The simplicial volume of manifolds covered by H^2xH^2

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Introduction

The main purpose of this dissertation is to introduce the concept of simplicial volume of a closed manifold and to study it for locally symmetric spaces following [BK08a]; we will also present explicit calculations in the case of manifolds covered by Hnor H2×H2_{, following [Thu78], [Gro82] and [BK08b].}

The notion of simplicial volume of a manifold Mn was first introduced by Gromov in his seminal paper “Volume and bounded cohomology” in 1982; it has been widely studied since, due to the number of applications in a variety of branches of geometry.

Being defined as the `1-norm of the fundamental class of M in its n-th real homology space, the simplicial volume kM k is a purely topological invariant; when kM k > 0, it provides an obstruction to the existence of maps from a given manifold to M with arbitrarily large degree (see Lemma 4.2.3). Nevertheless, the simplicial volume can often capture geometric properties of Riemannian manifolds. For example, Gromov originally introduced this concept to study the minimal volume of manifolds, i.e.:

MinVol M := inf{vol(M, g) | | sec(M, g)| ≤ 1} ;

namely he showed that, for arbitrary manifolds, the following inequality holds (see Corollary A in §0.5 in [Gro82]):

kM k ≤ (n − 1)nn! · MinVol M .

Moreover, Gromov’s Proportionality Principle surprisingly binds the simpli-cial volume to the usual Riemannian volume (a purely geometrical notion) for closed manifolds with isometric universal coverings (see Chapter 5). Furthermore, the possible Euler numbers of n-dimensional flat vector bun-dles (i.e. admitting a set of trivialisations with constant transition functions) over M are related to kM k. This was originally observed in 1958 by Milnor in the case of bundles over surfaces (see [Mil58]) and later extended to a more general setting by Sullivan ([Sul76], [Gro82]) and Smillie ([Smi]) who proved:

kM k ≥ 2n|χ(E)|

for all n-dimensional flat vector bundles E → M (for details, see also [IT85] and [BKM12]).

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These results provide the most interesting consequences in the case of non-vanishing simplicial volume. This happens for a wide variety of manifolds, such as negatively curved closed manifolds, closed locally symmetric spaces of noncompact type, products of these examples and connected sums of these with arbitrary manifolds, in dimension at least 3 (see Chapter 5). However, kM k vanishes whenever π1(M ) is amenable or, in particular, solvable.

In Chapters 1 and 2 we will recall some standard notions that will be use-ful in the subsequent treatment. In particular, Chapter 2 will be devoted to the general theory of symmetric spaces, which is closely related to the study of semisimple Lie groups and Lie algebras. The fact that simply con-nected symmetric spaces decompose in factors with nonpositive, nonnegative or vanishing curvature will be the main motivation for our restricted interest to the simplicial volume of these spaces.

In Chapter 3 we will introduce the continuous cohomology spaces of a topo-logical group G; this can be seen as a generalisation of the usual group cohomology, in that both theories yield the same result for countable dis-crete groups. This concept was first defined by S. Hu in 1954 ([Hu54]) and further studied by Van Est and Mostow who explored its relationship with the theory of Lie algebra cohomology of Chevalley and Eilenberg (see [Est55], [Mos61]). The categorical approach via relatively injective modules that we present is due to Hochschild and Mostow, who systematised the theory in 1962 ([HM62]); we will follow the treatment in Guichardet’s book ([Gui80]). When G is the isometry group of a contractible symmetric space S, the continuous cohomology of G will prove a useful tool, being related to the space of G-invariant differential forms on S by the Van Est isomorphism. We will also widely exploit the connections between continuous cohomology and continuous bounded cohomology, which we will define in Chapter 4. Bounded cohomology was first introduced for topological spaces and discrete groups by Gromov in 1982 ([Gro82]), as a tool for studying the simplicial volume of manifolds. This theory was then extended to topological groups by Burger and Monod, who defined continuous bounded cohomology in [Mon01]. The resulting interplay between the simplicial volume of closed locally sym-metric spaces, the cohomology and bounded cohomology of their fundamen-tal groups, the continuous cohomology and continuous bounded cohomology of the isometry group of their universal coverings will be decribed in Chap-ter 4 and it will be at the core of the results in ChapChap-ters 5 and 6.

In particular, after the proof of Gromov’s Proportionality Principle following Thurston ([Thu78]) and Löh ([L¨06]), Chapter 5 will be devoted to under-standing for which locally symmetric spaces the simplicial volume vanishes. We will also give an explicit computation of the simplicial volume of hy-perbolic manifolds, result originally due to Gromov and Thurston (see for example [BP92] or [Thu78]).

Finally, Chapter 6 will be entirely devoted to Bucher-Karlsson’s computa-tion of the simplicial volume of manifolds covered by H2× H2_{. We remark}

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that, to this day, this remains the only class of manifolds (except hyperbolic ones and those that can be obtained with Theorem 5.2.2) for which the exact value of the simplicial volume is known and nonzero.

Moreover, we will see that the ratio between the Riemannian volume and the simplicial volume of closed, oriented manifolds covered by H2 × H2 _differs

from the supremum of volumes of straight simplices in the universal covering space. This is a previously unnoticed and somewhat astonishing difference from the hyperbolic case.

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Chapter 1

Preliminaries

1.1 Topological groups

Definition 1.1.1. A topological group is a group G which is also equipped with a topology such that the map

G × G → G (x, y) 7→ x−1y is continuous.

In particular we have homeomorphisms:

Lg: G → G Rg: G → G Cg: G → G

x 7→ gx x 7→ xg x 7→ gxg−1

which we will keep denoting this way throughout the dissertation. Moreover, we also define the homeomorphisms

Lk_g: Gk → Gk Rk_g: Gk → Gk

(g0, ..., gk) 7→ (gg0, ..., ggk) (g0, ..., gk) 7→ (g0g, ..., gkg)

for later use. We will refer to Lg and Rg as left translations and right

translations, respectively.

A discrete, normal subgroup Γ of a connected topological group G is always contained in the centre of G. The next lemma easily follows from this fact.

Lemma 1.1.2. Let p : G → H be a covering map between connected topo-logical groups, which is also a homomorphism. Then the kernel of p is a discrete subgroup of the centre of G.

Remark 1.1.3. The notion of covering space is well-behaved with respect to that of topological group. Indeed, let p : G → H be a covering map between

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arbitrary topological spaces. If H is a topological group, then G inherits a structure of topological group that makes p a continuous homomorphism; this structure is unique once a lift of the identity element of H has been chosen.

Conversely, if G is a topological group and Γ is a discrete subgroup of its centre, endowing G/Γ with the quotient topology we obtain a topological group and the covering map p : G → G/Γ is a continuous homomorphism.

An open subgroup of a topological group is always closed as well (since all its cosets must be open as well). As a consequence, a connected topological group is always generated by any neighbourhood of its identity element. Moreover, a locally connected group always has exactly one connected, open subgroup and it is the connected component of the identity.

The following lemma will prove useful in the following chapters.

Lemma 1.1.4. Let G, H be locally compact, Hausdorff, second-countable topological groups and let φ : G → H be a continuous homomorphism. If φ is surjective, it is also open.

Proof. Let g ∈ G and let V be one of its neighbourhoods; we have to prove that φ(g) is an inner point of φ(V ). Without loss of generality we can assume that g = e.

Let U be a compact neighbourhood of e in G with the property that U−1U ⊆ V . We can find countably many elements gnof G so that the sets gnU cover

all G (since G is metrisable by the Nagata-Smirnov Theorem and hence Lindelöf).

Then H is covered by the compact sets φ(g_nU ) which cannot all be nowhere dense by the Baire Theorem; hence some φ(g_nU ) must have an inner point and the same holds for φ(U ). Let φ(u) be an inner point of φ(U ); then e is an inner point of φ(u−1U ). Since u−1U ⊆ V , the statement follows.

Finally, we recall the following well-known result, which we will exploit a huge number of times.

Theorem 1.1.5. Let G be a locally compact, Hausdorff topological group. Then, up to scalar multiplication, there exists a unique Borel measure µ on G such that:

1. µ is finite on compact sets and positive on open sets;

2. (L_g)∗µ = µ for each g ∈ G.

An analogous result holds if we replace left translations with right ones.

We will refer to µ as a left (right) Haar measure on G. We would also like to stress that by the term “Borel measure” we will always simply mean a positive measure defined on all Borel sets.

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Definition 1.1.6. A locally compact, Hausdorff topological group G is said to be unimodular if its left Haar measures are invariant also under right translations.

Lemma 1.1.7. Compact groups are unimodular.

Proof. Let µ be a left Haar measure for the compact group G. Since left and right translations commute, also (Rg)∗µ is a left Haar measure, for each

g ∈ G. By the uniqueness part of Theorem 1.1.5, there exists a real number λ(g) such that

(Rg)∗µ = λ(g) · µ .

Evaluating on G, we obtain:

0 < µ(G) = ((Rg)∗µ) (G) = λ(g) · µ(G) < +∞ ,

so it must be λ(g) = 1 for each g ∈ G. This means precisely that µ is a right Haar measure as well.

We will see other examples of unimodular groups in Section 4.5.

1.2 Lie groups and Lie algebras

Definition 1.2.1. A Lie algebra is a vector space g over a field K = R, C, which is also equipped with a bilinear symmetric map [·, ·] : g × g → g satis-fying the Jacobi identity:

[[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0, ∀X, Y, Z ∈ g. A morphism of Lie algebras is a bracket-preserving, linear map.

Given a vector space V over K, we can endow the space of endomorphisms of V with the usual commutator [f, g] = f ◦ g − g ◦ f , thus obtaining the Lie algebra gl(V ).

For each element X ∈ g we can consider the map

adX: g → g

Y 7→ [X, Y ] and consequently define a morphism of Lie algebras

adg: g → gl(g)

X 7→ adX

known as the adjoint representation of g. Its kernel is called the centre of g.

Definition 1.2.2. An ideal h of g is a vector subspace such that adX maps

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The image ad_g(g) of the adjoint representation of g is always an ideal of gl(g). The Killing form of g is the symmetric, bilinear map κg: g × g → R

defined by κg(X, Y ) := tr(adX ◦ adY); we will simply denote it by κ when

there is no ambiguity.

Definition 1.2.3. The Lie algebra g is said to be semisimple when its Killing form is non-degenerate.

A semisimple Lie algebra cannot have any nonzero abelian ideals. In particular adg provides an embedding g in gl(g).

Definition 1.2.4. A Lie group is a smooth manifold G which also has a group structure, with the requirement that the map

G × G → G (x, y) → x−1y

be smooth. A morphism of Lie groups is simply a smooth homomorphism.

If G is a Lie group, the homeomorphisms Lg, Rg, Cg are actually

diffeo-morphisms for each g ∈ G.

The tangent space of G at the identity element e can be endowed with a Lie algebra structure in the following way. Given two tangent vectors X, Y , it is possible to extend them to G-invariant vector fields eX, eY on G, i.e. vector fields that are preserved by all the differentials dL_g, g ∈ G. We then define

[X, Y ] := [ eX, eY ]e ,

where the second bracket represents a Lie bracket. We will simply call this Lie algebra the Lie algebra of G. The group G is said to be semisimple if its Lie algebra is.

The following standard results clarify the relationship between Lie groups and Lie algebras. They are Theorems 20.19 and 20.21 in [Lee12].

Theorem 1.2.5. Given Lie groups G, H with Lie algebras g, h and a mor-phism φ : G → H, the differential d_ef : g → h is a morphism of Lie algebras. If G is simply connected, every morphism g → h is the differential at e of a unique morphism G → H.

Theorem 1.2.6. Every finite-dimensional real Lie algebra is isomorphic to the Lie algebra of a simply connected Lie group, which is unique up to iso-morphism. In particular, connected Lie groups with isomorphic Lie algebras have isomorphic universal covering spaces.

A Lie subgroup of G is a (not necessarily injective) immersion i : H → G which is also a group homomorphism; with an abuse, we will also call “Lie subgroup” the image i(H). A Lie subgroup is said to be regular if the

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immersion is actually an embedding; some authors limit the definition of Lie subgroup to this case.

As an example, the torus has several one-dimensional Lie subgroups; the regular ones are embedded circles, whereas the non-regular ones are injective immersions of R with dense image.

When i : H → G is a Lie subgroup, the image of the differential of i at the identity element is always a subalgebra of the Lie algebra g of G. Conversely, each subalgebra h ⊆ g is the image of dei for some Lie subgroup i : H → G.

The following is Corollary 20.13 in [Lee12].

Theorem 1.2.7. Regular Lie subgroups are precisely the closed (algebraic) subgroups.

If G is a Lie group with Lie algebra g, it is possible to define an exponential map

exp : g → G

in the following way: given X ∈ g and eX the corresponding G-invariant field on G, we can consider the associated vector flow ΦX: G × R → G and define

exp X as ΦX(e, 1) (the flow is defined on all of G × R because eX has constant

norm if we endow G with a G-invariant Riemannian metric and any such metric is complete; see Lemma 1.4.4 in the next sections).

The exponential map is smooth and its differential at 0 is the identity of g. Moreover, exp sends one-dimensional subspaces of g to one-parameter subgroups of G (i.e. Lie subgroups of the form R → G) and all one-parameter subgroups of G are obtained this way.

For each vector space V , the group GL(V ) is a Lie group with Lie algebra gl(V ). Furthermore, exp : gl(V ) → GL(V ) is the usual matrix exponential. Whenever H < GL(V ) is a Lie subgroup, its Lie algebra h ⊆ gl(V ) can be recovered thanks to the relation

h_{= {X ∈ gl(V ) | exp(tX) ∈ H, ∀t ∈ R} .}

Theorem 1.2.8. A continuous homomorphism φ : G → H between Lie groups is always smooth.

Proof. The graph F of φ is a closed subgroup of G × H. Its Lie algebra is

f= {(X, Y ) ∈ g ⊕ h | φ(exp_G(tX)) = exp_H(tY ), ∀t ∈ R}, where g and h are the Lie algebras of G and H.

Let πG, πH be the projections of G × H onto its factors. Since πG|F is an

injective homomorphism, if we prove that (d_eπG)|f is an isomorphism, then

it is easy to see that πG|F is a diffeomorphism of F onto G and the map

φ = πH ◦ (πG|F)−1 must be smooth. The injectivity of (deπG)|f is obvious,

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that (X, Y ) ∈ f.

Take a neighbourhood U of the origin in g that is mapped diffeomorphically by the exponential map onto some neighbourhood U0 of the identity element of G; define similarly V ⊆ h and V0 ⊆ H. If U, V are small enough, exp_G×H maps (U × V ) ∩ f diffeomorphically onto (U0× V0) ∩ F ; moreover, we can assume that φ(U0) ⊆ V0. Now, for X ∈ g, there exists an integer n ∈ N so that _n1X ∈ U ; then, there must exist Y ∈ V with φ(exp_G(_n1X)) = exp_HY and a Z ∈ f with exp_G×HZ = (exp_G(_n1X), exp_HY ). So Z = (_n1X, Y ) and nZ = (X, nY ) ∈ f.

If g is a Lie algebra, we can consider the subgroup Aut(g) < GL(g) of linear automorphisms preserving the bracket. Since Aut(g) is closed, it admits a Lie group structure making it a regular Lie subgroup of GL(g). An endomorphism f of g is said to be a derivation if

f ([X, Y ]) = [f (X), Y ] + [X, f (Y )]

holds for all X, Y ∈ g. The set Der(g) of all derivations of g is a subalgebra of gl(g) containing adg(g). A derivation is said to be inner if it is of the form

adX for some X ∈ g.

It is easy to check that the Lie algebra of Aut(g) is precisely Der(g). We define the group Int(g) of inner automorphisms of g as the connected Lie sub-group of Aut(g) associated with the Lie algebra adg(g) of inner derivations.

Note that this subgroup might not be regular in general.

Lemma 1.2.9. If g is semisimple, we have adg(g) = Der(g). In particular

Der(g) is semisimple.

The adjoint representation of G is the morphism of Lie groups

AdG: G → GL(g),

where AdG(g) is the differential at e of the map

Cg: G → G

x 7→ gxg−1 .

The adjoint representations of g and G fit in the commutative diagram

G AdG //GL(g) g adg // exp OO gl(g) exp OO

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Lemma 1.2.10. Let G a be a Lie group and Z its centre (which is a closed subgroup). Then the Lie algebra z of Z is precisely the centre of the Lie algebra g of G.

Proof. The centre of g is contained in z. Indeed, if X, Y ∈ g satisfy [X, Y ] = 0, then exp X, exp Y will commute too (as is easy to check by observing that the G-invariant fields ˜X, ˜Y must have trivial Lie bracket at each point). Conversely, take X ∈ z; then exp(tX) lies in Z for each t ∈ R. In particular AdG(exp(tX)) = exp(adg(tX)) must be the identity of g and, deriving in t,

we obtain ad_gX = 0.

Corollary 1.2.11. A connected Lie group is abelian if and only if its Lie algebra is abelian. In particular it must be of the form Rn× (S1₎m_.

Proof. The first part follows from the previous lemma. Let G be a connected abelian Lie group with Lie algebra g. The simply connected Lie group with Lie algebra g must be Rk, with k = dim g, and it must cover G. So G = Rk/Γ for some lattice Γ ⊆ Rk. It is well-known that a lattice in Rk is always isomorphic to some Zm, with m ≤ k, and that it admits a basis over Z which can be extended to a basis of Rk (over R). The result follows.

1.3 Cartan decompositions

Definition 1.3.1. A real Lie algebra g is said to be compact if the Lie group Int(g) is compact.

A subalgebra h ⊆ g is said to be compactly embedded if the connected Lie subgroup of Aut(g) associated with adg(h) ⊆ gl(g) is compact.

Remark 1.3.2. Let i : H → G be a Lie subgroup; even assuming that i is injective, the topologies of H and i(H) are, in general, different (e.g. the torus has injective, dense one-parameter subgroups).

Still, there is no ambiguity in the notion of compact Lie subgroup. Sure enough, H is compact if and only if i(H) is compact (if i(H) is compact, then H is a closed, and hence regular, Lie subgroup of G and i must be an embedding; the other implication is trivial).

In particular, the notion of compactly embedded subalgebra is well-defined.

Given a vector space V , a bilinear form (·, ·) and an endomorphism A, we say that (·, ·) is A-invariant if

(Av, w) + (v, Aw) = 0

holds for all v, w ∈ V . This condition is easily seen to be equivalent to

(etAv, etAw) = (v, w), ∀t ∈ R, ∀v, w ∈ V,

where e : gl(V ) → GL(V ) denotes the matrix exponential. Indeed, one can check that the exponential of an antisymmetric matrix is always orthogonal.

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Lemma 1.3.3. Let H be a compact subgroup of GL(V ) for some finite-dimensional real vector space V . Then there exists a scalar product (·, ·) on V that is preserved by all elements of H.

Proof. Let µ be a left Haar measure on H; clearly µ is finite. Having fixed an arbitrary scalar product (·, ·)₀, we can define

(v, w) := Z

H

(h(v), h(w))0 dµ(h).

The fact that this scalar product is preserved by all elements of H follows from the invariance under left translations of µ.

Lemma 1.3.4. A compact real Lie algebra g always admits a scalar product which is g-invariant, i.e. adX-invariant for all X ∈ g.

Proof. The group Int(g) is compact, so the previous lemma provides an Int(g)-invariant, and hence g-invariant, scalar product on g.

We remark that by “scalar product” we mean a positive definite, sym-metric bilinear form.

Lemma 1.3.5. If a subalgebra h ⊆ g is compactly embedded, then there exists an h-invariant scalar product on g.

Moreover, a compactly embedded subalgebra is always compact.

Proof. Let Int_g(h) be the connected subgroup of Int(g) with Lie algebra adg(h). If h is compactly embedded, the group Intg(h) is compact and

Lemma 1.3.3 yields an Intg(h)-invariant, and hence adg(h)-invariant, scalar

product on g.

Let Auth(g) be the group of automorphisms of g that leave h invariant.

Then the restriction homomorphism r : Auth(g) → Aut(h) is continuous and

it maps the compact group Int_g(h) onto Int(h). Hence, Int(h) is compact. Lemma 1.3.6. A semisimple real Lie algebra g is compact if and only if its Killing form is negative definite.

Proof. If the Killing form κ is definite, the group O(κ) < GL(g) of linear automorphisms preserving κ is compact. Since Aut(g) and Int(g) are closed subgroups of O(κ), they are compact as well.

Assume now that g is compact. We can put a g-invariant, scalar product on g; considering an orthonormal basis of g (as a vector space), all the elements of adg(g) will be represented by antisymmetric matrices. In particular, if

adX is represented by the matrix A = (ai,j), then A 6= 0 since the centre of

g is trivial and we have:

κ(X, X) = tr(A2) = −X

i,j

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Definition 1.3.7. A real form in a complex Lie algebra g is a real subalgebra g0 such that g = g0⊕ (i · g0).

For example, any real Lie algebra g0is a real form of its complexification

g= g0⊗ C.

To each real form we can associate an R-linear automorphism σ : g → g by setting σ|g0 = id and σ|i·g0 = −id; with this definition, it is also clear that

σ preserves the bracket and σ2 = id. We shall say that σ is the conjugation relative to g₀_{. Conversely, the set of fixed points of any R-linear involution} anticommuting with the multiplication by i and preserving the bracket is a real form.

In the setting above, the two Lie algebras share the same Killing form (more precisely: κ_g₀ = κg|g0); hence, g is semisimple if and only if g0 is.

The following is a consequence of the existence of Chevalley generators for semisimple complex Lie algebras. See [Hel78] or [Hum72] for more details.

Theorem 1.3.8. Every semisimple complex Lie algebra contains a real form that is compact.

Two real forms g₁, g2 of g are said to be compatible if, given σ1, σ2 their

conjugations, g₁ is σ₂-invariant and vice versa (though the latter invariance always follows from the former). If this is the case, we obtain a decomposition

g₁= (g1∩ g2) ⊕ (g1∩ (i · g2)) .

Theorem 1.3.9. Every real form g₀ of a semisimple complex Lie algebra g is compatible with some compact real form.

Proof. Let u be a compact real form of g, τ its conjugation and σ the con-jugation with respect to g0. Let κ be the Killing form of g.

We are looking for some f ∈ Aut(g) such that the compact form f · u is compatible with g₀; equivalently, we desire that σ and f ◦ τ ◦ f−1 commute. Since κ|u= κu is real-valued and τ (iX) = −i · τ X for all X ∈ g, it is easy to

check that κ(τ X, τ Y ) =κ(X, Y ) for all X, Y ∈ g. Hence the R-bilinear form

κτ(X, Y ) := −κ(X, τ Y );

is clearly Hermitian on g. Moreover, κu being negative definite, it is easy to

check that κτ is positive definite.

Consider now the automorphism N = στ of g (it is C-linear and bracket-preserving since σ, τ are antilinear and bracket-bracket-preserving); it satisfies

κτ(N (X), Y ) = κτ(X, N (Y )), ∀X, Y ∈ g

and hence is represented by a diagonal matrix in some basis of g which is κ_τ-orthonormal. Moreover, the eigenvalues of N are real, so that P =

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N2 has only positive eigenvalues and can be written as P = eQ, for some diagonalisable operator Q with real eigenvalues.

It is easy to check that {etQ_{, t ∈ R} is a one-parameter subgroup of Aut(g).} Define τ0 = exp(1₄Q) ◦ τ ◦ exp(−1₄Q). Since τ N τ−1 = N−1, we obtain τ etQτ−1 = e−tQ for all t ∈ R and this easily implies στ0 = τ0σ after some straightforward computations.

Remark 1.3.10. Compatible compact real forms u1, u2⊆ g must coincide.

If this were not the case, the restriction of the Killing form of g to u1∩ (i · u2)

would have to be both positive and negative definite.

Corollary 1.3.11. All compact real forms in a semisimple complex Lie al-gebra are conjugated via inner automorphisms.

Proof. This follows immediately from the proof of Theorem 1.3.9 and the previous remark.

We can now define the Cartan decomposition of a semisimple real Lie algebra, which will be essential in the study of symmetric spaces.

Definition 1.3.12. Let g be a semisimple real Lie algebra; g can be seen as a real form in the complex semisimple Lie algebra gC_{:= g ⊗ C. Let u be a}

compact real form of gC_{which is compatible with g.}

A Cartan decomposition of g is a decomposition g = k ⊕ p, where k is the subalgebra g ∩ u and p is the vector space g ∩ (i · u).

Lemma 1.3.13. Let g = k ⊕ p be a Cartan decomposition. Then k is a maximal compactly embedded subalgebra in g.

Proof. Let u be the compact real form of gC _{whose intersection with g is k.}

Let Int_R(gC_{) be the group of inner automorphisms of g}C_{viewed as a real Lie}

algebra. Let σ be the conjugation associated to the real form g of gC_{. The}

group Int(g) can be viewed as the identity component of the closed subgroup of Int_R(gC_{) given by those automorphisms that commute with σ; in}

partic-ular, we can identify Int(g) with a closed subgroup of Int_R(gC_{). Similarly,}

Int(u) can be seen as a compact subgroup of Int_R(gC_).

Then Int(u) ∩ Int(g) is a compact subgroup of Int_R(gC_{) (and of Int(g)) with}

Lie algebra h; hence h is compactly embedded.

If k were not maximal, there would be a bigger compactly embedded subal-gebra k0, which would then be forced to contain some X ∈ p \ {0}.

Being τ X = −X, an easy computation shows that

κτ(adXY, Z) = κτ(Y, adXZ),

for all Y, Z ∈ g. Hence, adX is diagonalisable and nonzero (since semisimple

algebras have trivial centre). But then exp(t·ad_X_{), t ∈ R, is a one parameter} subgroup of GL(g) that cannot be contained in any compact Lie subgroup, contradicting the fact that the Lie subgroup of Aut(g) associated with adg(k0)

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Finally, Cartan decompositions are unique up to inner automorphisms, as shown by the next result.

Theorem 1.3.14. Let g be a real semisimple Lie algebra with Cartan de-compositions g = k1⊕ p1 = k2⊕ p2. Then there exists an inner automorphism

of g conjugating k₁, k2 and also p1, p2.

Proof. The two Cartan decompositions are associated with compact real forms u₁, u2 of gC; let τ1, τ2 be the corresponding conjugations.

Exactly as in the proof of Theorem 1.3.9, there exists a one-parameter sub-group {etQ_{, t ∈ R} of automorphisms of g}C _{such that e}Q _{= (τ}₁_τ₂₎2 _and

exp(1₄Q) must send u2 to a compact real form that is compatible, and hence

coincident, with u1 (see Remark 1.3.10).

We just have to check that the restriction of exp(1₄Q) to g0 is an inner

au-tomorphism. Since both τ₁, τ2 leave g0 invariant, so do τ1τ2 and each etQ.

The restriction to g0 of this one-parameter subgroup of Aut(g) is therefore

a one-parameter subgroup of Aut(g0) and has to lie in the identity

compo-nent. But, by semisimplicity and Lemma 1.2.9, this component is precisely Int(g₀).

1.4 Some basics of Riemannian geometry

Let (M, g) be a connected Riemannian manifold. Throughout this disserta-tion we will denote by I(M ) the group of isometries of M and by I+(M ) the subgroup of orientation-preserving transformations. Moreover, I0(M )

will be the identity component of I(M ) (with respect to the compact-open topology).

The following is an easy but useful lemma.

Lemma 1.4.1. Let f, g ∈ I(M ) and assume that, for some x ∈ M , we have f (x) = g(x) and dxf = dxg. Then f = g.

Proof. The subset of T M where df and dg coincide must be open and closed.

We recall the following standard result originally due to S. B. Myers and N. E. Steenrod ([MS39]). An easier proof in the case of symmetric spaces can be found as Lemma 3.2 in Chapter IV of [Hel78].

Theorem 1.4.2. Let M be any Riemannian manifold. The group I(M ) has a Lie group structure which is compatible with the compact-open topology. Moreover the map

I(M ) × M → M × M (f , x) 7→ (f (x), x) is proper and smooth with respect to this structure.

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Definition 1.4.3. A Riemannian manifold (M, g) is said to be homogeneous if I(M ) acts transitively on M .

Lemma 1.4.4. A homogeneous Riemannian manifold is complete.

Proof. Since the isometry group acts transitively, the injectivity radius is constant over the manifold. In particular, all geodesics are defined on the whole real line. The conclusion follows from the Hopf-Rinow Theorem.

Lemma 1.4.5. The isometry group of a connected, homogeneous Rieman-nian manifold M has finitely many connected components.

Proof. Let G = I(M ) and Gp the stabiliser of some point p ∈ M . The

space G/G_p with the quotient topology is clearly homeomorphic to M : the continuous bijection G/G_p→ M induced by the action of G on p, is proper because of Theorem 1.4.2 and hence a homeomorphism.

So G/Gp is connected and Gp must have at least as many connected

com-ponents as G; but G_p is compact, hence it has finitely many connected components.

We recall that, for a Riemannian manifold (M, g) with Levi-Civita con-nection ∇, the curvature tensor is the (1, 3)-tensor satisfying

R(X, Y )Z := (∇Y∇X− ∇X∇Y − ∇[Y,X])Z,

whenever X, Y, Z are vector fields on M . Equivalently, we can consider the (0, 4)-tensor

R(X, Y, Z, W ) := g(R(X, Y )Z, W ).

Given p ∈ M and a two-dimensional subspace V of TpM , the sectional

curvature of V is

sec V := R(u, v, u, v)

where (u, v) is any orthonormal basis of V . It is easy to check that this is well-defined by exploiting the symmetries of the curvature tensor.

The manifold M is said to have nonpositive curvature if sec V ≤ 0 for each V ⊆ TpM and each p ∈ M .

We will need the following celebrated results on nonpositively-curved mani-folds.

Theorem 1.4.6 (Cartan-Hadamard). Let (Mn, g) be a complete, connected Riemannian manifold with nonpositive curvature and let p be one of its points. Then the exponential mapping

exp_p: TpM → M

is a smooth covering map. In particular, the universal covering space of M is diffeomorphic to Rn.

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Theorem 1.4.7 (Cartan). Let (M, g) be a simply connected, complete Rie-mannian manifold with nonpositive curvature and let G be any subgroup of I(M ). If G has a bounded orbit, then G fixes a point.

Remark 1.4.8. Let M be a simply connected, nonpositively-curved man-ifold. Given a (k + 1)-tuple of points (x₀, ..., xk) ∈ Mk+1, we can define a

canonical, smooth singular simplex ∆k → M which we will call the straight simplex with vertices x0, ..., xk and which we will denote by ∆(x0, ..., xk).

The definition is obtained via an inductive process and goes as follows. First, for k = 0, the simplex ∆(x) is simply the point x. Then, sup-pose to have defined all straight simplices with at most k vertices and let f0: ∆k−1 → M be the simplex ∆(x1, ..., xk). View ∆k as the set

{(λ0, ..., λk) | λi ≥ 0, λ0+ ... + λk= 1} .

Define f : ∆k → M so that each path in ∆k _{of the form}

t 7→ (1 − t) · (λ0, 0, ..., 0) + t · (0, λ1, ..., λk)

is sent to the geodesic from x₀ to f₀(λ1, ..., λk) in M , parametrised by a

multiple of arc-length. Well-definition and smoothnes of f follow from the Cartan-Hadamard Theorem.

We will say that ∆(x0, ..., xk) is the geodesic cone on the simplex ∆(x1, ..., xk)

with vertex x₀.

Observe moreover that the faces of ∆(x0, ..., xk) are precisely the straight

simplices ∆(x0, ...,xbj, ..., xk). However, in general, permuting the vertices of ∆(x0, ..., xk) we will obtain different straight simplices.

Finally we recall the concept of totally geodesic submanifold, which we shall briefly need in Chapter 6.

Definition 1.4.9. A submanifold Σ of the Riemannian manifold M is said to be totally geodesic if, for each p ∈ Σ, the Riemannian exponential map of M takes a neighbourhood of the origin in TpΣ to a subset of Σ.

The following is Exercise 8 in Chapter 5 of [Pet06].

Proposition 1.4.10. Let M be a Riemannian manifold and Σ a totally geodesic submanifold, which we consider with the induced Riemannian met-ric. Then the curvature tensor of Σ is simply the restriction of the curvature tensor of M .

In particular, if V is a plane in a tangent space of Σ, the sectional cur-vature of V is the same whether we compute it in Σ or M .

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1.5 Some functional analysis

In this section we collected a few basic facts on Fréchet spaces and on some representations of topological groups that they allow us to construct. These results will be used mainly in Chapter 3 when we shall define the continuous cohomology spaces of a group.

In §1.5.3 we will also recall a few properties of the Bochner integral, a gen-eralisation of the Lebesgue integral to functions taking values into a Banach space. We will need this construction in Chapter 4.

1.5.1 Fréchet spaces

Definition 1.5.1. A topological vector space is a Hausdorff topological space E which also has a vector space structure (over R or C) such that vector addition and scalar multiplication are continuous.

In addition, we say that E is locally convex if the origin has a neighbourhood basis made of convex open sets.

We will always consider implicitly real topological vector spaces.

Definition 1.5.2. A locally convex topological vector space E is said to be a Fréchet space if its topology is induced by a complete metric that is invariant under translations.

In particular, all Banach spaces are Fréchet spaces. Throughout the rest of this section, E will be a Fréchet space and d will be a translation-invariant, complete metric inducing its topology.

We will exploit the following examples of Fréchet spaces in Chapter 3.

Example 1.5.3. Let X be a locally compact, Hausdorff, second-countable topological space. The space of continuous functions from X to E, endowed with the compact-open topology, is a Fréchet space; we will denote it by C(X, E).

Indeed, we can construct a metric on C(X, E) in the following way. Let X =S

n∈NKn be an exhaustion of X by compact sets and define for maps

f, g ∈ C(X, E) e d(f, g) :=X n≥0 1 2n+1 · min sup x∈Kn d(f (x), g(x)) , 1 .

It is easy to check that this defines a complete, translation-invariant metric and that E is a locally convex topological vector space with the associated topology.

Convergence with respect to ed is precisely uniform convergence on compact sets (see e.g. Theorem 46.8 in [Mun00]).

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Example 1.5.4. Let Mn be a smooth manifold. Also the space C∞(M, E) of smooth functions from E to M can be endowed with a structure of Fréchet space.

We will show it only for E = R, since this is the only case that we will need; however the general case is very similar.

Cover M with countably many compact sets Kα, α ∈ A, each endowed with

coordinates xα₁, ..., xα_n (i.e. each K_α is contained in a chart). Given α ∈ A and β1, ..., βn∈ N, we can define

e dα,β1,...,βn(f, g) := sup x∈Kα ∂β1+...+βn ∂xα,β1 1 ...∂x α,βn n f (x) − ∂ β1+...+βn ∂xα,β1 1 ...∂x α,βn n g(x) ,

for f, g in C∞_{(M, R). Then order all possibles (n + 1)-uples (α, β}1, ..., βn),

associating an integer i to each of them, and define:

e d∞(f, g) :=X i∈N 1 2i+1 · min e di(f, g), 1 .

Again it is easy to check that all the required properties are satisfied. In particular, completeness follows from standard results on the convergence of derivatives.

The notion of convergence defined by ed∞ coincides with that of uniform convergence on compact sets of f and all its partial derivatives. Hence, the topology induced on C∞_{(M, R) by e}d does not depend on the choices involved in the definition of the metric.

Let now F be a closed subspace of E. The vector space E/F becomes a topological vector space when we endow it with the quotient topology; the projection E → E/F is linear, continuous and open. Moreover, it is clear that E/F is locally convex if E is.

Lemma 1.5.5. The quotient E/F of a Fréchet space by a closed subspace is a Fréchet space as well.

Proof. If d is a complete, translation-invariant metric inducing the topology of E, we can define on E/F :

d(e1, e2) := d(e1− e2, F ).

It is immediate to check that d is well-defined and translation-invariant. Moreover, given e₁, e2, e3∈ E:

d(e1− e2, x) + d(e2− e3, y) ≥ d(e1− e3, x + y) ≥ d(e1− e3, F ), ∀x, y ∈ F,

so that d(e1, e2) + d(e2, e3) ≥ d(e1, e3) and d is a metric. Since the balls of

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topology of E/F .

Now we just have to check completeness. Let (en)n∈Nbe a Cauchy sequence

in E/F ; up to passing to a subsequence, we can assume that d(en, en+1) <

1/2n. Choosing appropriately each e_n∈ E, we can obtain d(e_n, en+1) < 1/2n

as well. Since (E, d) is complete, (en) converges and so does (en).

Definition 1.5.6. Let E, F be topological vector spaces and f : E → F a continuous linear map. If f is injective, we say that it is strong if it admits a left-inverse which is linear and continuous.

In the general case, f is said to be strong if both the inclusion ι : Ker f ,→ E and the quotient mapf : E/Ker f ,→ F are.

The previous definition is the correct notion of strong map in the category of topological vector spaces and we will refer to it repeatedly in Chapter 3. However, in Chapter 4 we will meet a slightly different notion of strong map, which will be appropriate in the category of Banach spaces. It will be always clear from the context, which notion of strength we are adopting.

Lemma 1.5.7. Let f : E → F be a strong, continuous, linear map between topological vector spaces. Then there exist closed subspaces E1 ⊆ E and

F1 ⊆ F with E = Ker f ⊕ E1 and F = Im f ⊕ F1. Moreover, the restriction

of f to E1 is a homeomorphism with Im f .

Proof. Let α, β be left inverses for ι and f , respectively. Then E1 = Ker α

and F1 = Ker β satisfy the first part of our statement.

It is also evident that the restriction of f to E1is a continuous bijection with

Im f . Let π the quotient projection from E to E/Ker f ; clearly π|_E₁ is a homeomorphism. Hence (π|E1)

−1_{◦ β is a continuous inverse for f .}

Proposition 1.5.8. Let E be a topological vector space and F a finite-dimensional subspace. Then F is closed.

Proof. This is 1.21 in [Rud91].

Theorem 1.5.9. Let f : E → F be a continuous, surjective linear map between Fréchet spaces. Then f is open.

Proof. This follows from 2.11 in [Rud91] and the Baire Theorem. The proof is identical to the one usually given for Banach spaces.

Corollary 1.5.10. Let f : E → F be a continuous, linear map between Fréchet spaces. If G = Imf has finite codimension in F , then it is a closed subspace.

Proof. Let V be a finite-dimensional subspace of F such that F = G ⊕ V ; the subspace V is closed by Proposition 1.5.8.

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with the product topology.

We can define a continuous, linear bijection (E/Kerf ) × V → F as f on the first summand and as the inclusion of V in F on the second summand. This has to be a homeomorphism by Theorem 1.5.9. Since E/Kerf is closed in (E/Kerf ) × V , this means that G is closed in F .

1.5.2 Fréchet G-modules

Let G be a topological group.

Definition 1.5.11. A Fréchet G-module is a Fréchet space E provided with a left action of G on E by linear homeomorphisms such that the map

G × E → E (g, e) 7→ g · e

is continuous. We will also call this a continuous representation of G.

Definition 1.5.12. A morphism of G-modules (or G-map) is a linear con-tinuous map f : E → F conjugating the actions of G, i.e satisfying

f (g · e) = g · (f (e)) for each g ∈ G and e ∈ E.

Definition 1.5.13. The subspace of invariants of a Fréchet G-module E is

EG:= {e ∈ E | g · e = e, ∀g ∈ G}.

Observe that the subspace of invariants is always a closed subspace of the G-module E; in particular, it inherits the Fréchet structure.

The following result will prove particularly useful for constructing G-modules; it can be found (in the more general context of barrelled spaces) in [Bou04] as Proposition 1 in Chapter VIII, §2, No.1.

Proposition 1.5.14. Let G be a locally compact, Hausdorff group acting on a Fréchet space E by linear homeomorphisms. Suppose that this representa-tion of G is separately continuous, i.e. for each e ∈ E the map

ve: G → E

g → g · e

is continuous. Then this is actually a continuous representation of G and E is a G-module with this action.

We will always assume G to be Hausdorff and locally compact in the following discussion.

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Example 1.5.15. Let E be a Fréchet G-module and X a locally compact, Hausdorff, second-countable topological space. Suppose that G acts on the left on X and that the map

G × M → M (g, x) → g · x

is continuous. Then G acts on the left also on the Fréchet space C(X, E) defined in Example 1.5.3 by

(g · f )(x) := g · f (g−1· x) .

We obtain a Fréchet G-module that we will keep denoting by C(X, E). Indeed, thanks to Proposition 1.5.14, we only have to check that this repre-sentation of G is separately continuous and that G acts by homeomorphisms. To prove the former assertion, assume that there exists a sequence g_n → g in G and some f ∈ C(X, E) such that gn· f 6→ g · f . Then we would be able

to find some compact set K ⊆ X where (gn· f ) does not converge uniformly;

so we could find some > 0 and points x_n∈ K with

d(gn· (f (gn−1xn)), g · (f (g−1xn))) ≥ , ∀n ∈ N.

Without loss of generality, we can assume that x_n→ x for some x ∈ K (since X is metrisable by the Nagata-Smirnov Theorem); but then both sequences in the previous inequality would converge to the point g · (f (g−1x)), which is absurd.

To prove the second assertion, we only have to show that, if g ∈ G and fn → f uniformly on compact sets, then g · fn → g · f as well. Take

K ⊆ X compact and observe that the union K of all fn(K) and f (K) must

be compact in E; hence, g|_K is uniformly continuous, which implies our statement.

We will be especially interested in the G-modules C(Gn, E) in the later chapters.

Example 1.5.16. Let G be a Lie group acting on the left on a smooth manifold M , so that the map

G × M → M

(g, x) → g · x := τ (g)(x)

is smooth. Then also the space C∞(M, E) defined in Example 1.5.4 is a Fréchet G-module, with the restriction of the action of G on C(M, E). In the case where E = R with the trivial action of G, whis is particularly easy to prove. One just has to proceed as in the previous example, with the additional observation that, given f ∈ C∞(M, E), each partial derivative of g · f can be expressed as a sum of products of partial derivatives of τ (g−1) and partial derivatives of f , each of them then composed with τ (g−1).

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Example 1.5.17. In the setting of the previous example, we can also define a left action of G on the space of p-forms Ωp_{(M, R) by}

g · ω := (τ (g−1))∗ω .

We can embed Ωp_{(M, R) in C}∞(ΛpT M, R) in an obvious way. If we equip the space C∞(ΛpT M, R) with the topology defined in Example 1.5.4, the image of this embedding is closed since it consists precisely of those maps that are linear on each fibre of ΛpT M → M . Therefore, endowing Ωp(M, R) with the subspace topology induced by this embedding, we obtain a Fréchet space.

The action of G on Ωp_{(M, R) is a continuous representation, since it is simply} the restriction of the action of G on ΛpT M given in Example 1.5.16 (an action of G on ΛpT M is induced canonically by the action of G on M ).

Finally, recall that the differential dp: Ωp(M, R) → Ωp+1(M, R), is defined as (dpω)q[X0, ..., Xp] =: p X j=0 (−1)j· X_jωq h X0, ..., cXj, ..., Xp i + +X i<j (−1)i+j· ωq h [Xi, Xj], X0, ..., cXi, ..., cXj, ..., Xp i ,

where X0, ..., Xp are vector fields near q ∈ M .

It is easy to check that dp is continuous with respect to our topologies.

Example 1.5.18. Let E, F be Fréchet G-modules, with F finite-dimensional, and let d be a complete, translation-invariant metric inducing the topology of E.

The vector space E ⊗ F has a structure of Fréchet space given by viewing it as a direct sum of a finite number of isometric copies of (E, d) and by considering the product metric.

The group G acts on E ⊗ F by g · (e ⊗ f ) := (g · e) ⊗ (g · f ); this way E ⊗ F becomes a Fréchet G-module.

1.5.3 Integration of vector-valued functions

We will need the following generalisation of the Lebesgue integral. See [Mon01] and [JU77].

Theorem 1.5.19. Let X be a metrisable, second-countable topological space, E a Banach space and f : X → E a continuous function. Let µ be a σ-finite Borel measure on X.

We say that f is Bochner integrable if

Z

X

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and in this case it is possible to define a (unique) elementR

Xf (x) dµ of E,

called Bochner integral, such that the result is the usual Lebesgue integral if E is finite-dimensional and furthermore, for a general E and for each linear, continuous operator to another Banach space T : E → F , we have

T Z X f (x) dµ = Z X T (f (x)) dµ. Moreover: 1. kR_Xf (x) dµk ≤R_Xkf (x)k dµ; 2. R X(αf + βg) = α R Xf + β R

Xg, whenever α, β ∈ R and g ∈ C(X, E);

3. the Fubini-Tonelli Theorem holds also for the Bochner integral;

4. if µ is finite, Y is a topological space and F : X × Y → E is continuous and bounded, then the map F : Y → E defined by

F (y) := Z

X

F (x, y) dµ(x) is continuous.

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Chapter 2

Symmetric Spaces

This chapter is devoted to the study of symmetric spaces. In Section 2.1 we will see how this geometric concept can be rephrased in purely algebraic terms. Geodesic symmetries correspond to involutions of Lie groups and Lie algebras and the Riemannian exponential map can be expressed in terms of the exponential map of a Lie group.

Then we will show that symmetric spaces decompose into factors of three different types, each with a clear behaviour in terms of sectional curvature and compactness. It will also become clear that we are mainly interested in semisimple Lie groups.

Finally, in Section 2.4 we will show that semisimple groups always contain maximal compact subgroups and originate symmetric spaces.

2.1 Globally symmetric spaces

Let (Mn, g) be a Riemannian manifold. An isometry f of M fixing a point x is called the geodesic symmetry at x if dxf = −id.

By Lemma 1.4.1, all geodesic symmetries are involutions. Moreover, x is always an isolated fixed point for the geodesic symmetry at x.

Definition 2.1.1. A connected Riemannian manifold M is said to be a globally symmetric space if there is a geodesic symmetry sx at x for every

x ∈ M .

A locally symmetric space is a Riemannian manifold covered by a globally symmetric space. We will usually refer to globally symmetric spaces simply as symmetric spaces.

Proposition 2.1.2. Symmetric spaces are homogeneous.

Proof. We start by proving completeness. Assume that (a, b) ⊆ R is the maximal domain of some geodesic γ in the symmetric space M . If b < ∞, choose some a+b₂ < t < b and consider the geodesic arc γ0 = sγ(t)◦ γ.

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A portion of γ0 can be glued to γ, yielding an extension of its domain to (a, 2t − a), since 2t − a > b. This violates the maximality of (a, b), forcing us to conclude that b = +∞ and, similarly, a = −∞. The Hopf-Rinow Theorem then guarantees completeness.

Now take x, y ∈ M and a geodesic arc γx,y connecting them; its existence

follows again from the Hopf-Rinow Theorem. If p is the midpoint of γx,y, it

is immediate that s_p exchanges x and y.

The presence of geodesic symmetries allows us to express the parallel transport in terms of one-parameter subgroups of the isometry group. We will see later in this section (Lemma 2.1.6) that even the actual geodesics are given by the action of one-parameter subgroups of the isometry group.

Proposition 2.1.3. Let M be a symmetric space, p one of its points and γ a geodesic with γ(0) = p. Call s_t the geodesic symmetry at γ(t) and define the isometries Tt:= st/2s0. Then:

1. d0Tt is the parallel transport from p to γ(t) along γ;

2. the map t 7→ Tt is a one-parameter subgroup of I(M ).

Proof. 1. Take a tangent vector v at γ(t/2) and let v0 and vt be the

tangent vectors at p and γ(t) obtained by parallel transport. Since dst/2(v) = −v and isometries respect the parallel transport, it is clear

that ds_t/2(v0) = −vt. Hence dTt(v0) = vt, that is dpTt is the parallel

transport along γ.

2. First, it is easy to check that T_tTt0 = T_t+t0 for each t, t0 ∈ R; indeed this

is equivalent to s_t/2s0st0_/2= s_(t+t0_)/2and this equality follows from the

fact that both sides of it are isometries fixing γ((t + t0)/2) and acting as −id on the tangent space at that point (which is a consequence of the description of the differentials of geodesic symmetries given in the first part of the proof).

Hence we have obtained a homomorphism t 7→ Tt of R into I(M ). It

is easy to check that this map is continuous so that we can appeal to Theorem 1.2.8 to prove that it is actually smooth.

A proof of the following easy proposition can be found in [Hel78] (see Theorem 4.2 and Proposition 4.3 in Chapter II).

Proposition 2.1.4. • Let G be a Lie group and H a closed subgroup. Then the space of left cosets G/H admits a unique smooth structure (compatible with the quotient topology) such that the projection

π : G → G/H is a smooth submersion.

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• Let G act transitively on a smooth manifold M so that the map G × M → M × M

(g, x) 7→ (g · x, x)

is smooth and proper. Then, denoting by Gp the set of all elements of

G fixing some p ∈ M , the map

πp: G/Gp→ M

g 7→ g · p is a diffeomorphism.

Theorem 1.4.2 and Proposition 2.1.4 clearly imply that any connected, homogeneous Riemannian manifold M is diffeomorphic to G/H where G is some connected Lie group and H is a compact subgroup of G.

Sure enough, it suffices to set G = I₀(M ) and to choose some p ∈ M ; setting H = Gp we obtain the desired diffeomorphism πp: G/H → M .

Remark 2.1.5. When M is a symmetric space, we can say more. The geodesic symmetry sp is an isometry and so it induces by conjugation an

order-two automorphism σp of I(M ) and hence of G. Since σp commutes

with all isometries fixing p, we have σ_p|_H = id and the diagram G/H πp // σp M sp G/H πp //M clearly commutes.

Furthermore, deσp is an involutive automorphism of the Lie algebra g of G;

its set of fixed points is precisely the Lie algebra h of H. Indeed, it is clear that d_eσp|h is the identity and, if deσp were to fix some other vector in g,

we would get a one-parameter subgroup of G acting nontrivially on p and commuting with sp; in particular sp would fix pointwise the orbit of p under

the action of this subgroup and p would not be isolated in Fix(s_p).

Lemma 2.1.6. Mantaining the notation of the previous remark, let p be the eigenspace of d_eσp relative to the eigenvalue −1. Then the curves

γX(t) := πp(exp(tX)),

with X ∈ p, are precisely the geodesics in M through p.

Proof. Choose a geodesic γ in M with γ(0) = p, call stthe geodesic symmetry

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subgroup of G; in particular there exists X ∈ g with T_t _{= exp(tX), ∀t ∈ R.} Observe that

σp(Tt) = s0Tts0= s0st/2 = Tt−1= T−t,

so that it must be deσp(X) = −X, i.e. X lies in p. It is immediate to check

that γ = γ_X.

Conversely, if G is a connected Lie group and H is a compact subgroup, we can always endow the smooth manifold G/H with a Riemannian metric that turns it into a homogeneous Riemannian manifold, as we will now show. Every g ∈ G induces a diffeomorphism

τ (g) : G/H → G/H xH → gxH

and the elements of G fixing the preferred point o = H ∈ G/H are precisely the elements of H. Via the map τ , the group G acts transitively on the manifold M = G/H.

Denote by g, h the Lie algebras of G, H respectively. We can endow g with an Ad_G(H)-invariant scalar product because of Lemma 1.3.3. So we can decompose g = h⊕h⊥, where both h and h⊥are Ad_G(H)-invariant subspaces. The differential of π : G → G/H at e is zero on h, while it identifies h⊥ and ToM via an isomorphism. In particular, ToM inherits the AdG(H)-invariant

metric that we have on h⊥. This metric is preserved by d_oτ (h) whenever h ∈ H since the diagram

G π Ch // G π G/H τ (h) ////G/H

commutes. As a consequence, we can transport this scalar product to each point of M using some τ (g) and obtain a well-defined, homogeneous, Rie-mannian metric on M such that τ (g) is an isometry for each g ∈ G.

Observe that only the compactness of Ad_G(H) (and not that of H) was needed for this construction.

Remark 2.1.7. If, in addition to our pair of Lie groups (G, H), we have an involution σ of G so that Fix(deσ) = h and σ|H = id, we can endow

M = G/H with a symmetric-space structure.

Since deσ is an involution of g whose 1-eigenspace is h, we can decompose

g = h ⊕ p, where p is the eigenspace of deσ associated with the eigenvalue

−1. Since h and p are both Ad_G(H)-invariant, we can obtain as above an AdG(H)-invariant metric on g where p = h⊥ and this defines a structure of

homogeneous Riemannian manifold on M with τ : G → I(M ).

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so, it suffices to check the existence of a geodesic symmetry at o. But σ is the identity on H and hence descends to a map s : M → M which fixes o; the fact that dos = −id follows immediately from the observation that deπ

identifies p with T_oM .

Observe that the geodesic symmetry at o (and hence at all points) does not depend on our choice of the scalar product on g.

Lemma 2.1.8. In the setting of the previous remark, the geodesics through o in G/H are precisely the curves

γX(t) = π(exp(tX)),

with X ∈ p. In particular the geodesics through o do not depend on the chosen metric on G/H.

Proof. Clearly we have a continuous homomorphism

τ : G → I(G/H) g 7→ τ (g) .

Theorem 1.2.8 guarantees that τ is smooth. Note that the diagram

I(G/H) πo //G/H G τ OO π _// G/H id OO

is commutative. Moreover, if σo is the involution of I(G/H) constructed in

Remark 2.1.5, it is easy to check that τ σ = σ_oτ ; indeed, this is equivalent to τ (σ(g)) = s_oτ (g)so holding for each g ∈ G and this is clear from the

definition of σ.

We can decompose the Lie algebra ˜g of I(G/H) as ˜g= ˜h⊕ ˜p, where ˜h, ˜p are the eigenspaces of σ_o. Since τ σ = σ_oτ , it is clear that deτ maps p into ˜p.

Actually, we see that it must be deτ (p) = ˜p, as π, ˜π map p, ˜p isomorphically

onto T_o(G/H).

We conclude by invoking Lemma 2.1.6.

Remarks 2.1.5 and 2.1.7 lead us to the following definition.

Definition 2.1.9. A Riemannian symmetric pair is a pair of Lie groups (G, H), where G is connected and H is a closed subgroup of G such that:

• Ad_G(H) is compact;

• there exists an involution σ of G satisfying σ|H = id and Fix(deσ) = h,

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The previous discussion shows that a symmetric space always defines a Riemannian symmetric pair (Remark 2.1.5) and a Riemannian symmetric pair always defines a symmetric space (Remark 2.1.7).

We can also give a slightly different notion involving only Lie algebras.

Definition 2.1.10. An orthogonal symmetric Lie algebra is a real Lie al-gebra g equipped with an involutive automorphism s, such that Fix(s) is a compactly embedded subalgebra.

If Fix(s) intersects trivially the centre of g, we speak of an effective orthog-onal symmetric Lie algebra.

The next lemmas clarify the interplay between the previous definitions.

Lemma 2.1.11. Let (G, H) be a Riemannian symmetric pair with involution σ. Assume either that G is semisimple or that (G, H) has been obtained with the procedure described in Remark 2.1.5.

Then the Lie algebra g of G becomes an effective orthogonal symmetric Lie algebra if we equip it with d_eσ.

Proof. The subalgebra Fix(deσ) = h is compactly embedded because AdG(H)

is compact and it is precisely the identity component of the Lie subgroup of Int(g) associated with adg(h).

If G is semisimple, the centre of g is trivial and so must be its intersection with h. If G = I₀(M ) for some symmetric space M and H is the stabiliser of some point, it is clear that H cannot meet the centre of G outside {e} (by transitivity of the action of G on M ). This implies that h intersects trivially the centre of g (thanks to Lemma 1.2.10).

Lemma 2.1.12. Let (g, s) be an effective orthogonal symmetric Lie algebra with Fix(s) = h. Let G be a Lie group with Lie algebra g and H the connected subgroup associated with h.

If G is simply connected, (G, H) is a Riemannian symmetric pair.

Proof. Since G is simply connected, Theorem 1.2.5 guarantees the existence of an involution σ of G with d_eσ = s. Since H is connected, we must have σ|_H = id, as this holds in a neighbourhood of the identity (being σ(exp X) = exp(deσ(X))). In particular H is the identity component of the

set of fixed points of σ and must therefore be closed.

Moreover, Ad_G(H) is precisely the connected subgroup of Int(g) associated with adg(Fix(s)) and so has to be compact.

Lemma 2.1.13. Let (g, s) be an effective orthogonal symmetric Lie algebra with Fix(s) = h. Let G be a Lie group with Lie algebra g and suppose that the connected subgroup H with Lie algebra h is closed.

Then there exist G-invariant Riemannian metrics on G/H. Endowing G/H with one such metric, one gets a locally symmetric space.

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Proof. Since H is connected and h is compactly embedded, it is clear that AdG(H) is compact. A G-invariant metric on G/H can then be constructed

as in Remark 2.1.7 by fixing an Ad_G(H)-invariant scalar product on To(G/H).

Now let π : eG → G be the universal covering space of G and call eH the iden-tity component of π−1(H). It is clear that eG/ eH → eG/π−1(H) ' G/H is a covering map; choosing an appropriate G-invariant metric on eG/ eH, this map becomes locally isometric. Moreover, since eG is simply connected, Lemma 2.1.12 guarantees that (G, H) is a Riemannian symmetric pair and Remark 2.1.7 implies that eG/ eH becomes a symmetric space.

Actually, the space eG/ eH is the universal covering of G/H: every loop in e

G/ eH can be lifted to an open path in eG (since eG → eG/ eH is a submersion) which can then be closed with a path in eH; in particular every loop in eG/ eH is the image of a loop in eG, which is simply connected.

Remark 2.1.14. The conclusion of Lemma 2.1.8 holds for G/H also in the setting of Lemma 2.1.13, as is easy to check by looking at eG/ eH.

Finally, we present the following fact without proof (see Theorem 4.1 in Chapter V in [Hel78]). It explains the relationship between Remark 2.1.7 and Remark 2.1.5 when G is semisimple. Furthermore, we will need to appeal to this result in the following section.

Theorem 2.1.15. Let (G, H) be a Riemannian symmetric pair. Assume that G is semisimple and that H does not intersect the centre of G. Then, en-dowing the quotient M = G/H with a G-invariant metric as in Remark 2.1.7, the group G acts on M by isometries and we have:

G = I0(M ) .

2.2 Decomposition of symmetric spaces

In Section 2.1 we have seen that a symmetric space M can always be rep-resented as a quotient G/H, where G is a connected Lie group and H is a compact subgroup. Moreover, G is endowed with an involution σ and d_eσ turns g into an effective orthogonal symmetric Lie algebra with Fix(deσ) = h.

Definition 2.2.1. Let (g, s) be an effective orthogonal symmetric Lie alge-bra. If h and p are the eigenspaces of s relative to the eigenvalues 1 and −1 respectively, we can write g = h ⊕ p.

• If g is compact and semisimple, we say that (g, s) is of compact type; • if g is semisimple, but not compact, we say that (g, s) is of noncompact

type if g = h ⊕ p is a Cartan decomposition;

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The proof of the following fact is purely algebraic and can be found in [Hel78] as Theorem 1.1 of Chapter V.

Theorem 2.2.2. Let (g, s) be an effective orthogonal symmetric Lie algebra and let h and p be the eigenspaces of s relative to the eigenvalues 1 and −1. Let g be any ad_g(h)-invariant scalar product on p.

Then there are ideals g₀, g+, g− of g so that:

• g = g₀⊕ g₊⊕ g− and each piece is s-invariant;

• (g₀, s|g0), (g+, s|g+) and (g−, s|g−) are effective orthogonal symmetric

Lie algebras, respectively of euclidean, compact and noncompact type.

Moreover, this decomposition is orthogonal with respect to both the Killing form of g and g.

Definition 2.2.3. Let M be a symmetric space. Then, setting G = I0(M )

and denoting by Gp the subgroup of G fixing p ∈ M , we have that (G, Gp) is

a Riemannian symmetric pair and the Lie algebra g of G inherits a structure of orthogonal symmetric Lie algebra thanks to Lemma 2.1.11.

According to the type of g, we say that M is of euclidean, compact or non-compact type. The same terminology applies to locally symmetric spaces, according to their universal covering space.

Corollary 2.2.4. A simply connected symmetric space M is always iso-metric to a product M0 × M+× M−, where M0 is an euclidean space and

M+, M− are simply connected symmetric spaces of compact and noncompact

types, respectively.

Proof. We can write M = G/G_p with the notation of Definition 2.2.3 and consider the associated orthogonal symmetric Lie algebra (g, s). Let π : eG → G be the universal covering of G and denote by eGp the identity component

of π−1(Gp).

It is clear that eG/π−1(Gp) ' G/Gp, so the natural map eG/ eGp→ G/Gp is a

covering map; since M is simply connected, this map must be a diffeomor-phism (in particular π−1(Gp) is connected).

The Lie algebra of G_p is h = Fix(s) and we denote by p the eigenspace of s relative to the eigenvalue −1. The Riemannian metric on M corresponds to an adg(h)-invariant scalar product g on p. Theorem 2.2.2 then yields a

decomposition g = g₀ ⊕ g₊ ⊕ g− associated with g; set hi = h ∩ gi and

pi = p ∩ gi, for i ∈ {0, +, −}.

If G0, G+, G− are simply connected Lie groups with Lie algebras g0, g+, g−

and H₀, H+, H− are their connected subgroups associated with h0, h+, h−,

then it is clear that eG is isomorphic to G0× G+× G− and the isomorphism

takes eGp to H0× H+× H−, since h = h0⊕ h+⊕ h−. In particular

e

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Since the restriction of g to each p_i is ad_g(hi)-invariant, it induces a

Rie-mannian metric on Mi = Gi/Hi that turns it into a symmetric space; see

Lemma 2.1.12 and Remark 2.1.7. Moreover, g(p_i, pj) = 0 if i 6= j, so that we

get an isometry between M and the Riemannian product M₀× M₊× M−.

We have to check that each Mi is a symmetric space “of type i”. First, M0

is simply connected (since M is) and we shall see in the next section that its sectional curvature vanishes identically (Theorem 2.3.2); therefore, M₀ is an euclidean space.

If i = ±, let Γi be the intersection of the centre of Gi with Hi; since Gi is

semisimple, Γ_i is discrete. Observe that (G_i/Γi, Hi/Γi) is again a

Rieman-nian symmetric pair inducing the symmetric space Mi. Moreover, the centre

of Gi/Γi is disjoint from Hi/Γi; indeed, if hΓi is in the centre of Gi/Γi, the

map g 7→ ghg−1 on G_i takes values in Γ_i, which is discrete, and hence must be constant, i.e. h lies in Γi.

Now we can invoke Theorem 2.1.15 to obtain I0(Mi) = Gi/Γi. In particular

the space M_i is of compact/noncompact type since the pair (G_i/Γi, Hi/Γi)

is.

Of course, each of the factors M₀, M₊, M− can happen to be simply a

single point.

Corollary 2.2.4 reduces the study of general symmetric spaces to those of compact and noncompact type; we will see in Section 2.4 that these corre-spond to compact and noncompact semisimple Lie groups, respectively. In particular, symmetric spaces of noncompact type are noncompact and those of compact type are compact.

2.3 Curvature of symmetric spaces

The proof of the following result is particularly technical and can be found in [Hel78] (Theorem 4.2, Chapter IV).

Theorem 2.3.1. Let (G, H) be a Riemannian symmetric pair with involu-tion σ; assume that H is connected. Endow the quotient M = G/H with a G-invariant Riemannian metric (as in Remark 2.1.7) and define o ∈ M as the projection of the identity element of G. Let g = h ⊕ p be the usual decomposition of the Lie algebra of G into eigenspaces of deσ.

Then, the curvature tensor at o is given by:

Ro(X, Y )Z = [[X, Y ], Z] ,

where X, Y, Z ∈ ToM can be seen as vectors in p ⊆ g.

Theorem 2.3.2. Let (G, H) be a Riemannian symmetric pair with involu-tion σ. Set s = deσ and assume that (g, s) is an efficient orthogonal

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with a G-invariant metric. Then, assuming that n ≥ 2:

1. if (g, s) is of compact type, M has nonnegative curvature;

2. if (g, s) is of noncompact type, M has nonpositive curvature;

3. if (g, s) is of euclidean type, M is flat, i.e. its universal covering space is isometric to Rn.

Proof. Since M is homogeneous, it suffices to compute the sectional curva-tures at o; here the curvature tensor is given by Theorem 2.3.1.

When (g, s) is of euclidean type, Theorem 2.3.1 trivially implies that the sec-tional curvature of M vanishes everywhere; this is well-known to be equiva-lent to being covered by an euclidean space. So let us assume that we are in one of the first two cases; in particular, we can assume g to be semisimple. The metric g on ToM corresponds to an adg(h)-invariant metric on p

(man-taining the notation of Definition 2.2.1). Hence there exists an endomor-phism f of p (as a vector space) satisfying

κ(X, Y ) = g(f (X), Y ), ∀X, Y ∈ p,

where κ is the Killing form of g. Since κ is symmetric, we have

g(f (X), Y ) = g(X, f (Y )), ∀X, Y ∈ p,

so that p is the direct sum of the eigenspaces pi of f and these spaces are

mutually orthogonal with respect to g and κ.

We have the easy relations [p, p] ⊆ h and [h, p_i] ⊆ pi, for each i; the latter

identities follow from the fact that f commutes with all the elements of adg(h), since g and κ are adg(h)-invariant.

Observe that [p_i, pj] = 0 whenever i 6= j. Sure enough, take Xi ∈ pi, Xj ∈ pj

and Y ∈ h and notice that κ([Xi, Xj], Y ) = κ(Xj, [Y, Xi]) = 0; now [Xi, Xj]

has to be zero as the restriction of κ to h is negative definite (in both our cases).

To conclude, let λi be the eigenvalue of f associated with the eigenspace pi;

it is immediate that λi < 0 in case 1 and λi > 0 in case 2, for each i. Now

take X =P Xi and Y =P Yi in p, with Xi, Yi ∈ pi; we have

Ro(X, Y )X = [[X, Y ], X] = [ X i [Xi, Yi], X] = X i [[Xi, Yi], Xi],

where the last equality is due to the fact that, if i 6= j:

The simplicial volume of manifolds covered by H^2xH^2

Contents

Introduction

Chapter 1

Preliminaries

1.1

Topological groups

1.2

Lie groups and Lie algebras

1.3

Cartan decompositions

1.4

Some basics of Riemannian geometry

1.5

Some functional analysis

Chapter 2

Symmetric Spaces

2.1

Globally symmetric spaces

2.2

Decomposition of symmetric spaces

2.3

Curvature of symmetric spaces