_{2}.

This is case two of Lemma 5.3. Here

Λ = Λ_{1} = {e_{1} − e_{3}, e_{2} − e_{4}, e_{1} − e_{4}, e_{2} − e_{3}}

and dim(W) = |Λ′_{1}| = 2 = l−1 only if the roots multiplicity is one. In this case (according to Fact 5.1) 𝔤 ≃ 𝔰𝔩(4, ℝ) ≃ 𝔰𝔬(3, 3). This possibility is realized in the action of SO(3, 3) on the standard model C^{2,2} (see §2.4).

l is arbitrary, and 𝔭 is the maximal parabolic associated with α_{1}.

In this case

Λ = {e_{1} − e_{j} | 2 ≤ j ≤ l + 1}

and |Λ| = l. Since the sums of distinct weights in Λ are distinct, Λ_{1} can contain at most one pair. It follows that |Λ′_{1}| + |Λ_{3}| ≥ l − 1 (whether Λ_{1} is empty or Λ_{2} is empty). The dimension of W is then greater or equal to l − 1 times the roots multiplicity. At the same time, by assumption, it is smaller or equal to l − 1 so the roots multiplicity is one, hence (by Fact 5.1) 𝔤 ≃ 𝔰𝔩(l + 1, ℝ).

From now on we will consider the form r(m) on 𝔤. First notice that in the case 𝔤 ≃ 𝔰𝔩(l + 1, ℝ), we can and will assume that the parabolic subalgebra 𝔭 actually coincides with 𝔮, by Lemma 3.2. In particular 𝔭 acts conformally on 𝔤 (by the definition of 𝔮). Equivalently, we have that ⟨[X_{α}, X_{β}], X_{γ}⟩ + ⟨X_{β}, [X_{α}, X_{γ}]⟩ is a constant multiple of ⟨X_{β}, X_{γ}⟩, for β, γ ∈ Λ and α ∉ Λ (namely when X_{α} ∈ 𝔭). We always assume X_{α} ∈ 𝔤_{α} and so on.

We now proceed to show that if l > 2 then the form has an ldimensional isotropic subspace, namely 𝔭̆ itself, which contradicts our assumption. We focus our attention on the radical of the form. First note that if l > 2 then Λ_{3} is nonempty, hence the radical is nonzero. Indeed, Λ has l weights, and |Λ_{1} ∪ Λ_{2}| is at most two. On the other hand, we claim that if the radical is nonzero, it must coincide with 𝔭.

Recall that, since any root space is one dimensional, we have that [𝔤_{α}, 𝔤_{β}] = 𝔤_{α+β} for α, β, α + β ∈ Φ.

Assume β ∈ Λ_{3} (and therefore, 𝔤_{β} is radical) and γ ∈ Λ is arbitrary. We will show that 𝔤_{γ} is radical. Let δ ∈ Λ be arbitrary root. If δ = β then ⟨X_{γ}, X_{δ}⟩ = ⟨X_{δ}, X_{γ}⟩ = 0. Assume δ ≠ β, and note that δ, β ∈ Λ, hence they are of the form e_{1} − e_{i}, e_{1} − e_{j}. So δ − β = e_{j} − e_{i} is in Φ − Λ. Denoting α = δ − β ∈ Φ − Λ, given X_{β} ∈ 𝔤_{β} and X_{δ} ∈ 𝔤_{δ} there exist X_{α} such that [X_{α}, X_{β}] = X_{δ}. Since α ∈ Φ − Λ, X_{α} is in 𝔭, hence acts conformally. Since X_{β} is radical we have

⟨[X_{α}, X_{γ}], X_{β}⟩ + ⟨X_{γ}, [X_{α}, X_{β}]⟩ = C · ⟨X_{γ}, X_{β}⟩ = 0

hence

⟨X_{γ}, X_{δ}⟩ = ⟨X_{γ}, [X_{α}, X_{β}]⟩ = −⟨[X_{α}, X_{γ}], X_{β}⟩ = 0

and similarly

⟨X_{δ}, X_{γ}⟩ = ⟨[X_{α}, X_{β}], X_{γ}⟩ = −⟨X_{β}, [X_{α}, X_{γ}]⟩ = 0.

This proves that 𝔭 is contained in the radical if l > 2.

We conclude that l = 2. Since the parabolic subalgebra we consider here is the one associated to α_{1}, we have Λ = Λ_{1} = {e_{1} − e_{2}, e_{1} − e_{3}}. This possibility is realized in the action of SL(3, ℝ) on the standard model S^{2}, (see §2.4).

l is arbitrary, and 𝔭 is the maximal parabolic associated with α_{l}.

The parabolic subalgebra associated with α_{l} is the image of the parabolic subalgebra associated with α_{1} under an outer automorphism of the Lie algebra. This implies that for l = 2 (and only in this case) there is another conformal action of SL(3, ℝ). This action is on the Grasmann variety of two planes in ℝ^{3} which is one of the standard models in §2.4.

This concludes the discussion of all possible cases, and the determination of the possible pairs (𝔤, 𝔭) promised in Step 1.

5.4 Steps 2 and 3: Existence of a compact orbit, finitely covering a standard model

We first summarize the notations and conclusion of the discussion so far. We are given a compact manifold with a bilinear structure M, and a simple group G of real rank l, acting conformally on M, and we assume that the dimension of a maximal isotropic subspace is bounded by l − 1. We have focused our attention on a certain point m ∈ M and denoted its stability group by H. We have seen that H is contained in a proper parabolic subgroup which we have denoted by P. The pair (𝔤, 𝔭) must be one of the pairs described in Step 1 (§5.2, 5.3). Furthermore, each of those pairs is realized by an action on a standard model.

We now show that the Lie algebras 𝔥 and 𝔭 coincide. Indeed, the dimension of the isotropic subspace W < 𝔭̆ is exactly l − 1. This follows from examination of all possible cases in step one. Another fact that follows from this examination is the following observation: in all possible cases λ − Λ = Λ, or in other words, α ∈ Λ if and only if λ − α ∈ Λ. As a result, if 𝔥 is a proper subalgebra of 𝔭 then there exists a root α such that 𝔤_{α} < 𝔥̆ and 𝔤_{λ−α} ∩ 𝔭̆ = 0. Since α + β ≠ λ ⇒ 𝔤_{α} ⊥ 𝔤_{β}, It follows that W ⊕ 𝔤_{α} is an isotropic subspace that contain W properly, hence its dimension is at least l, which is a contradiction.

We conclude that 𝔥 = 𝔭 and H is of finite index in the parabolic subgroup P of G. The orbit Gm is isomorphic to a finite cover G/P, and in particular is compact. The same conclusion holds for any closed orbit. To see that, assume M_{1} ⊂ M is a closed orbit. Note first that we have already seen that M contains no fixed points (see the proof of Lemma 4.2). Upon reflection one sees that the arguments used so far in the proof of Theorem 2 apply equally well to the subbundle of the tangent bundle which is tangent to the orbits. Hence they apply equally well when we consider the action of G on M_{1}. It follows that the stability group of a point in M_{1} is of finite index in P. This conclude the proof of step two, namely the fact that every closed G-orbit is isomorphic as a G-space to a finite cover of G/P.

It remains to prove step three, namely that a closed orbit is conformally equivalent to a finite cover of one of the standard models. First notice that one can substitute a standard model instead of M in the arguments of step one and two. This is permissible since the bound l − 1 on the dimension of an isotropic subspace is satisfied for the standard models, as we invite the reader to verify. Since the action on the standard model is transitive, we deduce that the model action is conformally and equivariantly isomorphic to G/P. Clearly, it remains to show that this conformal structure on G/P (the one which makes it conformally equivalent to the standard model) is unique.

This follows from the fact that 𝔭 < 𝔰𝔬(p+1, q +1) contains a copy, 𝔰, of 𝔰𝔬(p, q) (or 𝔰𝔭(1, ℝ) if 𝔤 ≃ 𝔰𝔩(3, ℝ)) whose action on T_{m}(G · m) ≃ 𝔤/𝔭 must be faithful. Indeed, otherwise, by Lemma 3.5, its action on the orbit is trivial, hence the action of G itself is trivial because it has a nontrivial kernel. The conformal action of 𝔰 on 𝔤/𝔥 = 𝔤/𝔭 ≃ ℝ^{p+q} (or the action of 𝔰𝔭(1, ℝ) on 𝔤/𝔥 = 𝔤/𝔭 ≃ ℝ^{2}) is actually orthogonal, by simplicity. It is easy to see that any 𝔰-invariant form on ℝ^{p+q} (or 𝔰𝔭(1, ℝ)-invariant form on ℝ^{2}) is a constant times the usual one. To see this fact, note that 𝔰 is a real form of 𝔰_{ℂ} = 𝔰𝔬(p + q, ℂ) (or 𝔰𝔭(1, ℂ)), which acts irreducibly on ℂ^{p+q} (or ℂ^{2}), and use the following:

Claim. If S is a semisimple subgroup of GL(n, ℝ) and S_{ℂ} acts irreducibly on ℂ^{n} then the space of S-invariant forms on ℝ^{n} is at most one dimensional.

The claim follows from the following three facts: Every two linearly independent complex forms have a degenerate (complex) linear combination, the radical of an invariant complex form is an invariant complex subspace, and finally, S is Zariski dense in S_{ℂ}.

We see that the conformal structure on G/P coincides with that of the standard model in the tangent space to a certain point, hence, by Lemma 2.6, it coincides everywhere.

This concludes the proof of Theorem 2.

6. Some further results and remarks

We list here for completeness some results that follow quite easily from the discussion above, and will be very brief in their explanation.

6.1 A generalization of Theorem 1

Notice that in the proofs of Lemma 4.1 and hence of Theorem 1, no harm is done if we assume that the form is not necessarily bilinear, but multilinear. We will define:

Definition 6.1. A manifold with a k-linear structure is a couple (M, s), such that M is a C^{∞}-manifold and s : M → T^{∗}M^{⊗k} is a C^{∞}- section. We call s the structure of (M, s).

A subspace of a vector space with a multilinear structure is called isotropic if the restriction of the form to this subspace is the zero form. The proof of Theorem 1 applies without change to show the following:

Theorem 5. Let G be a connected almost simple real Lie group with finite center. Assume G acts conformally (and nontrivially) on a compact manifold with a multilinear structure M. Then there exists some point m ∈ M, where the multilinear form on T_{m}(M) has an isotropic subspace of dimension at least rk_{ℝ}(G) − 1.

6.2 The isometric case

We now comment briefly on the case where the action of the almost simple group G on the manifold M is isometric. The result below should be compared with [7, 5.3] and [18] where it is obtained for pseudo-Riemannian manifolds under the assumption that the volume induced by the pseudo-Riemannian structure is finite, and by use of the Borel density theorem. Below, we do not need an invariant volume form, and we allow the bilinear (or multilinear) forms to be singular, for example.

Theorem 6. Let G be a connected almost simple noncompact real Lie group with finite center. Assume G acts isometrically (and nontrivially) on a compact manifold with a multilinear structure M. Then for every point m ∈ M, the form induces on 𝔤, r(m), is a G-invariant form (with respect to the adjoint representation of G on 𝔤).

Proof. Consider the equivariant map r : M → 𝔤^{∗⊗k} (see §2.3). Clearly U = span(r(M)) is a G invariant subspace, and every G-orbit in it is bounded. This property is established as follows. If u_{1} … u_{k} ∈ r(M) span U, then writing u = b_{1}u_{1} + … b_{k}u_{k}, we have that G · u is bounded since each G · u_{i} is bounded, being contained in the compact set r(M). Therefore given an ℝ-split torus, its connected component must act trivially (i.e., with eigenvalues equal to 1). It follows that the action of G itself is trivial. q.e.d.

Note that a similar proof can be given for any real algebraic group which does not have a compact factor.

Consider again the case of bilinear forms. Here it is easy to classify real G-invariant forms on 𝔤. For an almost simple complex group, there is evidently a unique (up to a scalar multiple) invariant complex form, that is the Killing form. If G is real there are two possibilities.

𝔤 has no complex structure: G_{ℂ} is then almost simple as well (see e.g., [10, 6.94]). Because G is Zariski dense in G_{ℂ} there is a unique (up to scalars) G-invariant complex bilinear form on 𝔤_{ℂ}. It follows that the space of real invariant forms on 𝔤 is also one dimensional, and so must be spanned by the Killing form on 𝔤.

𝔤 has a complex structure: Denote 𝔤 = 𝔰_{ℝ} for some complex simple Lie algebra 𝔰. 𝔤_{ℂ} is isomorphic to 𝔰⊕𝔰 (see e.g., [10, 6.94]). So the space of real invariant forms on 𝔤 is 2-dimensional and is obviously spanned by the real and imaginary parts of the Killing form of 𝔰. Observe that the real part of the Killing form of 𝔰 is equal the Killing form of 𝔤.

We call G absolutely simple if 𝔤 has no complex structure. We can now state (compare [7, 5.3] and [18]).

Corollary 7. Let G be a connected almost simple noncompact real Lie group with finite center. Assume G acts isometrically (and nontrivially) on a compact manifold with a bilinear structure M. If G is absolutely almost simple , then for every point m ∈ M, the form induced on 𝔤, r(m), is a scalar multiple of the Killing form. In general, the induced form is invariant, symmetric, nondegenerate and of the same signature as the Killing form, unless it is the zero form.

Finally, we observe that if the bilinear structure on M gives rise to an invariant volume form of finite total mass, then the foregoing result can easily be deduced from Borel’s density Theorem. Here one need not assume that M is compact. For the proof, one notes that the image of the measure on M under the map r is supported on G-fixed forms on 𝔤. Hence for almost every point in M the induced form on 𝔤 is G-invariant. The set of such points is obviously dense and closed. Hence the same conclusion applies to every point in M.

6.3 A lower bound for minimum codimension of closed subgroups

Lemma 3.4 gives the lower bound rk_{ℝ}(G) for the codimension of closed subgroups of an almost simple group. The proof shows also that the subgroup of minimal codimension is always a maximal parabolic subgroup. An elaborate list of the maximal parabolic subgroup of minimum codimension in every simple group appears in [16]. We need only an estimate which we establish directly by an elementary argument.

In the Appendix below we actually calculate a lower bound to the codimension of all maximum dimensional parabolic subgroups of every simple group. In particular, it turns out that the bound in Lemma 3.4 is realized only for groups locally isomorphic to SL(n, ℝ). We conclude:

Corollary 8. Let G be an almost-simple Lie group which is not locally isomorphic to SL(n, ℝ), G′ a proper closed subgroup of G. Then

codim(G′) ≥ 2 · rk_{ℝ}(G) − 2.

This bound is realized only in groups locally isomorphic SO(n, n).

We remark that the foregoing corollary implies that the map r : M → ℙ(𝔤^{∗} ⊗ 𝔤^{∗}) exists under weaker conditions than those stated in Lemma 4.2 (or the assumptions of Theorem 2). This map gives a nontrivial smooth projective factor defined on M.

Appendix

A. Standard parabolic subalgebras

Let us recall some facts about the structure of standard parabolic subalgebras first (compare e.g., [10, 9]). Let 𝔤 be a simple real Lie algebra. Let 𝔞 be a maximal ℝ-split abelian subalgebra of 𝔤, and Let Φ ⊂ 𝔞^{∗} be the root system related to 𝔞. Let Π be the set of positive roots associated to a fixed chosen positivity, and Δ be the related set of simple roots. For every subset I ⊂ Δ one can define Φ_{I} = Φ ∩ span(I) and Π_{I} = Π ∩ span(I). Define also

𝔭_{I} = 𝔤_{0} ⊕ ⨁_{−α∈Π} 𝔤_{α} ⊕ ⨁_{β∈Π_{I}} 𝔤_{β}

𝔭̆_{I}, the direct 𝔞-module complement to 𝔭_{I}, is then given by

𝔭̆_{I} = ⨁_{α∈Π−Π_{I}} 𝔤_{α}

Write Π̆_{I} = Π − Π_{I}. It is clear that dim(𝔭̆_{I}) ≥ |Π̆_{I}|. The latter is very easy to compute given the type of Φ and I, using the fact that if Γ is the Dynkin diagram of Φ then Γ_{I}, the diagram of the root system Φ_{I}, is obtained from Γ by erasing all vertices not in I, and all edges connected to them. |Π| is given by the following table (see e.g., [9, p. 66]):

Type  A_{l} B_{l}, C_{l} (BC)_{l} D_{l} E_{6} E_{7} E_{8} F_{4} G_{2}

Number of Positive Roots

\binom{l+1}{2} l^{2} l^{2} + l l^{2} − l 36 63 120 24 6

Notice that Φ_{I} is not irreducible in general, and we have Φ_{I} = ∏_{i} Φ_{J_{i}}, where Γ_{J_{i}} are the connected components of Γ_{I}. We deduce |Π_{I}| = Σ_{i}|Π_{J_{i}}|.

In what follows, the simple roots of a given root system are ordered according to the standard order as in [8, Ch. X, §3].

Lemma A.1. Let Φ be a root system of rank l, whose type is not A_{l}. Let Δ be the set of simple roots, and I a subset of Δ such that |Π̆_{I}| < 2l. It follows that Φ is of type B_{l}, C_{l} or D_{l}, and I = Δ − {α_{1}}, unless Φ = D_{4}. In the latter case I = Δ − {α_{3}}, and I = Δ − {α_{4}} occur also.

Proof. Denote I_{i} = Δ − {α_{i}}. It is enough to proof the Lemma assuming I = I_{i} for some i ∈ Δ, for then if I ≠ I_{1} then I ⊂ I_{i} for some i > 1 and |Π̆_{I}| ≥ |Π̆_{I_{i}}| ≥ 2l.

We do a case by case analysis.

B_{l}, C_{l} (l ≥ 2): Assume first that l ≥ 3 and I = I_{i} = Δ − {α_{i}} for some 2 ≤ i ≤ l. Define J_{1} = {1, …, i − 1}, J_{2} = {i + 1, …, l}. I_{i} = J_{1} ∪ J_{2} and Γ_{J_{1}}, Γ_{J_{2}} are the connected components of Γ_{I}. Γ_{J_{1}}, Γ_{J_{2}} are Dynkin diagrams of root systems of type A_{i−1} and B_{l−i} (or C_{l−i}) respectively, so Φ_{I} is of type A_{i−1} × B_{l−i} ( or A_{i−1} × C_{l−i}). From the table above we deduce |Π̆_{I}| = |Π| − (|Π_{J_{1}}| + |Π_{J_{2}}|) = l^{2} − (\binom{(i−1)+1}{2} + (l − i)^{2}) = \frac{1}{2}i(4l − 3i + 1) ≥ 2l.

For l = 2, |Π̆_{∅}| = 4 = 2l and |Π̆_{I_{1}}| = |Π̆_{I_{2}}| = 3 = 2l − 1.

Note that B_{2} and C_{2} are the same root system, but with a different order of the two simple roots. Therefore the pair B_{2}, I_{2} is equivalent to C_{2}, I_{1}, and the pair C_{2}, I_{2} is equivalent to B_{2}, I_{1}. Hence these extra solutions are redundant.

We summarize: |Π̆_{I}| ≤ 2l − 1 implies I = I_{1} (and |Π̆_{I_{1}}| = 2l − 1).

(BC)_{l} (l ≥ 1): Assume l = 1. Then Π = {α, 2α}, Δ = {α}, and I, which is a proper subset of Δ, must be the empty set. Then |Π̆_{I}| = 2 = 2l.

We now assume l ≥ 2. As before, we assume I = I_{i} for some i. The root systems of types B_{l} and C_{l} are subsystems of Φ so |Π̆_{I}^{(BC)_{l}}| is at least |Π̆_{I}^{B_{l}}| or |Π̆_{I}^{C_{l}}|. Recall that Π̆_{I}^{(BC)_{l}} is exactly the set of positive roots whose expansion as a linear combination of simple roots contains a nonzero multiple of α_{i}. α_{i} belongs to one of the subsystems B_{l} or C_{l}, so the claim follows. In light of the previous discussion of B_{l} and C_{l}, we may assume I = I_{1}. We conclude by exhibiting a root in Π̆_{I}^{(BC)_{l}}, which is not in B_{l}. Indeed, it is easy to check that 2α_{1} is a positive integral linear combination of simple roots, with a nonzero α_{1} component.

D_{l} (l ≥ 4): Assume first that I = I_{i} = Δ − {α_{i}} for some 2 ≤ i ≤ l − 2. Define J_{1} = {1, …, i − 1}, J_{2} = {i + 1, …, l}. I_{i} = J_{1} ∪ J_{2} and Γ_{J_{1}}, Γ_{J_{2}} are the connected components of Γ_{I}. Γ_{J_{1}}, Γ_{J_{2}} are Dynkin diagrams of root systems of type A_{i−1} and D_{l−i} correspondingly, so Φ_{I} is of type A_{i−1} × D_{l−i}. From the table above we deduce |Π̆_{I}| = |Π|−(|Π_{J_{1}}|+|Π_{J_{2}}|) = (l^{2}−l)−(\binom{(i−1)+1}{2}+(l−i)^{2}−(l−i)) = \frac{1}{2}i(4l − 3i − 1) ≥ 2l.

Assume l ≥ 5. If i = l − 1 or l then Φ_{I_{i}} is of type A_{l−1} and |Π̆_{I}| = (l^{2} − l) − \binom{l}{2} = \frac{1}{2}l(l − 1) ≥ 2l.

We summarize: For l > 4, |Π̆_{I}| ≤ 2l − 1 implies I = I_{1} (and |Π̆_{I_{1}}| = 2l − 2). For l = 4 the possibilities I = I_{3} and I = I_{4} also occur.

E_{6}: Deleting α_{1} or α_{6} leaves D_{5}, and |Π̆_{I}| = 14.

Deleting α_{2} leaves A_{5}. |Π̆_{I}| = 21.

Deleting α_{3} or α_{5} leaves A_{1} × A_{4}. |Π̆_{I}| = 25.
Deleting α_{4} leaves A_{2} × A_{1} × A_{2}, and |Π̆_{I}| = 29.
In any case |Π̆_{I}| > 12 = 2l.
E_{7}: \begin{matrix}|Π̆_{I_{1}}| = 33 & |Π̆_{I_{2}}| = 42 & |Π̆_{I_{3}}| = 47 \\ |Π̆_{I_{4}}| = 53 & |Π̆_{I_{5}}| = 50 & |Π̆_{I_{6}}| = 42 \\ |Π̆_{I_{7}}| = 27 \end{matrix}

In any case |Π̆_{I}| > 14 = 2l.

E_{8}: |Π̆_{I_{1}}| = 78 |Π̆_{I_{2}}| = 92 |Π̆_{I_{3}}| = 98

|Π̆_{I_{4}}| = 106 |Π̆_{I_{5}}| = 104 |Π̆_{I_{6}}| = 97
|Π̆_{I_{7}}| = 83 |Π̆_{I_{8}}| = 57
In any case |Π̆_{I}| > 16 = 2l.
F_{4}: \begin{matrix}|Π̆_{I_{1}}| = 15 & |Π̆_{I_{2}}| = 20 \\ |Π̆_{I_{3}}| = 20 & |Π̆_{I_{4}}| = 15 \end{matrix}
In any case |Π̆_{I}| > 8 = 2l

G_{2}: |Π̆_{I_{1}}| = |Π̆_{I_{2}}| = 5 > 4 = 2l

q.e.d.

Note that in all cases above |Π̆_{I}| ≥ 2l − 2. We deduce that if for some I, |Π̆_{I}| < 2l − 2 than Φ is of type A_{l}.

Lemma A.2. Let Φ be a root system of type A_{l}. Let Δ be the set of simple roots, and I a subset of Δ which is not equal to Δ − {α_{1}} or Δ − {α_{l}}. We then have the following:

• |Π̆_{I}| ≥ l + 1.

• Assume l ≥ 3. If |Π̆_{I}| = l + 1 then l = 3 and I = Δ − {α_{2}}.

Proof. We prove the first statement first. Denote I_{i} = Δ − {α_{i}}. |Π̆_{I_{1}}| = |Π̆_{I_{l}}| = l, hence it is enough to proof the Lemma assuming I = I_{i} for some 2 ≤ i ≤ l − 1, because for any other I, I ⊊ I_{i} for some i, and then |Π̆_{I}| > |Π̆_{I_{i}}|. Assuming I = I_{i} we can assume l ≥ 3.

Define J_{1} = {1, …, i−1}, J_{2} = {i+1, …, l}. I_{i} = J_{1}∪J_{2} and Γ_{J_{1}}, Γ_{J_{2}} are the connected components of Γ_{I}. Γ_{J_{1}}, Γ_{J_{2}} are Dynkin diagrams of root systems of type A_{i−1} and A_{l−i} correspondingly, so Φ_{I} is of type A_{i−1} × A_{l−i}. From the table above we deduce

|Π̆_{I}| = |Π| − (|Π_{J_{1}}| + |Π_{J_{2}}|)

= \binom{l + 1}{2} − \binom{(i − 1) + 1}{2} − \binom{(l − i) + 1}{2}

= i(l + 1 − i).

Assuming l ≥ 3, 2 ≤ i ≤ l − 1 we obtain

|Π̆_{I_{i}}| = i(l + 1 − i) ≥ 2(l − 1) ≥ l + 1. (2)

For proving the second statement observe first that for I = Δ − {α_{1}, α_{l}}, |Π̆_{I}| = 2l − 1 > l + 1. Therefore, it is enough to assume that I = I_{i} for some 2 ≤ i ≤ l − 1, because for any other I, I ⊊ I_{i} for some 2 ≤ i ≤ l − 1, and then |Π̆_{I}| > |Π̆_{I_{i}}|.

In order to have equality in equation (2), one must have 2(l − 1) = l + 1, which implies l = 3, and i(4 − i) = i(l + 1 − i) = 2(l − 1) = 4, which implies i = 2. q.e.d.

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