Abdelghani Zeghib
email: abdelghani.zeghib@ens-lyon.fr

HOMOGENEOUS SPACES, DYNAMICS, COSMOLOGY: GEOMETRIC FLOWS AND RATIONAL DYNAMICS

The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimensional dynamical systems, obtained by considering homogeneous metrics.

INTRODUCTION

These notes are variations around the homogeneous space

X n = Sym + n = GL(n, R)/O(n)
that is, the space of n × n positive definite symmetric matrices, and sometimes, its subspace of those matrices with determinant 1, Y n = SSym + n = SL(n, R)/SO(n) Besides their striking beauty, these spaces modelize many structures and support fascinating geometry and dynamics. It is surely very interesting to investigate interplays between these aspects. We will not do it systematically here, but rather briefly note some of them 1 . Our modest remark here is that "Bianchi" Ricci flows and Bianchi cosmologies are better seen as natural dynamical systems (i.e. differential equations) on X n , respectively of first and second orders. To be more precise, Bianchi spaces are special homogeneous Riemannian spaces, those given by left invariant metrics on Lie groups. For a given Lie group G, the space of such metrics is identified with X n , n = dim G. The Ricci flow acting of the space of Riemannian metrics on a manifold, becomes here a (gradient-like) flow on X n . Similarly, the Cauchy problem for the (vaccum) Einstein equations becomes here a dynamical system on T X n . The group G acts preserving all these dynamical systems.

References. There are now abundant references on the Ricci flow techniques, since their use by Perelman in his program on the geometrization conjecture on 3-manifolds. We may quote as an example [START_REF] Chow | The Ricci flow: an introduction[END_REF] as an interesting recent reference. The "toy" homogeneous case (with which we are dealing here) was in particular investigated in [START_REF] Isenberg | Ricci flow of locally homogeneous geometries on closed manifolds[END_REF][START_REF] Isenberg | Ricci flow on locally homogeneous closed 4-manifolds[END_REF].

The history of cosmology from a relativistic point of view, i.e. applying Einstein equations, is very old, and was in fact usually considered within a homogeneous framework, even an isotropic one, as in the standard big-bang model [START_REF] Hawking | The large scale structure of space-time[END_REF][START_REF] Wald | General relativity[END_REF][START_REF] Weinberg | Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity[END_REF]. Even, concrete interplay between cosmology and the mathematical theory of dynamical systems, were involved in the literature, but this seems to be not well known by "mathematicians" (which gives motivation for our text here). As recent references, we may quote [START_REF] Andersson | The global existence problem in general relativity[END_REF][START_REF] Ringström | The Bianchi IX attractor[END_REF][START_REF]Dynamical systems in cosmology[END_REF].

For the general material on homogeneous spaces, we quote [START_REF] Besse | Einstein manifolds[END_REF][START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF][START_REF] Milnor | Curvatures of left invariant metrics on Lie groups[END_REF][START_REF] O'neill | Semi-Riemannian geometry. With applications to relativity[END_REF]. Finally, in these proceedings, recommended references would include [START_REF] Kloeckner | Produits scalaires pseudo-euclidiens[END_REF][START_REF] Sari | Introduction aux systèmes dynamiques et applications à un modèle cosmologique[END_REF].

A MULTIFACETED SPACE

The general linear group GL(n, R) acts transitively on the space of positive scalar products on R n , the stabilizer of the canonical scalar product being the orthogonal group O(n). This space is therefore identified to the homogeneous space GL(n, R)/O(n). It will be sometimes more convenient to deal with a reduced variant:

Y n = SL(n, R)/SO(n)
which then represents conformal scalar products, or equivalently, scalar products with unit volume (i.e. their unit ball has volume 1 with respect to the canonical volume).

For a more intrinsic treatment, we start with a real vector space E, and consider Sym(E) the space of its quadratic forms. Inside it, we have the open subspace of all non-degenerate ones Sym * (E), and Sym + (E) (or X(E)) the open cone of positive definite ones. We also get spaces of conformal structures by taking quotient by R acting by homothety; in particular SSym + (E) (or Y (E)) will denote the space of conformal positive definite structures. In the case of E = R n , we use the notations:

Sym n , Sym * n , Sym + n (= X n ), SSym + n (= Y n ) They are identified to subspaces of symmetric matrices {A = A * ∈ M nn }. The last three spaces correspond respectively to: det A = 0, A has positive eigenvalues, and positive eigenvalues with det A = 1.

Exercise 2.1. Show that the GL(n, R)-action on Sym n is given by g.A = g * Ag (where g * is the transpose of g).

2.0.1.

A metric on the space of metrics. A Euclidean structure q on a vector space E induces similar ones on associated spaces, in particular on the dual E * and on E * ⊗ E * . If (e i ) is a q-orthonormal basis, then its dual basis (e * i ) is also orthonormal, and also is the basis (e * i ⊗ e * j ).

Now, since Sym

+ n is open in Sym n , its tangent space at any point q is naturally identified to Sym n , which is thus endowed with the scalar product , q . Therefore, Sym + n becomes (tautologically) a Riemannian space. Exercise 2.2. Show that p, p q = tr(q -1 pq -1 p) (= tr(pq -1 pq -1 )) [START_REF] Abraham | Foundations of mechanics[END_REF] (where p ∈ T q (Sym + n )is identified with a matrix ∈ Sym n ). -Show that Sym + n is isometric to the product R * + × Y n , more precisely, one has an isometry: A ∈ Sym + n → (log det A, A det A ) ∈ (R, ncan) × Y n (where can stands for the canonical metric of R).

-Show that q → q -1 is an isometry, that is, Y n as well as X n are Riemannian symmetric spaces.

Exercise 2.3. For n = 1, Y n is R * + endowed with dx 2 x 2 . -Y 2 is "a hyperbolic plane", i.e. a homogeneous simply connected surface with (constant) negative curvature. Compute this constant. Does this correspond to a classical model of the hyperbolic plane?

-Show that the SL(n, R)-action on Y n factors through a faithful action of P SL(n, R).

2.0.2. Symmetric matrices vs quadratic forms. We guess it is worthwhile to seize the opportunity and clarify the relationship between quadratic forms and their representations as matrices. Let E be a vector space, and as above Sym(E) and Sym + (E) its spaces of quadratic forms, and those which are positive definite, respectively. A basis (e i ) of E yields a matricial representation isomorphism

P ∈ Sym(E) → p = (P (e i ), P (e j )) ij ∈ Sym n (= Sym(R n ))
Justified by a latter use (see §8), the restriction to Sym + (E), will be denoted by:

Q ∈ Sym + (E) → q ∈ Sym + n .
In fact, these representations depend only on the scalar product on E for which the given basis is orthonormal. Now, given (Q, P ) ∈ Sym + (E) × Sym(E), that is a pair of a scalar product together with a quadratic form on E, one associates a Q-autoadjoint endomorphism f :

E → E, representing P by means of Q, that is P (x, y) = Q(x, f (y)) = Q(f (x), y),
where P and Q are understood here as symmetric bilinear forms. Fact 2.4. Given (Q, P ) and their associates (q, p), the endormorphism f has a matrix representation A = q -1 p.

Conversely, given Q and a Q-autoadjoint endomorphism f , its corresponding quadratic form P has a matrix representation p = qA.

2.1. Flats. Let B = (e i ) be a basis of E. The flat F B ⊂ Sym + (E) is the space of scalar products on E for which B is orthogonal. It is parametrized by n positive reals x i , its elements have the form:

Σx i e * i ⊗ e * i .
Exercise 2.5. Show that the metric induced on F B is given by Σ i

dx 2 i x 2 i
, and that

(t 1 , . . . , t n ) ∈ (R n , can) → Σ exp(t i )e * i ⊗ e * i ∈ F B is an isometric immersion.
Prove that: -F B is totally geodesic in Sym + (E).(Hint: make use of the isometries of Sym + n , p → σ * i pσ i , where σ i is the reflection fixing all the e j , j = i, and σ i (e i ) = -e i ). -GL(E) acts transitively on the set of flats of the form F B (Hint: relate this to the simultaneous diagonalization of quadratic forms).

-Any geodesic of Sym + (E) is contained in some F B .

AN INDIVIDUAL LEFT INVARIANT METRIC

Standard references for this section and the following one are [START_REF] Besse | Einstein manifolds[END_REF][START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF][START_REF] Milnor | Curvatures of left invariant metrics on Lie groups[END_REF][START_REF] O'neill | Semi-Riemannian geometry. With applications to relativity[END_REF].

3.0.1. Equation of Killing fileds. Let (M, , ) be a pseudo-Riemannian manifold.

A Killing field X is a vector field that generates a (local) flow of isometries. Any vector field X has a covariant derivative which is an endomorphism of T M defined by: D

x X : u ∈ T x M → ∇ u X(x) ∈ T x M , where ∇ is the Levi-Civita connection of the metric. Fact 3.1. X is a Killing field, iff, D x X is skew-symmetric with respect to , x , for any x ∈ M .
Proof. Let us first recall that this is the case for the Euclidean space: in fact, this is equivalent to that the Lie algebra of the orthogonal group is the space of skewsymmetric matrices.

In the general case, assume x generic, that is, X(x) = 0, and consider N a small transverse submanifold (to X at x). Let Y and Z be two vector fields defined on N , and extend them on open neighborhood of x, by applying the flow of X, that is by definition:

[X, Y ] = [X, Z] = 0. If X is Killing, then Y, Z is constant along X: X. Y, Z = 0. Thus, by definition of the Levi-Civita connection, ∇ X Y, Z + Y, ∇ X Z = 0. Apply the commutations ∇ X Y = ∇ Y X, ∇ X Z = ∇ Z X, to get: 0 = ∇ Y X, Z + Y, ∇ Z X , that is D x X is skew-symmetric.
It is also easy to use those arguments backwards, that is, if D x X skew-symmetric for any x, then X is a Killing field.

3.0.2. Three Killing fields. Let now X, Y and Z be three Killing fields. Apply skew-symmetry for all their covariant derivatives, and get (at the end of substitutions) the following formula:

(

2) 2 ∇ X Y, Z = [X, Y ], Z + [Y, Z], X -[Z, X], Y Remark 3.2.
Observe the beauty (= symmetry) and easiness of the formula! Remark 3.3. Recall Koszul's formula for the Levi-Civita connection:

2 ∇ Y Z, X = Y Z, X + Z X, Y -X Y, Z - Y, [Z, X] + Z, [X, Y ] + X, [Y, Z] (3)
It implies the previous formula of three Killing fields. Conversely, this last formula yields Koszul's one for any combination of Killing fields with coefficients (nonnecessarily constant) functions on M . In particular (2) yields (3) if M is homogeneous. In fact, as (2) holds as X, Y and Z are pointwise Killing at order 1, (2) yields (3) also in the general non-homogeneous case.

3.1. Left invariant metrics. We are interested now on Riemannian metrics on a Lie group G which are left invariant, i.e any left translation x → gx is isometric. This is in particular the case of any flow φ t (x) = g t x, where {g t } is a one-parameter subgroup of G. Its infinitesimal generator X(x) = ∂φ t ∂t (x)| t=0 is a Killing field. This is a right invariant vector field: X(g) = X(1)g.

Therefore, a left invariant metric is exactly a metric admitting the right invariant fields as Killing fields.

Another characterization is that a left invariant metric is one for which left invariant fields have a constant length. (but they are not necessarily Killing).

Such a metric is equivalent to giving a scalar product on the tangent space of one point in G, say T 1 G, i.e. the Lie algebra of G. We keep the same notation , for both the metric on G and the scalar product on its Lie algebra G.

Here, to be precise, we define the Lie algebra G as the space of right invariant vector field on G. (The other choice, i.e. that of left invariant vector fields, would induce modification of signs in some formulae).

From the formula of three Killing fields, one sees that the connection on G is expressed by means of the scalar product on G and the Lie bracket. In other words, one can forget the group G and see all things on the Lie algebra. For instance the Riemann curvature tensor is a 4-tensor on G. The Ricci curvature is just a symmetric endomorphism of G.

The connection.

Any Lie group has a canonical torsion free connection defined for right invariant vector fields by:

∇ X Y = 1 2 [X, Y ].
Any other left invariant connection can be written

∇ X Y = 1 2 [X, Y ] + C(X, Y )
, where C : G × G → G is a bilinear map, and the connection is torsion free, iff, C(X, Y ) = C(Y, X). In the case of the Levi-Civita connection of a metric, one deduces from the formula of three Killing fields

2 C(X, Y ), Z = [X, Z], Y + X, [Y, Z] = ad * X Y, Z + ad * Y X, Z (4)
(Here as usually ad u v = [u, v], and ad * is its adjoint with respect to , ). It then follows:

(5)

∇ X Y = 1 2 ([X, Y ] + ad * X Y + ad * Y X 3.2.1. Sectional curvature.
The following formulae for various curvatures follow from Eq. ( 4), see for instance [START_REF] Besse | Einstein manifolds[END_REF] for detailed proofs.

R(X, Y )Y, X = - 3 4 [X, Y ], [X, Y ] - 1 2 [X, [X, Y ], Y - 1 2 [Y, [Y, X], X] + C(X, Y ), C(X, Y ) -C(X, X), C(Y, Y ) + Y, [[X, Y ], X]
3.2.2. Ricci curvature. The tensor C disappears in the expression of the Ricci and scalar curvatures!

Ric(X, X) = - 1 2 B(X, X) -[Z, X], X - 1 2 Σ i |[X, e i ]| 2 + 1 4 Σ ij [e i , e j ], X 2 (6)
where,

• B is the Killing form of G, that is the bilinear form (X, Y ) → tr(ad X ad Y ),

• (e i ) is any orthonormal basis of (G, , ),

• and finally, Z is the vector of G defined by Z, Y = trace(ad Y ), that is, it measures the unimodularity defect of G by means of , .

3.2.3. Scalar curvature. r = - 1 4 Σ ij |[e i , e j ]| 2 - 1 2 Σ i B(e i , e i ) -|Z| 2 (7)
3.3. Warning: left vs right. A metric is bi-invariant i.e. invariant under both left and right translations of G if and only if its associated scalar product is invariant under the Ad representation. In this case, the connection is the canonical one given by ∇

X Y = 1 2 [X, Y ],
for X and Y right-invariant vector fields. As said above, this torsion free connection exists on any Lie group, but does not in general derive from a Riemannian or a pseudo-Riemannian metric (i.e. it is not a Levi Civita connection). Its geodesics trough the neutral element are one-parameter groups, and its curvature is given by R(X, Y )Z = (1) For any right invariant (Killing) field X, X, X is constant on G.

(2) The orbits of any such X are geodesic (3) The orbit of 1 (∈ G) under any such X is a one parameter group.

All this suggests the possibility to define other quantities essentially equivalent to the Ricci curvature. Say, in the bi-invariant case, the Ricci curvature is essentially the Killing form, and hence, in the general left invariant case, the remaining part Re (of the Ricci curvature) is an obstruction to the constancy of X, X , or (equivalently) an obstruction for one parameter groups to be geodesic. A naive construction goes as follows. For X a right invariant vector field, let l X : x ∈ G → X(x), X(x) , its length function, and dl X 1 its differential at 1. Define Re(X, X) = tr(dl X 1 ⊗ dl X 1 )...

CURVATURE MAPPINGS ON X n

4.1. All scalar products together: The space Sym + (G) et al. We are now considering all left invariant Riemannian metrics on G. As said above the space of such metrics can be identified with Sym + (G), the space of positive definite scalar products on G. Let Sym(G) be the space of all quadratic forms on G. Then, the above formula for the Ricci curvature determines a map:

Ric : Sym + (G) → Sym(G)
4.2. Aut(G)-action on Sym + (G). Not only interior automorphisms of G, but also "exterior" ones, i.e. general automorphisms act on Sym(G). Their group Aut(G) is sometimes identified with Aut(G), assuming implicitly that G is simply connected. Of course, this action is compatible with all mappings that will be discussed below.

All Lie algebras together.

We can now go a step further and deal with all Lie algebras of a given dimension n. As vector spaces they are identified with R n . The space of quadratic forms and the positive definite one are Sym n and Sym + n . In short, for any Lie algebra G (endowed with a basis allowing one to identify it with R n ), we have a Ricci and a scalar curvature mapping, involving the bracket structure of G:

Ric G : Sym + n → Sym n r G : Sym + n → R 4.4.
Formulae. The Lie algebra G has a basis (e i ). An element of Sym n is denoted by p = (x ij ) 1≤,i,j,≤n . The structure constants C k ij , are defined by [e i , e j ] = C k ij e k . (Here and everywhere in this paper, if a letter is repeated as a lower and an upper index, like k is the last equation, we use the Einstein summation convention, for example, a j i b j stands for Σ j a j i b j , etc...). From Formula (6), we have:

Ric G : q = (x ab ) ∈ Sym + n → p = (X ab ) ∈ Sym n
Where:

X ab = - 1 2 B ab - 1 2 C k ai C l bj x kl x ij + 1 4 C p ik C q jl x pa x qb x ij x kl + 1 2 E ab (8)
and, B ab = C j ai C i bj is the matrix representing the Killing form, (x ij ) is the inverse matrix of (x ij ), and

E ab = C j ij x is (C l sa x lb + C l sb x la )
This last term depends on q, but vanishes identically if G is unimodular. So, assuming G unimodular will simplify and shorten the formula. 4.5. Parameter. To simplify, we will restrict ourselves to unimodular algebras, and so (E ab ) disappears. Any Lie algebra is defined by a system (C k ij ) which furthermore satisfies the Jacobi identity. We can then consider a mapping,

Ric : ( -→ C , q) = ((C k ij ), (x ab )) ∈ R n 3 × Sym + n → p = (X ab ) ∈ Sym n X ab = - 1 2 C j ai C i bj - 1 2 C k ai C l bj x kl x ij + 1 4 C p ik C q jl x pa x qb x ij x kl (9)
This map is equivariant with respect to the GL(n, R) × GL(n, R)-action on the source and the GL(n, R)-action on the target.

THREE-DIMENSIONAL CASE: BIANCHI GEOMETRIES (OF CLASS A)

A Bianchi space (or geometry) is a homogeneous Riemannian 3-manifold together with a transitive free action of a Lie group. This is therefore equivalent to giving a left invariant Riemannian metric on a 3-dimensional Lie group. These groups have been classified by Bianchi (see for instance [START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF][START_REF]Dynamical systems in cosmology[END_REF]). They split into classes A and B, according to they are unimodular or not. To simplify we will consider here only class A.

Milnor (or Bracket cyclic) bases of a 3-dimensional Lie algebra.

Let G (a,b,c) be the Lie algebra generated by {u, v, w} with relations:

[u, v] = aw, [v, w] = bu, [w, u] = cv
It is easy to show that this is actually a Lie algebra, i.e. that the Jacobi identity is satisfied.

Conversely, a basis B = {u, v, w} of a Lie algebra G is called a Milnor basis ( [START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF]) if it satisfies the previous relations:

[u, v] = aw, [v, w] = bu, [w, u] = cv. In particular, G is then isomorphic to G (a,b,c) .

Invariance of Milnor flats under

Ric. Let us recall here that in dimension 3, giving the Ricci curvature is equivalent to giving the full Riemann curvature tensor (in higher dimension, Ricci is too weaker than Riemann).

Recall that the flat F B determined by B is the space of scalar products on G for which B is orthogonal.

We will say that

F B is a Milnor flat if B is a Milnor basis.
The flat F B is parametrized by 3 positive reals x, y and z, where:

x = u, u , y = v, v , and z = w, w .

The corresponding metric will be denoted by (x, y, z) ∈ (R + ) 3 Define the extended flat F B to be the set of all quadratic forms which are diagonal in the basis B, i.e. they have the same form as elements of F B , but x, y and z are allowed to be any real numbers.

One uses the orthonormal basis { u √ x , v √ y , w √ z } for the metric (x, y, z), and computes from Formula (6) (or (9)),

                 Ric(u, u) = Ric(v, v) = Ric(w, w) = 1 2 (b 2 x 2 yz -a 2 z y -c 2 y z ) + ac 1 2 (c 2 y 2 xz -a 2 z x -b 2 x z ) + ab 1 2 (a 2 z 2 xy -b 2 x y -c 2 y x ) + bc (10)
In fact, Ric is diagonal in the basis B, i.e. Ric(u, v) = . . . = 0. We can on the other hand perform some simplification, for instance, for Ric(u, u) we have:

1 2 (b 2 x 2 yz -a 2 z y -c 2 y z ) + ac = 1 2yz (b 2 x 2 -c 2 y 2 -a 2 z 2 + 2acyz) = 1 2yz (b 2 x 2 -(cy -az) 2 )
Proposition 5.1. Consider a 3-Lie algebra G, and a Milnor flat F B ⊂ Sym + 3 . Then, the Ricci map sends:

(x, y, z) ∈ F B → (X, Y, Z) ∈ F B
and is given by:

               X = Y = Z = 1 2yz (b 2 x 2 -(cy -az) 2 ) 1 2xz (c 2 y 2 -(az -bx) 2 ) 1 2xy (a 2 z 2 -(bx -cy) 2 ) (11)
Remark 5.2. Observe the complete symmetry of these equations: there is for instance "a duality" x → b: everywhere the coefficient of x (resp. x 2 ) is b (resp. b 2 ). Recall that the coordinate x correponds to the vector u, and they are both related to the coefficient b, by the fact that the unique bracket proportional to u is [v, w] = bu. The same observation applies to the other variables, following a same correspondence: (x, y, z) → (b, c, a).

Scalar curvature.

Similarly, from Formula (7), we infer: In fact, this is equivalent to the more precise: Corollary 6.2. There is six isomorphism classes of unimodular 3-algebras, represented as follows, where below + (resp. -) means any positive (resp. negative) number, e.g. +1 (resp. -1).

r = 1 2xyz (-b 2 x 2 -c 2 y 2 -a 2 z 2 + 2acyz + 2abxz + 2bcxy). (12
(1) (0, 0, 0): the abelian algebra R 3 .

(2) (0, 0, +): the Lie algebra of G = Heis, the Heisenberg group.

(3) (0, -, +): the Lie algebra of G = Euc, the group of rigid motions of the Euclidean plane (i.e. the isometry group of the affine Euclidean plane). It is a semi-direct product R ⋉ R 2 , where R acts on R 2 , by cos t -sin t sin t cos t .

(4) (0, +, +): the algebra of the group G = SOL, the group of rigid motions of the Minkowski plane (i.e. the Minkowski space of dimension 1 + 1), G = R ⋉ R 2 , where R acts on R 2 by e t 0 0 e -t .

(5) (-, +, +): sl(2, R).

(6) (+, +, +): so(3). They are labeled respectively, Bianchi: I, II, V II 0 , V I 0 , V III and IX.

Proof. (see [START_REF] Milnor | Curvatures of left invariant metrics on Lie groups[END_REF]). In dimension 3, there are exactly two semi-simple Lie algebras, so(3) and sl(2, R), and there are no semi-simple Lie algebra of dimension ≤ 2. Therefore, if a 3-algebra G is not semisimple, then it contains no semisimple algebra, that is, G is solvable.

Assume G contains an (abelian) ideal isomorphic to R 2 . Take any supplementary one dimensional subspace (hence a subalgebra) R. Then, G writes as a semi-direct product of R acting on R 2 . Since G is assumed to be unimodular, this action is via a one parameter group of SL(2, R). These one parameter groups split into: elliptic, parabolic and hyperbolic types. We obtain respectively: Euc, Heis and SOL.

It suffices therefore to show the existence of such an ideal R 2 . For this, let us remark that there exists always an abelian ideal of dimension 1, say A ∼ = R. Indeed, if the commutator ideal has dimension 2, then since it is solvable, its commutator has dimension 1 or 0... Now, since G acts on A, the Kernel L has at least dimension 2 and contains A. Let us consider here the case dim L = 2, since the dimension 3 case is easier. By definition (of the Kernel) this is an ideal, and since it has dimension 2 and has a non-trivial center (it contains A), then it is abelian.

SUMMARY; FURTHER COMMENTS

In this section, we present a general setup where the previous constructions can be defined. In other words, we "summarize" how one associates to a Lie group various rational dynamical systems. 7.1. A rational map. ( §4.4). Let G be a Lie algebra of dimension n, with a basis (e i ), such that [e i , e j ] = C k ij e k , and assume to simplify that it is unimodular. Then, we have a rational map:

Ric G : (x ab ) ∈ Sym + n → (X ab ) ∈ Sym n X ab = - 1 2 B ab - 1 2 C k ai C l bj x kl x ij + 1 4 C p ik C q jl x pa x qb x ij x kl (13)
as well as a rational (scalar curvature) function:

r G : (x ab ) ∈ Sym + n → X ab x ab (14)
(recall that (x ij ) is the inverse matrix of (x ij )).

7.2. Aut(G)-equivariance. Both Ric G and r G are respectively equivariant and invariant under the Aut(G)-action on Sym n (identified with Sym(G), the space of quadratic forms on G). 7.3. Extensions. Actually, everything extends to left invariant pseudo-Riemannian metrics on G, or equivalently to Sym * (G) the space of non-degenerate quadratic forms on G. The same formulae allow one to calculate the Ricci and scalar curvature of such metrics.

• Now, the formulae may have sense even for some degenerate quadratic forms! • We can also consider complex quadratic forms on the complexification of G. They form a space Sym(G) ⊗ C identified with Sym n (C), the space of symmetric complex matrices. In other words, to a Lie group G of dimension n is associated an Aut(G)-equivariant rational transformation Ric G on Sym n (C). We also have an Aut(G)-invariant meromorphic function r G .

• Ric G can be written as a rational vectorial map

( P i Q i ) i : C n(n+1)/2 → C n(n+1)
/2 where P i and Q i are homogeneous polynomials of a same degree 2n (on n(n + 1)/2 variables).

• From this, one gets a rational transformation of the projective space CP n(n+1)/2-1 ∼ = P Sym n (C). We will denote it by Ric G , to emphasize that it is the extension to the complex projective space. For instance, if n = 3, then we have a rational map on CP 5 .

• Ric G is equivariant under scalar multiplication. In fact, as this follows from its defining formula, Ric G is polynomial when restricted to matrices with det = 1. Therefore, a representative of Ric G (on the projective space) is the polynomial map (P i ) i (of degree 2n).

• Ric G is invariant under the (algebraic) action of (the complexification of) Aut(G) on CP n(n+1)/2-1 ( ∼ = P Sym(G)⊗C). It then determines a map on the quotient space. It depends however on the meaning to give to such a quotient space (by Aut(G)). As an algebraic action, it has a poor dynamics, and a nice quotient space can be thus constructed for it. There is in particular a notion of "algebraic quotient". In dimension n = 3, the algebraic quotient space has dimension 5 -3, more exactly, it is a (singular) compact complex surface S G , say. We have then associated to a 3-dimensional Lie group a rational map on a compact complex surface S G . This map seems to have a "poor dynamics", for instance, it has in general a vanishing entropy. We guess nevertheless that (other) "dynamical invariants" of it can characterize the group G (i.e. two different groups have different invariants). It is also worthwhile to see what happens in higher dimension case. 7.4. Forget invariance. In the formula defining Ric G , we can consider any system of parameters (C k ij ), not necessarily satisfying the Jacobi identity of Lie algebras. We obtain a big family of rational transformations generalizing those associated to Lie groups. In this case, various dynamical types may appear. We think it is worthwhile to investigate the structure of this parameter space, and to understand inside it, the (algebraic) set of Lie algebras, the algebraic actions on it... 7.5. Cross sections, Flats. Let us call a cross section S for Ric G or Ric G a submanifold in Sym + n (resp. P Sym n (C)) which is invariant under Ric G (resp. Ric G ) and such that S meets any G-orbit in a non-empty discrete set. The last condition implies in particular that S is transversal (at least in a topological sense) to the G-orbits. In fact there are weaker variants of this definition which can be useful, in particular in a geometric algebraic context.

• It is in the case where G is unimodular and has dimension 3 that cross sections occur easily. In this case, one can in fact find them, as affine (resp. projective) subspaces for Ric G (resp. Ric G ) of dimension 3 (resp. 2). Indeed, as explained in §6, the Lie algebra of such a group has a Milnor basis B = {u, v, w} ( §5.0.1), and the flat F B ( §2.1), or more formally its "extension" F B , i.e. the space of quadratic forms diagonalizable in B, is invariant under Ric G . In order to see that one gets in this way a cross section, it remains to show the abundance of Milnor bases as in the following statement, Exercise 7.1. Prove that any quadratic form can be digonalized in some Milnor basis of G. (Hint: this can be done by checking case by case. For instance, for the group SOL, its Lie algebra is generated by X, Y, Z, with relations [X, Y ] = Y, [X, Z] = -Z and [Y, Z] = 0. Consider u = X +T , where T belongs the the plane P generated by Y and Z. Then, the restriction of ad u on P, satisfies ad 2 u = -1. Choose u to be orthogonal to P (with respect to the given metric). Consider a non-vanishing vector v ∈ P, and let w = ad u (v). Then {u, v, w} is a Milnor basis, because of the fact ad 2 u = 1. We claim that v can be chosen such that w is orthogonal to v. This is a calculation in the basis {Y, Z}).

• Let us point out the following polynomial presentation of Ric G . As was said above, Ric G is invariant under scalar multiplication, that is, it suffices to consider its restriction on unimodular matrices SSym n , in which case, it becomes polynomial. In particular, from §5.1, Ric G has the following form as a cubic homogeneous polynomial map:

Ric G : C 3 → C 3 (15) (x, y, z) → 1 2 (x(b 2 x 2 -(cy -az) 2 ), y(c 2 y 2 -(az -bx) 2 ), z(a 2 z 2 -(bx -cy) 2 )).
7.6. Formula in each case.

7.6.1. Case of SO(3): a = b = c = 1. Ric(x, y, z) = 1 2 (x(x 2 -(y -z) 2 ), y(y 2 -(z -x) 2 ), z(z 2 -(x -y) 2 )). 7.6.2. Case of SL(2, R): a = b = 1, and c = -1. Ric(x, y, z) = 1 2 (x(x 2 -(y + z) 2 ), y(y 2 -(z -x) 2 ), z(z 2 -(x + y) 2 )).
∂g t ∂t = -2Ric(g t ) + 2 < r(g t ) > n g t
where n = dim M and < r(g) >= r(g)dv g /V ol(M, g) is the average scalar curvature of g [START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF]. 7.7.1. The vector field Ric G . Now, if G is an n-dimensional Lie group, then this gives a classical differential equation on the space of its left invariant Riemannian metrics, where one takes a punctual value of the scalar curvature instead of its average (since this scalar curvature is constant). Equivalently, this is a vector field on Sym + n . In fact, all this is derived from our previous rational map Ric G . Since Sym +

n is an open set in the vector space Sym n , the vectorial map Ric G : Sym + n → Sym n can be alternatively seen as a vector field on Sym + n , say, Ric G . Observe that Ric G is invariant under the Aut(G)-action on Sym + n (which is equivalent to the fact that Ric G is equivariant under the (linear) action of Aut(G)).

Let us denote a generic point of Sym + n by q, and consider the radial vector field V(q) = q. The previous differential equation, which we will call the Bianchi-Ricci flow associated to G is the vector field

-2Ric G + 2 r G n V 7.7.2. Commutation. Consider the bracket [Ric G , V] = D V Ric G -D Ric G V
, where D u denotes the usual derivation in the u-direction. This equals 0 -Ric G , since, D V Ric G = 0, i.e. the map Ric G is invariant under multiplication; and DV = Identity, everywhere. Therefore, [Ric G , V] = -Ric G . Because of this commutation rule (that is, the two vector fields generate a local action of the affine group), the essential dynamics of the Bianchi-Ricci flow comes from the Ric G -part. 7.7.3. Bianchi-Hilbert-Ricci flow. The remark applies to any combination of Ric G and V: understanding one combination allows one to understand the others. A famous one is Ein = Ric G -r 2 V, which can be called in this context the "Bianchi-Einstein flow", since the tensor Ric(g) -r 2 g of a Riemannian manifold (M, g) is called Einstein tensor (this is, essentially, the unique combination of Ric(g) and g which is divergence free). However, in order to prevent confusion with "Einstein equations" and some related flows which will be considered below, Ein could be better called Bianchi-Hibert flow. Indeed, the function

H : p ∈ Sym + n → r(p) det(p) ∈
R is the substitute of the classical Hilbert action in the case of left invariant metrics. Indeed: Exercise 7.2. Show that Ein is a gradient vector field. More exactly, Ein = ∇H, where the gradient ∇ is taken with respect to the metric of Sym + n . Remark 7.3. The computation can be handled in a more explicit way on a Milnor flat F B ( §5.1), where the Hilbert action has the form:

H(x, y, z) = 1 2 √ xyz (-b 2 x 2 -c 2 y 2 -a 2 z 2 + 2acyz + 2abxz + 2bcxy)
and the metric is

dx 2 x 2 + dy 2 y 2 + dz 2 z 2 7.7.4. Restriction on SSym + n .
The interest of the normalization in the definition of the Ricci flow is to let it preserving the volume of the Riemannian metric, that is, the total volume remains constant under evolution. In the case of left invariant metrics, this is equivalent to the fact that the vector field Ric G -r n V is tangent to SSym n . This in turn is equivalent to the fact, that for any q ∈ SSym n , Ric G (q) -r(q) n q is trace free, which follows from the very definition of the scalar curvature r. This allows one to justify the following simplification: write equations assuming q ∈ SSym n , i.e. det(q) = 1, which gives polynomial equation. However, in order to keep this polynomial natural, do not take reduction of variables from the equation det(q) = 1. To be more concrete, consider a flat F B , then instead of the rational forms of Ric G (x, y, z) and r G (x, y, z), we assume xyz = 1 which leads to polynomial forms: § §5.1 and Formula 12 (but we do not go further and eliminate one variable, say z = 1 xy ). We can then write the Bianchi-Ricci flow as follows ( 16)

-2Ric G +2 r 3 V(x, y, z) =                x( 2 3 bx(-2bx + cy + az) + 2 3 c 2 y 2 + 2 3 a 2 z 2 -4 3 acyz) y( 2 3 cy(-2cy + bx + az) + 2 3 b 2 x 2 + 2 3 a 2 z 2 -4 3 bcxz) z( 2 3 az(-2az + bx + cy) + 2 3 b 2 x 2 + 2 3 c 2 y 2 -4 3
bcxy) 7.7.5. Differential equations on a projective space. In the same way, we associate to a Lie group G an Aut(G)-invariant one dimensional complex algebraic foliation on the projective space P Sym n (C). Here, among combinations of the vector fields Ric G and V, only Ric G is relevant, since the radial vector field V becomes trivial on the projective space. In the case of a unimodular 3-group, we have the following homogeneous cubic differential system on C 3 :

(17)                dx dt = dy dt = dz dt = x(b 2 x 2 -(cy -az) 2 ) y((c 2 y 2 -(az -bx) 2 ) z(a 2 z 2 -(bx -cy) 2 )
7.7.6. Dynamics, compactifications. It is the dynamics of the Bianchi-Ricci flow -2Ric G + 2r 3 V which was investigated in the literature [START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF][START_REF] Isenberg | Ricci flow of locally homogeneous geometries on closed manifolds[END_REF]. As we argued above, this is essentially the same as that of the Einstein-Hilbert field ∇H. But, as a gradient flow, its dynamics is completely trivial on Sym + n ... The point is to study the behavior of orbits when they go to an infinity boundary ∂ ∞ Sym + n . There is however several ways to attach such a boundary to (the non-positively curved Riemannian symmetric space) Sym + n . One naturally wants to interpret ideal points as collapsed Riemannian metrics. With respect to this, the Hadamard compactification seems to be the most pertinent (see for instance [START_REF] Kloeckner | Symmetric spaces of higher rank do not admit differentiable compactifications[END_REF]). On the other hand, the advantage of algebraic compactifications (e.g. the projective space) is to extend the dynamics...

HAMILTONIAN DYNAMICS ON Sym + n

After consideration of some maps and vector fields, we are now going to study second order differential equations on Sym + n , the prototype of which is the geodesic flow of Sym + n , and then the "Einstein flow" associated to a Lie group.

8.1. Geodesic flow. Write the metric on Sym + n as: L(q, p) = p, p q = tr(q -1 pq -1 p).

Since Sym +

n is open in Sym n , its tangent bundle trivializes T Sym + n = Sym + n × Sym n . We will use the usual notations ∂L ∂q , ∂L ∂p for the horizontal and vertical differentials d p L and d q L.

We have:

∂L ∂q (δq) = tr(-q -1 (δq)q -1 pq -1 p-q -1 pq -1 (δq)q -1 p) = -2tr((δq)q -1 pq -1 pq -1

)
where δq is a horizontal tangent vector, i.e. an element of Sym n .

∂L ∂p (δp) = 2tr((δp)q -1 pq -1 ).

Now, write: q = q(t), p(t) = q = ∂q ∂t , and compute ∂ ∂t ∂L ∂p (δp) = 2tr((δp)q -1 [-2 qq -1 q + q]q -1 ).

The Euler-Lagrange equation is obtained by taking δq = δp = A, and writing for any A,

∂ ∂t ∂L ∂p A - ∂L ∂q A = 0.
This reads:

tr(2Aq -1 (-qq -1 q + q)q -1 ) = 0, ∀A ∈ Sym n .

and therefore, Fact 8.1. The equation of geodesics of Sym + n is the second order matricial equation on Sym + n : q = qq -1 q or equivalently (in the phase space):

(18) q = ṗ = p pq -1 p.

Other pseudo-Riemannian and Finsler metrics on Sym +

n . There is a canonical GL(n, R)-invariant form ω on Sym + n : ω q (p) = tr(q -1 p).

We can then associate to any reals α and β a Lagrangian: L α,β (q, p) = α(ω q (p)) 2 + β p, p q = α(tr(q -1 p)) 2 + βtr(q -1 pq -1 p).

For generic α and β, this is a homogeneous pseudo-Riemannian metric, but it can degenerate for some values.

Similarly, there are homogeneous Finsler metrics:

F α,β (q, p) = αω q (p) + β p, p q = α(tr(q -1 p)) + β tr(q -1 pq -1 p.

Exercise 8.2. Write the Euler-Lagrange equation for L α,β and F α,β .

-Solve the geodesic equation for Sym 2 .

EINSTEIN EQUATIONS IN A GAUSS GAUGE

Cylinders. Let M be a differentiable n-manifold endowed with a family of Riemannian metrics g t , t is a "time" parameter lying in an interval I. Consider the Lorentz manifold M = I × M endowed with the metric

, = ḡ = -dt 2 + g t , i.e. ḡ(t,x) = -dt 2 + (g t ) x
Such a structure is sometimes called a cylinder. Our purpose is to relate geometric (e.g. curvature) quantities on M and M . For a fixed point (t, x), R, Ric, and r will denote the Riemann, Ricci and scalar curvatures of (M, g t ) at x and ∇ its Levi-Civita connection. The corresponding quantities for M are noted by R, Ric and r and ∇.

9.1. Second fundamental form. The (scalar) second fundamental form of {t} × M is denoted k t (or sometimes simply k). Actually, the second fundamental form is defined as a vectorial form: II(X, Y ) equals the orthogonal projection of ∇X Y on Re 0 , where e 0 = ∂ ∂t . The scalar second fundamental form is defined by

k(X, Y ) = II(X, Y ), e 0 = ∇X Y, e 0 .
The Weingarten map a = a e 0 is defined by: a(X) = -∇X e 0 .

We have:

k(X, Y ) = ∇X Y, e 0 = X Y, e 0 -∇X e 0 , Y = 0 + a(X), Y .
In other words, a is the symmetric endomorphism associated to k by means of the metric g (we will use sometimes the notation a t as well as g t and k t , in order to emphasize the dependence on t).(The definition of k and a coincides with that in the Riemannian case. The unique difference is that here, II = -ke 0 , since e 0 is unit timelike, i.e. e 0 , e 0 = -1). 9.2. Geometry of the product. Consider e 1 , . . . , e n a frame of vector fields on M , that we also consider as horizontal vector fields on M . By definition, they commute with e 0 (= ∂ ∂t ). Fact 9.1. We have:

∇e 0 e 0 = 0 (the trajectories of e 0 are geodesic), [START_REF] Ryan | Homogeneous relativistic cosmologies[END_REF]

k t = (-1/2) ∂ ∂t g t , (20)
R(e 0 , e i )e i , e 0 = ∂ ∂t a t (e i ), e i + a 2 t (e i ), e i , ( 21)

Ric(e 0 , e 0 ) = ∂ ∂t tr(a t ) + tr(a 2 t ). ( 22
)
Proof. • We have 0 = ∂/∂t e 0 , e i = ∇e 0 e 0 , e i + e 0 , ∇e 0 e i .

But

e 0 , ∇e 0 e i = (1/2)e i . e 0 , e 0 = 0, since e 0 and e i commute. Therefore ∇e 0 e 0 , e i = 0, ∀i.

• We have ∂ ∂t g t (e i , e j ) = e 0 e j , e j = ∇e 0 e i , e j + ∇e 0 e j , e i Since e 0 commutes with e i and e j , this also equals:

-(a(e i ), e j -a(e j ), e i = -2k t (e i , e j )

• Computation of R(e 0 , e i )e i , e 0 : Because of the commutation relations, and because e 0 is geodesic, we have, by definition of the curvature:

R(e 0 , e i )e 0 = -∇e 0 a(e i ),

and thus: R(e 0 , e i )e 0 , e i = -∇e 0 a(e i ), e i = -e 0 a(e i ), e i -a(e i ), a(e i ) , and since a is symmetric, this also equals:

- ∂ ∂t a(e i ), e i -a 2 (e i ), e i
And hence, R(e 0 , e i )e i , e 0 = ∂ ∂t a t (e i ), e i + a 2 t (e i ), e i

• We can assume that at a fixed point (t, x), the basis (e i ) i≥1 is orthonormal, and taking the sum (over i > 0) we get:

Ric(e 0 , e 0 ) = ∂ ∂t tr(a t ) + tr(a 2 t ).

Remark 9.2. In fact, the meaning of "Gauss gauge" is nothing but that e 0 is unit and has geodesic orbits. 9.3. Gauss equation. It describes the relationship between the sectional curvatures for R and R: R(e i , e j )e j , e i = R(e i , e j )e j , i + k(e i , e i )k(e j , e j ) -k(e i , e j )k(e i , e j )

(observe this difference of sign of the k-term, in comparison with the Riemannian case). 9.4. Einstein evolution equation for k t . Again, assume (e i ) orthonormal, fix i, and take the sum over j > 0. We first have: Σ j k(e i , e i )k(e j , e j ) = k(e i , e i )tr(a) = tr(a) (a(e i ), e i ) and Σ j k(e i , e j )k(e i , e j ) = a 2 (e i ), e i ) (Indeed in matricial notations, a ij = a ji = k(e i , e j ), and thus (a 2 ) ii = Σ j a ij a ji ). Therefore, if we consider the quadratic form l, defined by: l(e i , e i ) = Σ j (k(e i , e i )k(e j , e j ) -k(e i , e j )k(e i , e j ))

then its associated endomorphism is:

tr(a)a -a 2
• Ric(e i , e i ) equals the trace of u → R(u, e i )e i . Remember, (e i ) is a Lorentz orthonormal basis, i.e. e i , e j = 0, for i = j, e 0 , e 0 = -1 and e j , e j = +1, for j > 0. It then follows that R(e i , e i ) = j>0 R(e i , e j )e j , e i -R(e 0 , e i )e i , e 0

• Returning to the Gauss equation [START_REF] Hooft | Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness[END_REF], and taking the sum over j > 0, we get:

Ric(e i , e i ) + R(e 0 , e i )e i , e 0 = Ric(e i , e i ) + (tr(a)a -a 2 )(e i ), e i

• Replacing R(e 0 , e i )e 0 , e i by its previous value:

Ric(e i , e i ) + ∂ ∂t a(e i ), e i + a 2 (e i ), e i = Ric(e i , e i ) + (tr(a)a -a 2 )(e i ), e i

Equivalently, for any

X, Y ∈ T M : ∂ ∂t k t (X, Y ) = -Ric(X, Y ) + Ric(X, Y ) + (tr(a t )a t -2a 2 t )(X), Y .
Fact 9.3. Define the square power k tg t k t to be the quadratic form associated by means of g t with the matrix a 2 t (where a t is the matrix associated to k t via g t ). Then:

∂ ∂t k t = -Ric + Ric + tr gt (k t )k t -2k tg t k t .
9.5. Gauss constraints. Consider again the Gauss equation and take the sum over i, j > 0:

r -2 Ric(e 0 , e 0 ) = r + (tr gt k t ) 2 -tr(k tg t k t ) = r + (tr gt k t ) 2 -|k t | 2 gt .
9.6. Matrix equations. We are now going to write equations by means of symmetric matrices associated to the quadratic forms g t and k t (see §2.0.2). For this, we fix x and a time t 0 and choose an orthonormal basis (e i (t 0 )) of T x M . We denote by q t (resp. p t ) the matrix associated with g t (resp. -2k t ), and by r ic t and ric t (or simply r ic and ric) those associated with Ric and Ric (recall they are the Ricci curvatures of respectively M , at (t, x), and (M, g t ), at x). With this, we have:

• Evolution equations: (24)    q = ṗ = p -ric + ric + 1 4 tr(q -1 p)p -1 2 q -1 pq -1 p
• Gauss constraints (actually said Hamiltonian constraints)

r -2 r ic(e 0 , e 0 ) = r + (tr(q -1 p)) 2 -tr(q -1 pq -1 p) = r + tr(q -1 p) 2 -p, p q = r -L -1,1 (q, p)
where L -1,1 is the pseudo-Riemannian metric defined in §8, for the value (α, β) = (-1, 1)

BIANCHI COSMOLOGY

We will now restrict ourselves to the vacuum case, i.e. M is Ricci-flat: Ric = 0, and thus also r = 0. We will also assume M is a Lie group G and the metrics on it (i.e. g t ) are left invariant. Therefore such a metric is identified with an element q ∈ Sym + n (n = dim G, the identification of Sym + (G) with Sym + n comes from a choice of a basis). Now, ric becomes a map ric : Sym + n → Sym n , and r : Sym n → R. We get the (beautiful) ODE system with constraints:

(25)            q = ṗ = L -1,1 (q, p) = p ric(q) + 1 4 tr(q -1 p)p -1 2 q -1 pq -1 p r(q) (Hamiltonian constraint)
Remark 10.1. Observe that ric and r are basic functions, they depend only on q (and not on p).

10.1. Isometric G-action on M . Here M = I × G, with ḡ = -dt 2 + g t . A left translation x ∈ G → hx is isometric for all the metrics g t , and therefore is isometric for ḡ as well.

10.2. The Bianchi-Einstein flow along and on a flat. Actually, there are other constraints to add to the ODE system above, in order to get what we will call the Bianchi-Einstein flow. These (momentum) constraints will be considered below. Before, let us consider a subsystem of it, the restriction (of everything) to a Milnor flat F B . The following proposition derives from Formulae ( 11) and ( 12).

Proposition 10.2. Let F B be a Milnor flat, and T F B its tangent bundle, a point of which is denoted by (q, p), q = (x, y, z), p = (x ′ , y ′ , z ′ ). The Bianchi-Einstein flow on F B is the following system of ODE on T F B , together with one algebraic constraint defined by a Lorentz metric on T F B and a basic function on F B . (The phase space has thus dimension 5, and is a fiber bundle over F B ):

(26)                                            ẋ = ẏ = ż = ẋ′ = ẏ′ = ż′ = x ′ y ′ z ′ 1 2yz (b 2 x 2 -(cy -az) 2 ) + 1 4 ( x ′ x + y ′ y + z ′ z )x ′ -1 2 x ′2 x 2 1 2xz (c 2 y 2 -(az -bx) 2 ) + 1 4 ( x ′ x + y ′ y + z ′ z )y ′ -1 2 y ′2 y 2 1 2xy (a 2 z 2 -(bx -cy) 2 ) + 1 4 ( x ′ x + y ′ y + z ′ z )z ′ -1 2 z ′2 z 2
The phase space is a hypersurface (maybe singular) N (in T F B ) defined by the Hamiltonian equation:

l (x,y,z) (x ′ , y ′ , z ′ ) = - r (x, y, z) 2 (27)
Where l is the Lorentz metric (on F B ):

(28) l (x,y,z) (x ′ , y ′ , z ′ ) = x ′ y ′ xy + x ′ z ′ xz + y ′ z ′ yz ,
and r is given by Formula (12):

r(x, y, z) = 1 2xyz (-b 2 x 2 -c 2 y 2 -a 2 z 2 + 2acyz + 2abxz + 2bcxy)
Exercise 10.3. Show explicitly that the constraint is preserved by the dynamics, i.e. the vector field determined by the differential equations is tangent to the "submanifold" N ⊂ T F B defined by the constraint.

Codazzi (or Momentum) constraints.

The Codazzi equation establishes a relation between the intrinsic and extrinsic curvatures of a submanifold M in a Riemanniann manifold M , and is in fact valid in the general background of pseudo-Riemannian manifolds provided the induced metric on the submanifold is also pseudo-Riemannian, i.e. it is not degenerate. More precisely, it states that some "partial symmetrisation" of the covariant derivative of the second fundamental form (all this depends only upon data on M ) equals the normal part of the Riemann curvature tensor (this depends on M ). The equation gives obstructions for a (vectorial) 2-tensor to be the second fundamental form of a submanifold.

In the case where M is a CMC spacelike hypersurface (i.e. with a constant mean curvature) in a Ricci flat Lorentz manifold M , one can deduce from Codazzi equation, by taking a trace, that the second fundamental form k is a divergence free 2tensor. This applies in particular to our case: our hypersurfaces are G-orbits and thus are CMC.

Let us recall some definitions. Firstly, if k is a symmetric 2-tensor on M , then its covariant derivative ∇ X k with respect to a vector X, is a 2-tensor:

(∇ X k)(Y, Z) = Xk(Y, Z) -k(∇ X Y, Z) -k(Y, ∇ X Z)
Now divk is a 1-form, the trace of ∇k (with respect to the metric of M ), i.e. if (e i ) is an orthonormal basis: divk(X) = Σ i ∇ e i k(e i , X) 10.3.1. Divergence of left invariant quadratic forms on Lie groups. At first glance one can guess that left invariant objects are divergence free (with respect to left invariant Riemannian metrics). This is however false (apart from some trivial cases).

Let G be a 3-dimensional unimodular Lie group, endowed with a left invariant metric , = q ∈ Sym + (G), with a Milnor q-orthonormal basis {u, v, w}: Fact 10.5. For the basis {u, v, w}, we have: Then, for a generic Milnor basis, any divergence free left invariant quadratic form (with respect to the metric for which this basis is orthonormal) is diagonalizable in this basis.

• ∇ u u = ∇ v v = ∇ w w = 0, • 2∇ u w = (-c + a -b)v, 2∇ v w =
In other words (keeping the previous notation), along a Milnor flat F B , an element p ∈ T q Sym + 3 satisfies the momentum constraints, iff, p ∈ T q (F B ) (or in more linear words, p ∈ F B ) 10.4. Cross sections for the Bianchi-Einstein flow. On T Sym + 3 , the group Aut(G) acts, preserving the Bianchi-Einstein flow (determined by G). A cross section ( §7.5) will play the role of a flow on a quotient space (for the Aut(G)-action).

Proposition 10.8. The Bianchi-Einstein flow on a generic Milnor flat is a cross section of the full Bianchi-Einstein flow (with constraints) on Sym 3 endowed with the Aut(G)-action. Generic flats exist except in the abelian and nilpotent cases, i.e. when G is R 3 or the Heisenberg group Heis.

Proof. Firstly, one easily sees that if there exists a Milnor basis for which a and b = 0, then after re-scaling, this basis becomes generic. This exists exactly when G is different from R 3 , and Heis. If we are not in these cases, then we can assume, after re-scaling if necessary, that all Milnor bases are generic. Let B be such a basis, and (q, p) ∈ T Sym + n , then up to application of an element of Aut(G), q ∈ F B . But since B is generic, if p satisfies the momentum constraints, then p ∈ T F B , which means that T F B is a cross section. 10.4.1. Case of G = R 3 . In the case of the Heisenberg group, there is exactly one momentum constraint which gives rise to invariant sets of the system. There is no such constraint in the case of R 3 , where we obtain the following system:

(29)            q = ṗ = L -1,1 (q, p) = p 1 4 tr(q -1 p)p -1 2 q -1 pq -1 p 0 (the lightlike cone bundle of L -1,1 )
The spacetime M has a metric

ḡ = -dt 2 + t 2p 1 du 2 + t 2p 2 dv 2 + t 2p 3 dv 2 .
This is called a Kasner spacetime (observe that in some cases, e.g. p 1 = p 2 = p 3 , this is just the Minkowski space) [START_REF] Andersson | The global existence problem in general relativity[END_REF][START_REF] Bogoyavlensky | Methods in the qualitative theory of dynamical systems in astrophysics and gas dynamics[END_REF][START_REF] Weinberg | Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity[END_REF].

Exercise 10.9. Prove the previous form of ḡ and solve the same problem in the case of the Heisenberg group. 10.5. Isometry group of M . As said previously, the left action of G on itself induces, by definition of its metric, an isometric action on M . In fact, if for some level (M, g t ), there are extra-isometries (i.e. other than left translations), then they extend to M . More precisely, if the metric at some level, say t = 0, is identified with q 0 ∈ Sym + (G), and M corresponds to a point (q 0 , p 0 ) ∈ T Sym + n = Sym + n × Sym n , and K ⊂ Aut(G) is the stabilizer of (q 0 , p 0 ), then, on the one hand, K acts as an isometric isotropy group for (M, g 0 ) (g 0 corresponds to q 0 ). On the other hand, K preserves the Bianchi-Einstein trajectory of (q 0 , p 0 ), and thus acts isometrically on M (as isotropy for any point identified with 1 ∈ G). 10.6. An example: Bianchi IX. This means G = SO(3), or more precisely its universal cover the sphere S 3 . In this case, there are Milnor bases with a = b = c = 1. Any other Milnor basis satisfies these equalities, up to re-scaling. Also, all such bases are equivalent up to conjugacy and re-scaling. Yet, this is the most challenging case of Bianchi cosmologies (see for instance [START_REF] Ringström | The Bianchi IX attractor[END_REF]). As an example, TAUB-NUT spacetimes are exact solutions of the Bianchi-Einstein equations of class IX. They are characterized among Bianchi IX spacetimes as those having extra-symmetries, i.e. a non-trivial isotropy, which then must be SO(2) (and thus their isometry group is S 3 ×SO(2), up to a finite index). Nevertheless, their high complexity (at least among exact solutions) led people to describe them as "counter-examples to everything"! In a Milnor flat where a = b = c = 1, these spacetimes correspond (up to isometry) to x = y, and x ′ = y ′ . The left invariant metric on G (at any time) corresponds to a Berger sphere, i.e. (up to isometry) a metric on the sphere derived from the canonical one, by rescaling the length along the fibers of a Hopf fibration. In other words, the set of solutions of the Bianchi-Einstein flow, which are Berger spheres at any time, is closed and invariant, say, the TAUB-NUT set. 10.7. Effect of a non-vanishing cosmological constant. Instead of requiring M to be Ricci-flat, let us merely assume it to be Einstein, i.e. Ric = Λḡ. Its effect is essentially an additive constant (related to Λ) in all equations and constraints. This situation does not seem to be systematically investigated in the literature. In particular, one can wonder whether the introduction of Λ is "catastrophic" or in contrary produces only a moderate effect. A similar situation is that of the paradigmatic example in holomrophic dynamics, of the quadratic family z → z 2 + c. Here the variation of the parameter c generates a chaotic dynamics as well as a fractal geometry [START_REF] Beardon | Iteration of rational functions. Complex analytic dynamical systems[END_REF]. 10.8. Wick rotation. Here M = I × M is endowed with the Riemannian metric ĝ = +dt 2 + g t . Writing Ric = 0, yields:

(30)            ∂ ∂t g t = -2k t ∂ ∂t k t = -Ric + tr gt (k t )k t -2k tg t k t 0 = r + |k t | 2 gt -(tr gt k t ) 2 (Constraint)
For instance, this allows one to construct examples of Riemannian Ricci flat manifolds, of co-homogeneity 1, i.e. their isometry group has codimension 1 orbits.

Notice that the (true) Einstein equations (i.e. without symmetries) can not be solved in a Riemannian context (they cannot be transformed to a hyperbolic PDE system). Maybe, this Bianchi situation can give insights on the reasons behind this fact.

Finally, it does not seem there exists a "true Wick rotation", i.e. some correspondence between solutions of Bianchi-Einstein equations in the Lorentzian and Riemannian cases. (Compare with [START_REF] Benedetti | Canonical Wick rotations in 3-dimensional gravity[END_REF]). 10.9. Orthonormal frames approach vs Metric approach. As a result of a search on fundamental references in this area, "dynamical systems and cosmology", one can get at least [START_REF] Bogoyavlensky | Methods in the qualitative theory of dynamical systems in astrophysics and gas dynamics[END_REF][START_REF] Ryan | Homogeneous relativistic cosmologies[END_REF] and [START_REF]Dynamical systems in cosmology[END_REF] which are surely the most known and recent synthesis in this "emerging" domain. The authors adopted there an "orthonormal frames approach" in opposite to our "metric approach" here (see explanations therein). They obtained the following system of quadratic polynomial differential equations on R 5 .

(31)

Σ ′ + = -(2 -q)Σ + -S + Σ ′ -= -(2 -q)Σ --S - N ′ 1 = (q -4Σ + )N 1 N ′ 2 = (q + 2Σ + + 2 √ 3Σ -)N 2 N ′ 3 = (q + 2Σ + -2 √ 3Σ -)N 3
where:

S + = 1 6 (N 2 -N 3 ) 2 -N 1 (2N 1 -N 2 -N 3 ) S -= 1 2 √ 3 (N 3 -N 2 ) (N 1 -N 2 -N 3 ) q = 1 2 (3γ -2)(1 -K) + 3 2 (2 -γ)(Σ 2 + + Σ 2 -) K = 1 12 N 2 1 + N 2 2 + N 2 3 -2 (N 1 N 2 + N 2 N 3 + N 3 N 1 )
Here γ is a parameter: 2/3 < γ < 2. (See for instance [START_REF] Sari | Introduction aux systèmes dynamiques et applications à un modèle cosmologique[END_REF]).

10.9.1. Comparison. This system of differential equations must be "equivalent" to our equations ( §10.2) on the tangent bundle of a Milnor flat, which was a rational differential system on R 6 with one constraint. A formal definition of equivalency of approaches is that the two systems are "bi-rationally equivalent". However, the transformation of our system to this polynomial system is by no means obvious. This last system was not a priori motivated by simplifying our more "naive" one, but rather by considering another point of view in considering Einstein equations. Instead of studying the evolution with time of the metrics on spacelike slices, one considers the evolution of brackets of orthonormal frames on these slices. The gauge freedom is more subtle in this case, but still this method is very clever, as shown by the simplified form of the equations here. In our Bianchi case, i.e. where spacelike slices are Lie groups with left invariant metrics, one can very roughly say that the bi-rational equivalence comes from the projection map M il(G) → Sym(G), where M il(G) is the space of Milnor bases of G. The next step is to lift the Einstein equation (including a gauge choice) to M il(G) (more precisely an associated bundle) and to take the quotient by the G-action! 10.10. Further remarks. This beauty of Sym n appeals one to go beyond..., but as we said, our contribution here is essentially preliminary and expository. Let us mention some facts that were not considered here (with the hope to give details on some of them in the future). 10.10.1. Variants of Sym + n . First, one can generalize the discussion from Sym + n to Sym * n , the space of all pseudo-Euclidean products, i.e. non-degenerate quadratic forms. Everything extends there, a pseudo-Riemannian metric (on Sym * n ), Ricci maps, Bianchi-Einstein flows... The components of Sym * n are spaces of quadratic forms of a given signature. As for Sym + n , each component is a pseudo-Riemannian symmetric space and plays a universal role in its class.

• Complex case. The same is true for complex spaces: Sym * n (C), the space of complex non-degenerate quadratic forms on C n , is a holomorphic symmetric space... •Projectivization. Taking the associated projective spaces will send all these spaces into compact ones, and hence compactify them, by attaching various boundaries, with more or less nice interpretations. A natural requirement is that ideal points correspond to collapsing of Riemannian metrics, say in the Gromov sense [START_REF] Gromov | Metric structures for Riemannian and non-Riemannian spaces[END_REF] (restricted here to homogeneous spaces). By algebraicity, all differential equations extend to the projective spaces.

• Fiberwise constructions. If E → B is a vector fiber bundle, then one can associate to it Sym + (E)...

• Configuration spaces. Another interesting aspect of Sym +

n is its configuration space aspect. We mention here the case of " hydrodynamics", where a geometric formalism (a Riemannian metric, its geodesic flow...) was developed (see for instance [START_REF] Arnold | Topological methods in hydrodynamics[END_REF]) following similar ideas as those presented here. There are also other non-linear and infinite dimensional situations, in particular a space Sym * (E) associated to a Hilbert space E could be exciting! 10.10.2. Geodesic flows of left invariant metrics. For a Lie group G, Sym + (G) plays a role of a parameter space of its left invariant metrics. The geodesic flow of any such metric on G is a second order quadratic ODE system on G [START_REF] Abraham | Foundations of mechanics[END_REF]. It is intersting to study the dependence on parameter of the qualitative properties of these geodesic flows. 10.10.3. (Locally) Homogeneous, but "non-simply homogeneous" spaces. Instead of left invariant metrics on Lie groups, one can consider general homogeneous spaces, say, those endowed with an isometric transitive, but not necessarily free action of a given group G. More important is the case of locally homogeneous spaces, i.e. when the metric varies in the space of all those locally modeled on a fixed space X endowed with a (non-fixed) G-invariant metric (but G is fixed). Here, G does not act (it acts only locally, as a pseudogroup). As an example, we have the Robertson-Walker-Friedman-Lemaitre spacetimes [START_REF] Hawking | The large scale structure of space-time[END_REF][START_REF] O'neill | Semi-Riemannian geometry. With applications to relativity[END_REF][START_REF] Wald | General relativity[END_REF], which are warped products M = T × w N , ḡ = -dt 2 + w(t)g, where N has a constant curvature. 10.10.4. Dimension 2+1. So far, only the Gauss gauge has been considered. Maybe, this is because of its "deterministic character", i.e. it gives rise to autonomous differential equations, instead of non-autonomous ones, as in the generic case. There are however other situations where interesting gauges are available. As an example, in 't Hooft's theory of systems of particles in dimension 2 + 1 [START_REF] Hooft | Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness[END_REF], one has a flat polyhedral surface with singularities, evolving (locally) in a Minkowski space. The gauge here is fixed by the fact that time is locally equivalent to a "linear time" in the Minkowski space. In particular the time levels remain flat polyhedral. By consideration of suitable spaces of such surfaces, one may be convinced there is a configuration space approach similar to our situation here. 10.10.5. Non-empty spaces. Recall that M = I × M is a perfect fluid, if Ric = (p + ρ)dt 2 + pḡ, where p is the pressure and ρ is the density [START_REF] Hawking | The large scale structure of space-time[END_REF][START_REF] O'neill | Semi-Riemannian geometry. With applications to relativity[END_REF][START_REF] Wald | General relativity[END_REF]. A Bianchi-Einstein flow can be defined in this case, when p and ρ are functions on T Sym + n , or more reasonably, when they are basic functions, i.e. they depend on the coordinate q ∈ Sym + n alone. Robertson-Walker spacetimes are examples of perfect fluids (strictly speaking, they would be covered by our approach, once we consider general locally homogeneous spaces, as discussed above). 10.10.6. Quantization of the Bianchi-Einstein flow. We strongly believe this is a natural case that can be treated by a quantum gravity theory (see for instance [START_REF] Carlip | Quantum gravity in 2+1 dimensions[END_REF]), that is, a reasonable quantization of the Bianchi-Einstein flow should be possible... 10.10.7. A modified Einstein equation. World would be perhaps simpler if the Einstein equation on T Sym + n were given by the mechanical system determined by the Riemannian metric , on Sym + n as a kinetic energy, and the Hilbert action H as a potential energy. Recall [START_REF] Abraham | Foundations of mechanics[END_REF] that solutions of such a mechanical system are curves q(t) ∈ Sym + n , satisfying ∇ q ′ (t) q ′ (t) = -∇H(q(t))

(∇ is the Riemannian-connection and ∇H is the Riemannian gradient of H). Other more "realistic" modified equations are obtained by replacing the Riemannian metric by a pseudo-Riemannian or a Finsler one of the form L α,β or F α,β ( §8).

) 6 . 3 Proposition 6 . 1 .

 6361 STRUCTURE OF UNIMODULAR LIE ALGEBRAS IN DIMENSION Any unimodular 3-Lie algebra has a Milnor basis. More precisely (a, b, c) ∈ R 3 → G (a,b,c) ∈ L gives a parametrization of the space of unimodular 3-Lie algebras L. The diagonal action of R * 3 on R 3 , and that of the permutation group S 3 by permutation of coordinates, preserve the isomorphism classes of algebras.

7. 6 . 3 . 2 ) 7 . 6 . 5 . 1 Ric. 7 . 7 .

 632765177 Case of the Heisenberg groupHeis: a = b = 0, c = 1 Ric(x, y, z) = 1 2 y 2 (-x, y, -z). 7.6.4. Case of Euc: a = 0, b = -1, c = 1 Ric(x, y, z) = 1 2 (x(x 2 -y 2 ), y(y 2 -x 2 ), -z(x + y) Case of SOL: a = 0, b = c = x 2 -y 2 ), y(y 2 -x 2 ), -z(x -y) 2 ) Bianchi-Ricci flow.Recall that the Ricci flow associated to a compact manifold M (of finite volume) is an evolution equation on its space M et(M ) of Riemannian metrics:

Fact 10 . 4 .

 104 [u, v] = aw, [v, w] = bu and [w, u] = cv (see §5.0.1). The proof of the following facts and corollaries is left as exercise. Let p ∈ Sym(G) represent a left invariant quadratic form. Let X, Y and Z be right invariant vector fields, with X(1) = e ∈ G. Let g t = exp te. Then the derivative X.p(Y, Z) at 1 ∈ G is given by: X.p(X, Y ) = ∂ ∂t p(Adg t (Y ), Adg t (Z)) = p([X, Y ], Z) + p(Y, [X, Z))

Corollary 10 . 6 .Corollary 10 . 7 .

 106107 (b -a + c)u ... (Use Formula (2)) Consider the left invariant quadratic form, p 12 = du⊗dv +dv ⊗du. Then: • u.p 12 (u, e) + v.p 12 (v, e) + w.p 12 (w, e) = -c -a, for e = w = 0, for e = u, or e = v • p 12 (u, ∇ u w) + p 12 (v, ∇ v w) + p 12 (w, ∇ w w) = 0 • It then follows that ω = divp 12 is such that ω(u) = ω(v) = 0, and ω(w) = -(c + a), that is ω = -(c + a)dw. Let us say a Milnor basis is generic if (a + c)(a + b)(b + c) = 0.

We would like to emphasis on that this is a preliminary short non-finished work. This explains in particular why many proofs are left as exercises. Also, this text may be considered as expository, although the method here does not follow any existing approach.1