NOTE ON HIRZEBRUCH’S PROPORTIONALITY PRINCIPLE

Toshiyuki Kobayashi^{∗} and Kaoru Ono

Abstract. A θ-stable homogeneous space G/H is introduced with the associated Riemannian space of compact type G_{U}/H_{U}. The equation among the characteristic classes over Γ\G/H inherits from the corresponding one over G_{U}/H_{U}. As an application we also obtain a certain necessary condition for the existence of a uniform lattice in a pseudo-Riemannian homogeneous space.

1. Introduction

In [8], Hirzebruch showed

Fact(Hirzebruch’s proportionality principle). Let D be a bounded Hermitian symmetric domain, Γ a torsionless discrete cocompact subgroup of the automorphism group Aut(D) of D, and M the compact Hermitian symmetric space dual to D. Then there is a real number A = A(Γ) such that c^{α}(Γ\D)[Γ\D] = Ac^{α}(M)[M] for any c^{α}, where α = (α_{1}, …, α_{k}) is a multi-index and c^{α} = c_{1}^{α_{1}} ∪ ⋯ ∪ c_{k}^{α_{k}} is a monomial of Chern classes.

The purpose of this note is to clarify this principle by eliminating unnecessary conditions. Our strategy for comparing characteristic classes of two manifolds is to use the following restriction maps:

O (⋀^{p} T M_{ℂ})

↙ rest.  ↘ rest.
A (⋀^{p}_{ℝ}T M_{1})  A (⋀^{p}_{ℝ}T M_{2}) ,

where M_{1}, M_{2} are real forms of a complex manifold M_{ℂ}. Let us explain the idea briefly in the above case. We shall take a common complexification of M_{1} = M and M_{2} = D, a covering manifold of Γ\D, which we look upon as real manifolds by forgetting the original complex structures. The groups G_{U} and G acting on M_{1} and M_{2} respectively have the common complexification G_{ℂ}, which acts on M_{ℂ}. As G_{U} is compact and acts on M_{1} transitively, the above diagram induces a homomorphism on the cohomology level: H^{∗}(M; ℂ) → H^{∗}(Γ\D; ℂ) (see Proposition(3.9)).

∗Partially supported by Grant-in-Aid for Scientific Research (No.63740074).

2 TOSHIYUKI KOBAYASHI^{∗} AND KAORU ONO

This enables us to treat non-Riemannian cases and non-complex cases as well as Hermitian cases. In this paper, the setting is generalized as follows:

Hermitian symmetric spaces ⇒ θ-stable homogeneous spaces
tangent bundles ⇒ homogeneous vector bundles.
characteristic numbers ⇒ characteristic classes

Here, a θ-stable homogeneous space (see §3 for definition) is a wide class of homogeneous spaces of reductive type, containing the cases where the isotropy subgroup is the group fixed by an automorphism of finite order (eg. semisimple symmetric spaces), compact (homogeneous Riemannian spaces), or a Levi part of a parabolic subgroup (semisimple orbits in a reductive Lie algebra), etc.

Formulation and our main theorem are stated in §4, asserting that equations among characteristic classes (ℝ-coefficient) of a homogeneous vector bundle over a θ-stable homogeneous space inherit from those of an associated Riemannian space of compact type. Our approach is elementary alike Weyl’s unitary trick or Flensted-Jensen duality in representation theory (see [6]), and the results lead to interesting corollaries:

Corollary 1. Let X be a Riemannian manifold of constant curvature. Then all the Pontrjagin class vanishes in H^{∗}(X; ℝ).

Corollary 2. Let G be a semisimple Lie group contained in a connected complexified Lie group G_{ℂ}, Γ be any discrete subgroup of G_{ℂ} acting on G_{ℂ}/G freely and properly discontinuously. Then all the Pontrjagin class of Γ\G_{ℂ}/G vanishes in H^{∗}(Γ\G_{ℂ}/G; ℝ).

Corollary 3. Let G/H be a (not necessarily Riemannian) semisimple symmetric space and G_{U}/H_{U} the associated Riemannian symmetric space of compact type. Let Γ be any discrete subgroup of G acting on G/H freely and properly discontinuously. If ∑ a_{α}p^{α}(G_{U}/H_{U}) = 0 in H^{∗}(G_{U}/H_{U}; ℝ), then ∑ a_{α}p^{α}(Γ\G/H) = 0 in H^{∗}(Γ\G/H; ℝ). Here p^{α} denotes a monomial of Pontrjagin classes. Furthermore, if H is connected, the above result holds when we replace p^{α} by a monomial of Pontrjagin classes and the Euler class.

Corollary 4. Let H be the centralizer of a toral subgroup of a connected semisimple Lie group G, and G_{U}/H_{U} an associated Riemannian space of compact type (generalized flag variety). Then there is an embedding G/H ↪ G_{U}/H_{U}, through which G/H carries a G-invariant complex structure induced from a G_{U}-invariant complex structure on G_{U}/H_{U}. Let Γ be any discrete subgroup of G acting on G/H freely and properly discontinuously. If ∑ a_{α}c^{α}(G_{U}/H_{U}) = 0 in H^{∗}(G_{U}/H_{U}; ℝ), then ∑ a_{α}c^{α}(Γ\G/H) = 0 in H^{∗}(Γ\G/H; ℝ). Here c^{α} denotes a monomial of Chern classes.

Corollary 1 can also be deduced from the following Fact(1.1)([16]). If X is a Riemannian manifold of constant curvature, then X ×S^{1} admits a flat affine connection.

The proof of this fact is not given in [16], so we show it for the sake of completeness (see Appendix B).

Note that H is not necessarily compact in our setting. G/H is a bounded Hermitian symmetric domain in Corollary 4 if and only if Ad_{G}(H) is a maximal compact subgroup of the adjoint group Int(𝔤) ≡ Ad_{G}(G).

If Γ is a uniform lattice in G/H and H is connected, the converse statement of Corollary 3 and Corollary 4 also holds. It is well-known that there exists a uniform lattice in G/H when H is a compact and G is linear ([2]). On the other hand, when H is noncompact, a discrete subgroup of G does not necessarily act on G/H properly discontinuously. Various aspects arise about the discrete subgroup which can act properly discontinuously on G/H: some of homogeneous spaces admit uniform lattices, others admit only finite groups. Applying our argument to Euler class, we have

Corollary 5. Let (G, H) be a linear θ-stable pair. Denote by K a maximal compact subgroup of G such that H ∩ K is also a maximal compact subgroup of H. If rankG = rankH and dim K/H ∩ K is odd, then G/H admits no uniform lattice, that is, there exists no discrete subgroup Γ of G such that Γ\G/H is a compact smooth manifold.

For example, let

G/H = SO(i + k, j + l)/SO(i, j) × SO(k, l).

Then there is no uniform lattice of G/H when three elements among i, j, k, l are odd and the other is even.

The authors are very grateful to Professor Akio Hattori for his constant stimulation and encouragement.

2. Preliminaries

In this section, we review the notion of invariant connection of reductive homogeneous space and the reduction of connections to real forms (cf. [15], [10]).

Let π: P → X be a smooth principal H-bundle. A connection on P → X is a splitting of the tangent bundle T P → P into an H-equivariant Whitney sum T P = V er(P) ⊕ Hor(P), where V er(P) = Ker(dπ: T P → T X) is the tangent bundle along fibers, and Hor(P) is so called a horizontal subbundle. The connection form α ∈ E^{1}(P, 𝔥) is defined by the composition of T P → V er(P), the first projection of the splitting T P = V er(P) ⊕ Hor(P), and V er(P) → 𝔥, the inverse of 𝔥 ∋ X ↦ X_{p}^{∗} ∈ V er(P)_{p}, where X^{∗} denotes the fundamental vector field on P. The curvature form Ω ≡ Dα is the horizontal 𝔥-valued 2-form on P given by Ω(X, Y) ≡ Dα(X, Y) =^{def} dα(prX, prY) (X, Y ∈ T P), where pr: T P → Hor(P) stands for the second projection of T P = V er(P) ⊕ Hor(P).

Let H′ be a subgroup of H, P′ → X a smooth principal H′-bundle. P′ → X is called a reduction of P → X if P = P′ ×_{H′} H. If Y is a submanifold of X and a

smooth principal H′-bundle π′: Q → Y is a reduction of π_{|Y} : P_{|Y} → Y satisfying

(2.1)  (T Q)_{p} ⊂ Ker(dπ′_{p}) ⊕ Hor(P)_{p},

for any p ∈ Q, we have a connection on Q induced from the one on P. Namely, let Hor(Q)_{p} =^{def} Hor(P)_{p} ∩ (T Q)_{p}, then the subbundle Hor(Q) of T Q determines a connection on Q → Y.

For E = P ×_{ρ}V, the vector bundle associated to a representation ρ: H → GL(V),

we have a connection induced from a connection on P. The curvature form Ω^{E} of this connection is a End(E) valued 2-form described as follows:

Ω^{E}(u, v) =^{def} [p, dρ(Ω(ũ, ṽ))],

via the identification P ×_{Ad(ρ)} End(V) = End(E). Here for x ∈ X, p ∈ P with

π(p) = x, ũ, ṽ ∈ (T P)_{p} are lifts of u, v ∈ (T X)_{x} respectively, and dρ is a Lie algebra homomorphism 𝔥 → 𝔤𝔩(V) induced from ρ.

We call a homogeneous space G/H is reductive when there exists an Ad(H)- stable vector subspace 𝔮 complementary to 𝔥 in 𝔤. For a reductive homogeneous space G/H, a connection on a principal H-bundle G → G/H is defined as follows: for g ∈ G,

Hor(G)_{g} =^{def} L_{g∗}𝔮.

This connection is called the canonical connection of the second kind on G/H in [15]. The curvature form is given by Ω_{o}(X, Y) = −[X, Y]_{|𝔥} where Z_{|𝔥} denotes the 𝔥 component of Z ∈ 𝔤 = 𝔥 + 𝔮, o is the origin corresponding to the identity element of G and X, Y ∈ 𝔮.

For any reductive homogeneous space G/H contained in its complexification G_{ℂ}/H_{ℂ}, the canonical connection of the second kind on G_{ℂ}/H_{ℂ} induces the one on G/H. In fact, 𝔤 ⊂ 𝔥 ⊕ 𝔮_{ℂ} implies the condition (2.1). Thus the principal bundle G → G/H inherits the connection from G_{ℂ}_{| G/H} → G/H.

3. θ-stable pair

In this section we introduce a notion of a θ-stable pair (G, H) and construct an algebra homomorphism between the cohomology rings of Γ\G/H and of the associated Riemannian space of compact type G_{U}/H_{U}.

Let 𝔤 be a semisimple Lie algebra defined over ℝ. We call a subalgebra 𝔥 in 𝔤 is θ-stable when there exists a Cartan involution θ of 𝔤 such that θ𝔥 = 𝔥. Then the following lemma is proved by standard arguments (see [18] Ch.1 §1).

Lemma(3.1). Let 𝔥 be a θ-stable subalgebra in 𝔤, 𝔮 the orthogonal subspace of 𝔥 in 𝔤 with respect to the Killing form. Then 𝔤 = 𝔥+𝔮 gives a direct decomposition as a 𝔥-module. Furthermore, the adjoint representation ad_{|𝔥}: 𝔥 → 𝔤𝔩(𝔤) is semisimple. Especially, 𝔥 is a reductive Lie algebra, that is, 𝔥 is decomposed into a direct sum of the center and the semisimple ideal [𝔥, 𝔥].

Example(3.2). Let 𝔤 be a semisimple Lie algebra over ℝ. The following subalgebras are θ-stable in 𝔤.

1) The centralizer (or normalizer) of a θ-stable subalgebra in 𝔤.

2) The subalgebra fixed by a linear automorphism of 𝔤 of finite order ([7] p.277).

3) A semisimple subalgebra ([14]).

Now we introduce a notion of a ‘θ-stable pair’.

Definition(3.3). Let G be a connected semisimple Lie group, H a closed subgroup of G. We call (G, H) a θ-stable pair when the following two conditions are satisfied:

a) There is a Cartan involution θ of G such that H has a polar decomposition H = (H ∩ K) exp(𝔥 ∩ 𝔭), where 𝔤 = 𝔨 + 𝔭 is the corresponding Cartan decomposition of 𝔤 and K is the connected subgroup of G with Lie algebra 𝔨.

b) The connected Lie subgroup corresponding to the Lie algebra 𝔥_{ℂ} = 𝔥 ⊗ ℂ is closed in the adjoint group Int(𝔤_{ℂ}).

When (G, H) is a θ-stable pair, we call G/H a θ-stable homogeneous space.

When G has a faithful finite dimensional representation, we call (G, H) is a linear θ-stable pair. In this case, the connected components of H are finite because K is compact.

The condition a) in the above definition implies θ𝔥 = 𝔥, and so 𝔥 is a θ-stable subalgebra in 𝔤. Conversely if H is connected, the condition a) can be replaced by the condition that 𝔥 is a θ-stable subalgebra in 𝔤.

Let (G, H) be a θ-stable pair. Then there is a closed subgroup H_{ℂ} of a connected Lie group G_{ℂ} with Lie algebras 𝔥_{ℂ} = 𝔥 ⊗ ℂ and 𝔤_{ℂ} = 𝔤 ⊗ ℂ respectively such that the inclusion 𝔤 ↪ 𝔤 ⊗ ℂ induces the following commutative diagram:

G ⟶^{ι} G_{ℂ}
∪  ∪
H ⟶^{ι} H_{ℂ},

and that

(3.4)  H_{ℂ} = ι(H) · (H_{ℂ})_{o}

(Say, choose G_{ℂ} the adjoint group Int(𝔤_{ℂ}) and put H_{ℂ} by (3.4).)

Let (G, H) be a θ-stable pair, θ a Cartan involution of 𝔤 which makes 𝔥 stable, and 𝔤 = 𝔨 + 𝔭 be the corresponding Cartan decomposition of 𝔤. Then we have a direct sum decomposition

𝔤 = 𝔥 ∩ 𝔨 + 𝔥 ∩ 𝔭 + 𝔮 ∩ 𝔨 + 𝔮 ∩ 𝔭,

as a vector space. Let G_{U} be a connected Lie subgroup of G_{ℂ} with Lie algebra 𝔤_{U} = 𝔨 + \sqrt{−1}𝔭. Set H_{U} = H_{ℂ} ∩ G_{U}. Then H_{U}, G_{U} are compact real forms of H_{ℂ}, G_{ℂ} respectively, and we have a natural map

G/H ⟶_{covering} ι(G)/H_{ℂ} ∩ ι(G) ↪_{complexify} G_{ℂ}/H_{ℂ} ↩_{complexify} G_{U}/H_{U}.

We call G_{U}/H_{U} (resp. G_{ℂ}/H_{ℂ}) an associated Riemannian space of compact type (resp. a complexification) for a given θ-stable homogeneous space G/H.

Remark(3.5). Each connected component of H_{U} meets H_{ℂ}. Moreover the cohomology ring H^{∗}(G_{U}/H_{U}; ℝ) is independent of the choice of the above complex Lie group G_{ℂ}. This notice is sometimes convenient for actual calculation.

Example(3.6). Let G be a connected semisimple Lie group. (G, H) is a θ-stable pair in either of the following cases:

1) H is the centralizer (or normalizer) in G of a θ-stable subalgebra 𝔱. When 𝔱 is a θ-stable abelian subspace, an associated Riemannian space of compact type G_{U}/H_{U} is called a (generalized) flag variety (cf. Lemma(6.1)).

2) H is an open subgroup in the group of the fixed points of an automorphism σ of finite order of G. When σ is involutive, the homogeneous space G/H is called a semisimple symmetric space.

3) H is a semisimple connected subgroup in G ([19] guarantees that H_{ℂ} is closed in G_{ℂ}).

4) H is compact.

Let (G, H) be a θ-stable pair. Then G/H, G_{U}/H_{U} and G_{ℂ}/H_{ℂ} are reductive homogeneous spaces in the sense of §2 with complementary subspaces 𝔮, 𝔮_{U} = 𝔮 ∩ 𝔨 + \sqrt{−1}𝔮 ∩ 𝔭 and 𝔮 ⊗ ℂ respectively. Therefore invariant forms are identified with the invariant elements in the exterior algebra of the cotangent space at the origins. Namely,

E^{∗}(G/H; ℝ)^{G} ≃ (⋀ 𝔮^{∗})^{H} ,
E^{∗}(G_{U}/H_{U}; ℝ)^{G_{U}} ≃ (⋀ 𝔮_{U}^{∗})^{H_{U}} ,

and

E^{∗}(G_{ℂ}/H_{ℂ}; ℂ)^{G_{ℂ}} ≃ (⋀ 𝔮_{ℂ}^{∗})^{H_{ℂ}} .

Define a linear map d: (⋀ 𝔮^{∗})^{H} → (⋀ 𝔮^{∗})^{H} by

(dh)(X_{1}, …, X_{n}) = ∑_{i<j}(−1)^{i+j}h([X_{i}, X_{j}]_{|𝔮}, X_{1}, …^{^{i} ∨ ^{j} ∨}, X_{n}), (X_{i} ∈ 𝔮),

d: (⋀ 𝔮_{U}^{∗})^{H_{U}} → (⋀ 𝔮_{U}^{∗})^{H_{U}} by

(dh)(X_{1}, …, X_{n}) = ∑_{i<j}(−1)^{i+j}h([X_{i}, X_{j}]_{|𝔮_{U}} , X_{1}, …^{^{i} ∨ ^{j} ∨}, X_{n}), (X_{i} ∈ 𝔮_{U}),

and d: (⋀ 𝔮_{ℂ}^{∗})^{H_{ℂ}} → (⋀ 𝔮_{ℂ}^{∗})^{H_{ℂ}} by

(dh)(X_{1}, …, X_{n}) = ∑_{i<j}(−1)^{i+j}h([X_{i}, X_{j}]_{|𝔮_{ℂ}}, X_{1}, …^{^{i} ∨ ^{j} ∨}, X_{n}), (X_{i} ∈ 𝔮_{ℂ}).

Then it is easy to see that these d’s correspond to the exterior derivatives under the above isomorphisms. Finally, define a linear isomorphism ϕ: 𝔮 → 𝔮_{U} by

ϕ(X + Y) = X + √\overline{−1}Y (X ∈ 𝔮 ∩ 𝔨, Y ∈ 𝔮 ∩ 𝔭).

Then we have the following

Lemma(3.7). With notation as above, let ξ and ξ_{U} be invariant differential forms on G/H and G_{U}/H_{U} respectively. Assume ξ and ξ_{U} satisfy the following condition:

ξ_{U}(ϕ(X_{1}), …, ϕ(X_{a}), ϕ(Y_{1}), … ϕ(Y_{b})) (3.8)

= (√\overline{−1})^{b}ξ(X_{1}, …, X_{a}, Y_{1}, … Y_{b}),

for any X_{i} ∈ 𝔮 ∩ 𝔨, Y_{j} ∈ 𝔮 ∩ 𝔭.

Then if ξ_{U} is an exact form, there is a G-invariant form η on G/H such that ξ = dη. If ξ_{U} is a closed form, ξ is also closed.

Proof. The natural isomorphism (see Remark(3.5))

(⋀ 𝔮^{∗})^{H} ⊗ ℂ ≃ (⋀ 𝔮_{ℂ}^{∗})^{H_{ℂ}} ≃ (⋀ 𝔮_{U}^{∗})^{H_{U}} ⊗ ℂ

induces

E^{∗}(G/H; ℝ)^{G} ⊗ ℂ ≃ E^{∗}(G_{ℂ}/H_{ℂ}; ℂ)^{G_{ℂ}} ≃ E^{∗}(G_{U}/H_{U}; ℝ)^{G_{U}} ⊗ ℂ.

The assumption (3.8) imply that ξ and ξ_{U} are the same images in the middle term. Suppose ξ_{U} is an exact form. Then there exists a G_{U}-invariant form η_{U} on G_{U}/H_{U} such that dη_{U} = ξ_{U}, because ξ_{U} is G_{U}-invariant and G_{U} is compact. Let η ∈ E^{∗}(G/H; ℝ)^{G} ⊗ ℂ be the corresponding element of η_{U} under the above isomorphism. Then we have ξ = dη. The second claim is similar and easy. Thus the lemma is proved. □

Proposition(3.9). With notation as above, let Γ be a discrete subgroup of G acting on G/H freely and properly discontinuously. Then

E^{∗}(G_{U}/H_{U}; ℝ)^{G_{U}} ⊗ ℂ →^{∼} E^{∗}(G/H; ℝ)^{G} ⊗ ℂ ↪ E^{∗}(Γ\G/H; ℝ) ⊗ ℂ

induces a ℂ-algebra homomorphism

Υ: H^{∗}(G_{U}/H_{U}; ℂ) → H^{∗}(Γ\G/H; ℂ).

If Γ\G/H is compact and H is connected, then Υ is injective.

Proof. The first claim is an immediate consequence of the preceding lemma. If Γ\G/H is compact and H is connected, G_{U}/H_{U} and G/H have G and G_{U} invariant orientation respectively. Therefore Υ is injective from the Poincaré duality. □

Remark(3.10). For the injectivity of Υ, the assumption of connectedness of H can be replaced by Ad(H)_{|𝔮} ⊂ SL(𝔮), which means that G/H admits a G-invariant orientation. But in general Υ is neither injective nor surjective. As the proof of our theorem in §5 shows, Υ transfers the characteristic classes on G_{U}/H_{U} to the corresponding ones on Γ\G/H.

Remark(3.11). It is easy to see that G_{U}/H_{U} is a compact symmetric space if and only if (G, H) is a semisimple symmetric pair. It is a well known fact due to É.Cartan that H^{∗}(G_{U}/H_{U}; ℂ) ≃ E^{∗}(G_{U}/H_{U}; ℝ)^{G_{U}} ⊗ ℂ if G_{U}/H_{U} is a symmetric space.

4. Statement of results

Let (G, H) be a θ-stable pair. Retain notations in §3. Let ρ: H → GL(V, ℝ), ρ_{U}: H_{U} → GL(V_{U}, ℝ) be finite dimensional representations. We call ρ and ρ_{U} has the same complexification when there are a complex vector space V_{ℂ}, a representa-

tion ρ_{ℂ}: H_{ℂ} → GL(V_{ℂ}, ℂ) and isomorphisms ψ: V ⊗ℂ →^{∼} V_{ℂ} and ψ_{U}: V_{U} ⊗ℂ →^{∼} V_{ℂ}

such that the following diagram commutes.

\begin{matrix}H & →^{ι}  H_{ℂ}  ↩ & H_{U} \\ ^{ρ}↓ & ↓^{ρ_{ℂ}} & ↓^{ρ_{U}} \\ GL(V, ℝ) & ↪_{ψ_{♯}} GL(V_{ℂ}, ℂ) ↩_{ψ_{U}_{♯}} & GL(V_{U}, ℝ). \end{matrix}

Now we are ready to state our main theorem.

Theorem. Let (G, H) be a θ-stable pair, G_{U}/H_{U} an associated Riemannian space of compact type, G_{ℂ}/H_{ℂ} a complexification. Let Γ be any discrete subgroup of G acting on G/H freely and properly discontinuously from the left.

1) Let ρ: H → GL(V, ℝ), ρ_{U}: H_{U} → GL(V_{U}, ℝ) be finite dimensional representations with the same complexification. Set ^{Γ}E =^{def} Γ\G ×_{ρ} V, E_{U} =^{def} G_{U} ×_{ρ_{U}} V_{U}.

If there is a relation ∑ a_{α}p^{α}(E_{U}) = 0 in H^{∗}(G_{U}/H_{U}; ℝ), then the equation ∑ a_{α}p^{α}(^{Γ}E) = 0 in H^{∗}(Γ\G/H; ℝ) holds. Here α = (α_{1}, …, α_{k}) is a multi-index and p^{α} = p_{1}^{α_{1}} ∪ ⋯ ∪ p_{k}^{α_{k}} is a monomial of Pontrjagin classes.

2) Let V be a finite dimensional vector space over ℂ, ρ: H_{ℂ} → GL(V, ℂ) a representation of H_{ℂ}. Set ^{Γ}E =^{def} Γ\G ×_{ρ_{|H}} V, E_{U} =^{def} G_{U} ×_{ρ_{|H_{U}}} V.

If there is a relation ∑ a_{α}c^{α}(E_{U}) = 0 in H^{∗}(G_{U}/H_{U}; ℝ), then the equation ∑ a_{α}c^{α}(^{Γ}E) = 0 in H^{∗}(Γ\G/H; ℝ) holds.

3) If Γ\G/H is compact and H is connected, the converse statement of 1) and

2) also holds.

Example(4.1). Let D = SO_{o}(n, 2)/SO(n) × SO(2) be the complex quadric, Γ a discrete cocompact subgroup of the automorphism group Aut(D) of D, and M the compact Hermitian symmetric space dual to D. Then c_{j}(Γ\D) ≠ 0 for any j with 1 ≤ j ≤ n = dim_{ℂ} D because we know that the corresponding result for M holds.

Example(4.2). The total Chern class c(ℂP^{n}) = 1 + c_{1}(ℂP^{n}) + ⋯ + c_{n}(ℂP^{n}) of the complex projective space ℂP^{n} is given by

c(ℂP^{n}) ≡ (1 + x)^{n+1} mod x^{n+1},

where x is the first Chern class of the hyperplane section bundle. Let X(p, q) = U(p + 1, q)/U(1) × U(p, q) (p + q = n). Then X(n, 0) = ℂP^{n} and X(0, n) be the dual Hermitian symmetric domain of noncompact type (ref. [7] for the terminology). Let Γ be a discrete subgroup of U(p + 1, q) acting on X(p, q) freely and properly discontinuously. Then we have

c_{j}(Γ\X(p, q)) = (∏_{l=0}^{j−1} \frac{n + 1 − l}{n + 1} ) c_{1}(Γ\X(p, q))^{j} (1 ≤ j ≤ n).

If Γ is a uniform lattice, c_{j}(Γ\X(p, q)) ≠ 0 for any j with 1 ≤ j ≤ n. It can be proved that there exists a uniform lattice for X(0, n), X(n, 0) (Riemannian case) and X(1, 2r), whereas any discrete subgroup acting properly discontinuously on X(p, q) with p ≥ q is finite.

Remark(4.3). We do not require that Γ is cocompact, so Theorem holds even when Γ = 1.

5. Proof of Theorem

Let (G, H) be a θ-stable pair. We retain notations in §2 and §3. We shall first prove the part (2) of Theorem, namely, the case of Chern classes. As we prepared in §2, the curvature forms Ω and Ω_{U} of G → G/H and G_{U} → G_{U}/H_{U} are given by

Ω_{o}(X, Y) = −[X, Y]_{|𝔥} (X, Y ∈ 𝔮),
Ω_{U}_{o}(X_{U}, Y_{U}) = −[X_{U}, Y_{U}]_{|𝔥_{U}} (X_{U}, Y_{U} ∈ 𝔮_{U}),

where Z_{|𝔥}, Z_{|𝔥_{U}} and Z_{|𝔥_{ℂ}} denote the second projections with respect to the decompositions 𝔤 = 𝔥 + 𝔮, 𝔤_{U} = 𝔥_{U} + 𝔮_{U} and 𝔤_{ℂ} = 𝔥_{ℂ} + 𝔮_{ℂ} respectively. So the curvature forms Ω_{o}^{E}, Ω_{o}^{E_{U}} of homogeneous vector bundles E → G/H and E_{U} → G_{U}/H_{U} are given by

Ω_{o}^{E}(X, Y) = −ρ([X, Y]_{|𝔥}) ∈ 𝔤𝔩(V),
Ω_{o}^{E_{U}} (X_{U}, Y_{U}) = −ρ([X_{U}, Y_{U}]_{|𝔥_{U}}) ∈ 𝔤𝔩(V),

where we identify 𝔤𝔩(V) with End(E)_{o} and End(E_{U})_{o} respectively.

Define a linear isomorphism ϕ: 𝔮 → 𝔮_{U} by

ϕ(X + Y) = X + √\overline{−1}Y (X ∈ 𝔮 ∩ 𝔨, Y ∈ 𝔮 ∩ 𝔭).

Since [X, Y]_{|𝔥} = [X, Y]_{|𝔥_{ℂ}} and [ϕ(X), ϕ(Y)]_{|𝔥_{U}} = [ϕ(X), ϕ(Y)]_{|𝔥_{ℂ}}, we have

[ϕ(X), ϕ(Y)]_{|𝔥_{U}} = (√\overline{−1})^{δ(X)+δ(Y)}[X, Y]_{|𝔥},

and so

Ω_{o}^{E_{U}} (ϕ(X), ϕ(Y)) = (√\overline{−1})^{δ(X)+δ(Y)}Ω_{o}^{E}(X, Y),

where X and Y are elements of 𝔮 ∩ 𝔨 or 𝔮 ∩ 𝔭 and we set δ(W) = 0 if W ∈ 𝔮 ∩ 𝔨, δ(W) = 1 if W ∈ 𝔮 ∩ 𝔭.

By Chern-Weil theory ([5],[10]), characteristic classes are represented by using curvatures. Namely, there is a ℂ-algebra homomorphism

w: Inv(L) ⟶ H^{∗}(BL; ℂ),

for a Lie group L, where Inv(L) denotes the ring of ℂ-valued invariant polynomials of the Lie algebra 𝔩 of L, and BL denotes the classifying space of a Lie group L. When L is a complex Lie group, we denote by Inv_{ℂ}(L) the subring of Inv(L) consisting of holomorphic polynomials. The Chern classes are considered as elements of H^{∗}(BGL(n, ℂ); ℝ). For f ∈ Inv(L) and a principal L bundle P → X, the characteristic class is represented by the differential form on X corresponding to the tensorial (i.e. L-invariant and horizontal) form f(Ω, …, Ω) on P where f is identified with its polarized multilinear form. If L is compact and connected, the homomorphism w is an isomorphism.

For a complex vector bundle F → X of rank n, the k-th Chern form c_{k} of F is represented by f_{k}(Ω^{F} , …, Ω^{F}) on X, where f_{k} is the homogeneous part of degree k in t of the real valued polynomial

f̃(A)(t) = det (I − \frac{t}{2π\sqrt{−1}}A) (A ∈ 𝔲(n)).

This formula is also applicable for GL(n, ℂ) vector bundles and gives a representative of the total Chern class via the identification:

Inv_{ℂ}(GL(n, ℂ)) ≃ Inv(U(n)) ≃ H^{∗}(BU(n); ℂ) ≃ H^{∗}(BGL(n, ℂ); ℂ).

Considering f as a multilinear form as before, we have,

f(Ω_{o}^{E_{U}} , …, Ω_{o}^{E_{U}})(ϕ(X_{1}), …, ϕ(X_{a}), ϕ(Y_{1}), … ϕ(Y_{b})) (5.1)

= (√\overline{−1})^{b}f(Ω_{o}^{E}, …, Ω_{o}^{E})(X_{1}, …, X_{a}, Y_{1}, … Y_{b}),

where X_{i} ∈ 𝔮 ∩ 𝔨, Y_{j} ∈ 𝔮 ∩ 𝔭, and f ∈ Inv(GL(n, ℂ)).

If [(wf)(E_{U})] = 0 in H^{∗}(G_{U}/H_{U}; ℝ), there exists a G-invariant form η on G/H such that dη = (wf)(E) owing to Lemma(3.7). Since both f(Ω_{o}^{E}, …, Ω_{o}^{E}) and η are locally invariant (that is, their pullbacks are G-invariant on G/H), the characteristic class [(wf)(^{Γ}E)] = 0 in H^{∗}(Γ\G/H; ℂ).

Applying this to the case that [wf] is the image of ∑ a_{α}c^{α} under the homomorphism

H^{∗}(BGL(n, ℂ); ℝ) → H^{∗}(BGL(n, ℝ); ℝ),

we get (2) in Theorem.

As Pontrjagin classes of a real vector bundle F are determined by Chern classes of its complexification F_{ℂ} = F ⊗ \underline{ℂ}, the comparison of the curvatures of E ⊗ \underline{ℂ} and E_{U} ⊗ \underline{ℂ} results in the part (1) of Theorem. The part (3) is derived from the last statement of Proposition(3.9).

6. Proof of Corollaries

Proof of Corollary 1.

A Riemannian manifold of constant negative (otherwise the statement is obvious) curvature is a quotient of the n-dimensional hyperbolic space form ℍ^{n} = SO_{o}(n, 1)/SO(n) by a torsion free discrete group of isometries. Thus from the knowledge of G_{U}/H_{U} = S^{n}, we obtain Corollary 1.

Proof of Corollary 2.

The associated Riemannian space of compact type for the θ-stable pair (G_{ℂ}, G) is G_{U} × G_{U}/ΔG_{U}, where G_{U} = {(g, g) ∈ G_{U} × G_{U}; g ∈ G_{U}}. Since this space is diffeomorphic to a group manifold G_{U}, all the Pontrjagin class vanishes. Now, Corollary 2 is deduced from Theorem.

Proof of Corollary 3.

Corollary 3 holds when (G, H) is a θ-stable pair in general. Corollary 3 is almost proved by applying 1) in Theorem to the adjoint representations Ad_{|H}: H → GL(𝔮) and Ad_{|H_{U}} : H_{U} → GL(𝔮_{U}). We only have to take account of Euler classes.

As G/H admits an indefinite metric by the Killing form restricted to 𝔮, the structure group of the tangent bundle can be reduced to SO_{o}(p, q) for some p, q ∈ ℕ, where p + q = dim 𝔮. To deal with Euler classes, we treat the complexified vector bundles again. From the fact that the rings of invariant polynomials of SO_{o}(p, q), SO(p + q) and the ring of the invariant holomorphic polynomials of SO(p + q, ℂ) are isomorphic, there is P ∈ Inv_{ℂ}(SO(p + q, ℂ)) such that

P_{|𝔰𝔬(p+q)} = P^{e},

where P^{e} ∈ Inv(SO(p + q)) is the invariant polynomial corresponding to the Euler class. Therefore in this case, we can calculate the Euler class by using P and the SO_{o}(p, q)-connection on G/H.

Proof of Corollary 4.

Let G be a connected semisimple Lie group and G_{ℂ} a complex Lie group with complexified Lie algebra of G. Let θ be a Cartan involution of 𝔤, and 𝔤 = 𝔨 + 𝔭 the corresponding Cartan decomposition. Let G_{U} be the compact real form of G_{ℂ} whose Lie algebra is given by 𝔤_{U} = 𝔨 + \sqrt{−1}𝔭. Fix an abelian subspace 𝔱 (≠ 0) in 𝔨. Let H, H_{U}, and H_{ℂ} be the centralizers of 𝔱 in G, G_{U}, and G_{ℂ} respectively. Fix a parabolic subgroup R of G_{ℂ} with Levi part H_{ℂ}. Then we have a generalized Borel embedding^{1} :

Lemma(6.1) (Folklore). With notation as above, both G_{U}/H_{U} and G/H are simply connected, and especially H_{U} and H are connected. Furthermore, there exists a G_{U}-invariant complex structure on a compact manifold G_{U}/H_{U} = G_{ℂ}/R, and G/H is realized in an open G-orbit of the identity coset of G_{ℂ}/R.

1[Griffiths-Schmid] (Acta Math. 1969) treated when H is compact and called G/H dual manifolds of Kähler C-space. [Shapiro] (Comment. Math. Helv. 1971) treated when G/H is a semisimple symmetric space which was classified on the Lie algebra level in [1].

We shall give a proof of this fact in Appendix A for the reader’s convenience.

It is known that there is a Levi decomposition R = H_{ℂ} · N, where N is the unipotent radical of R. As N is a normal subgroup in R, any representation ρ_{ℂ}: H_{ℂ} → GL(V, ℂ) is extended to R by letting N act on V trivially. We also denote this extension by ρ for brevity.

As we define a complex structure on G_{U}/H_{U} by the isomorphism G_{U}/H_{U} ≃ G_{ℂ}/R, the holomorphic tangent bundle of G_{U}/H_{U} is given by G_{ℂ} ×_{Ad_{|R}}

𝔤_{ℂ}/𝔯 ≃ G_{U} ×_{Ad_{|H_{U}}} 𝔤_{ℂ}/𝔯, and the holomorphic tangent bundle of G/H is given by

(G_{ℂ} ×_{Ad_{|R}} 𝔤_{ℂ}/𝔯)_{|G/H} ≃ G ×_{Ad_{|H}} 𝔤_{ℂ}/𝔯.

On the other hand, we have the following commutative diagrams.

𝔯 ⟶^{ad} 𝔤𝔩(𝔤_{ℂ}/𝔯)
∪  ↑^{≃}
𝔥 ⟶^{ad} 𝔤𝔩(𝔮)

via the isomorphism 𝔮 ≃ 𝔤_{ℂ}/𝔯 induced from the inclusion 𝔮 ↪ 𝔤_{ℂ}, and

𝔯 ⟶^{ad} 𝔤𝔩(𝔤_{ℂ}/𝔯)
∪  ↑^{≃}
𝔥_{U} ⟶^{ad} 𝔤𝔩(𝔮_{U})

via the isomorphism 𝔮_{U} ≃ 𝔤_{ℂ}/𝔯 induced from the inclusion 𝔮_{U} ↪ 𝔤_{ℂ}. Therefore Corollary 3 is reduced to 2) in Theorem.

Proof of Corollary 5.

As G has a faithful finite dimensional representation, the connected components of H is finite. Therefore the non-existence of a uniform lattice in G/H is derived from the case where H is connected. When H is connected, H_{U} is also connected and of maximal rank in G_{U} owing to (3.4) and the Euler number χ(G_{U}/H_{U}) does not vanish ([9]). This is also deduced from Hirsch’s formula^{2} of the Poincaré polynomial. On the other hand, χ(Γ\G/H) = 0 because the tangent bundle T (Γ\G/H) splits according to the H ∩ K module decomposition 𝔮 = 𝔮 ∩ 𝔨 + 𝔮 ∩ 𝔭 and because dim_{ℝ} 𝔮 ∩ 𝔨 is odd.

Remark (6.2). When rankG = rankH, it is easy to see that dim_{ℝ} 𝔮 is even. Therefore dim K/H ∩ K = dim_{ℝ} 𝔮 ∩ 𝔨 is odd if and only if dim_{ℝ} 𝔮 ∩ 𝔭 is odd.

2H.Cartan, J.-L.Koszul, and J.Leray, Colloque de Topologie, Bruxelles, 1950

7. Appendix

A. Proof of Lemma (6.1).

From definition we have

(A.1)  G ∩ H_{ℂ} = H, G_{U} ∩ H_{ℂ} = H_{U}.

Since 𝔥_{ℂ} = (𝔥_{ℂ} ∩ 𝔨_{ℂ}) + (𝔥_{ℂ} ∩ 𝔭_{ℂ}), both H_{U} and H are real forms of H_{ℂ}. Then H and H_{U} are connected because K ∩ H_{ℂ} = Z_{K}(𝔱) and G_{U} ∩ H_{ℂ} = Z_{G_{U}} (𝔱) are connected (see [7] Ch.7 Corollary 2.8.). As H and H_{U} contain the center of G and G_{U} respectively, G/H and G_{U}/H_{U} do not depend on the choice of coverings of G and G_{ℂ}. Thus both G/H and G_{U}/H_{U} are simply connected, and from now on we may assume that G is contained in its simply-connected complexification G_{ℂ}.

Fix a general element Z in \sqrt{−1}𝔱 so that 𝔥_{ℂ} = \{X ∈ 𝔤_{ℂ}; [Z, X] = 0\}. Then 𝔤_{ℂ} is decomposed into the negative, 0, and the positive eigenspaces of ad(Z), namely, 𝔤_{ℂ} = 𝔫^{−} +𝔥_{ℂ} +𝔫. Let R (resp. R^{−}) be a parabolic subgroup of G_{ℂ} with Lie algebra 𝔥_{ℂ} + 𝔫 (resp. 𝔥_{ℂ} + 𝔫^{−}). The natural inclusions G ⊂ G_{ℂ} ⊃ G_{U} induce

G/G ∩ R ⊂ G_{ℂ}/R ⊃ G_{U}/G_{U} ∩ R.

We will show that

(A.2)  𝔤 + (𝔥_{ℂ} + 𝔫) = 𝔤_{U} + (𝔥_{ℂ} + 𝔫) = 𝔤_{ℂ}.
(A.3)  G ∩ R = G ∩ H_{ℂ}, G_{U} ∩ R = G_{U} ∩ H_{ℂ}.

Then (A.2) implies G/G ∩ R and G_{U}/G_{U} ∩ R are open sets in G_{ℂ}/R, and since G_{U} is compact we have G_{ℂ}/R = G_{U}/G_{U} ∩ R. Using (A.3), we have G/H ⊂ G_{ℂ}/R = G_{U}/H_{U} which will complete the proof of the lemma.

Now let us show (A.2), (A.3). let τ be a conjugation of 𝔤_{ℂ} with respect to a real form 𝔤 (or 𝔤_{U}). Since Z ∈ 𝔨 = 𝔤 ∩ 𝔤_{U}, we have

τ(𝔫^{−}) = 𝔫, τ(𝔫) = 𝔫^{−},

and

τ(𝔥_{ℂ}) = 𝔥_{ℂ}.

We also denote by τ its lifting to an automorphism of a simply connected Lie group G_{ℂ}. Let X be any element of 𝔫^{−}. Then X = −τ(X) + (X + τ(X)) ∈ 𝔫 + 𝔤 (or ∈ 𝔫 + 𝔤_{U}), which shows (A.2). Let g be any element of G ∩ R (or G_{U} ∩ R). Acting τ to the equation g R g^{−1} = R, we get g R^{−} g^{−1} = R^{−}. Because R and R^{−} are self-normalizing, g ∈ R ∩ R^{−} = H_{ℂ}, proving (A.3).

Remark (A.4). With notation as above, G/H is a semisimple symmetric space if and only if the nilradical 𝔫 is abelian, and a bounded Hermitian symmetric domain if and only if H is maximal compact in G. These symmetric spaces are called ‘\frac{1}{2}-Kähler’ and ‘Kähler’ respectively in Berger’s classification ([1]).

B. Proof of Fact (1.1).

The simply connected hyperbolic space form ℍ^{n} can be embedded into ℝ^{n,1}, which is ℝ^{n+1} = {(x_{0}, …, x_{n}); x_{i} ∈ ℝ} equipped with the indefinite metric dx_{0}^{2} + ⋯ + dx_{n−1}^{2} − dx_{n}^{2}. As the isometry group of ℍ^{n} is a subgroup of index 2 in O(n, 1), M can be written as Γ\ℍ^{n} where Γ is a discrete subgroup of O(n, 1). For any fixed r > 0 (r ≠ 1), we define φ = φ_{r}: ℝ^{n,1} → ℝ^{n,1} by the scalar multiplication of r. M × S^{1} is diffeomorphic to Γ × ⟨φ⟩\ℝ_{+}^{n,1}, where ℝ_{+}^{n,1} is {(x_{0}, …, x_{n}); x_{n} > 0} and ⟨φ⟩ is the group generated by φ in GL(n + 1, ℝ). As Γ × ⟨φ⟩ is a subgroup of GL(n + 1, ℝ), the standard flat affine connection on ℝ^{n+1} is preserved under the action of Γ × ⟨φ⟩. Therefore M × S^{1} admits a flat affine connection.

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Toshiyuki Kobayashi Department of Mathematics University of Tokyo Hongo, Tokyo 113 Japan.

Kaoru Ono Mathematical Institute Tohoku University Sendai 980 Japan.