Stretched non-positive Weyl connections on solvable Lie groups

We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie groups admitting and not admitting such connections. In dimension 4 we fully classify solvable Lie groups and compact solvmanifolds which admit invariant SNP connections.


Introduction
In [TW] the authors deal with Weyl connections.By definition [F], a Weyl connection ∇ on a manifold M is determined by a 1-form ϕ satisfying the equality ∇X g = −2ϕ(X)g.
Since the difference between two connections is a tensor field, one derives the formula where E is dual to ϕ and ∇ is the Levi-Civita connection.Weyl connections locally coincide with the Levi-Civita connections of the metrics in the same conformal class.It was shown by Gauduchon [G] that for a compact manifold M there exists a unique metric in the conformal class such that the respective vector field is divergent free.In this case, one cosiders the Gauduchon representation (g, E) of the Weyl connection.Weyl connections are used in the study of Gaussian thermostats.A Gaussian thermostat is an ordinary differential equation on the tangent bundle to a Riemannian manifold of the form where x(t) is a parametrized curve in M , v ∈ T M .The trajectories of the Gaussian thermostat are geodesics of a Weyl connection [PW], they are also called thermostat geodesics [AR].Gaussian thermostats were introduced in [H] and were used (among other studies) in creating interesting models in nonequilibrium statistical mechanics.In these models, the vector field E contributes to the kinetic energy, while the second term of the thermostat equation retains the kinetic energy to be constant.In this paper we treat Gaussian thermostats from the geometric point of view, as a geometric structure (M, g, E) satisfying (1).
In [TW] the authors pose the problem of studying invariant Weyl connections on homogeneous spaces.One of the motivations for this study came from [BR], [R] and [W].It was shown in [W], that if the curvature of the Weyl connection is negative, then the Gaussian thermostat has some kind of "hyperbolic properties" of the geodesic flow (see also [PW]), and this was in accordance with models of nonequilibrium statistical mechanics and even more general dynamical properties, like A-axiom [BR], [R].Thus, looking for examples of such thermostats, it is natural to begin with the invariant case.The method described in [TW] is a s follows.Let G/H be a homogeneous Riemannian manifold with an orthogonal decomposion g = h + p.Let Note that if h = 0, then p 0 = g and this is the case considered in [TW] in greater detail.The projection of an invariant vector field on G determined by E ∈ p 0 defines a Weyl connection on G/H (see [TW]).Note, however, that there are no known examples of Weyl connections of negative curvature on compact manifolds M of dimension ≥ 3. On the other hand, Gaussian thermostats were studied on surfaces of negative curvature [AR].Therefore, one looks for conditions with features of non-positivity.One of such assumptions is the SNP condition.
Definition 1.Let (M, g) be a Riemannian manifold.Let E be a vector field and γE, γ > 0 be a family of Weyl connections defined by vector fields γE.A vector field E is called streched non-positive (SNP) if there exists γ 0 ≥ 0 such that the Weyl connections defined by the fields γE are non-positive for all γ ≥ γ 0 .A Weyl connection on a compact Riemannian manifold M is called streched non-positive (SNP) if its Gauduchon representation E is SNP.
Note that the SNP condition is the property of the Gaussian thermostat itself.It says that there is a non-isolated family (M, g, γE) of Gaussian thermostats which yields nonpositivity for the values of parameter γ exceeding some γ 0 .In this sense, the classification of homogeneous spaces with invariant SNP Weyl connections answers the question: what are Gaussian thermostats of the form (G/H, g, γE) which are non-positive for γ ≥ γ 0 ?Some necessary conditions (for general Riemannian manifolds) which imply SNP are found in [TW], Propositions 3.1 and 3.2.Note that some of them are subject to the condition div E = 0, and some not.We will use the following.
Proposition 1.If a Weyl connection determined by a divergence-free vector field E on a compact manifold is SNP, then In [TW] the following problem is considered.Let G be a unimodular Lie group with an invariant Riemannian metric g.Find invariant SNP connections on G, that is, find invariant vector fields E ∈ g such that (g, E) determines an invariant SNP Weyl connection.The following is a summary of results obtained in this direction.
Theorem 1 ( [TW], Theorem 4.3).For a unimodular Lie group G, if a left-invariant vector field E ∈ g satisfies (W1) and (W2), then E is parallel.Also, in [TW] the authors looked for isolated examples of Gaussian thermostats (G, g, E) with non-positive Weyl curvature.The following cases were analyzed: 1. g = a ⊕ ϕ ã, where a and ã are abelian Lie algebras of dimesnions n + 1 and n, respectively, and ϕ : a → gl(ã).The non-positivity of the Weyl connection determined by E ∈ a is expressed in terms of the eigenvalues of ϕ(E) ([TW],Theorem 5.1).
2. the case of a 3-dimensional Lie group, where non-positive Weyl connections were found on the 3-dimensional solvable Lie group Sol 3 , which is semidirect product R ⋊ ϕ R 3 determined by Note that 1) is a consequence of the results of [TW] which we summarize as follows.
Theorem 2. For a unimodular Lie group G endowed with an invariant Riemannian metric g a left-invariant vector field E ∈ g is SNP if and only if ad E is skew-symmetric and E ⊥ [g, g].
Proof.The proof follows from [TW], Proposition 4.1, Theorem 1 and a straightforward observation that a parallel vector field is SNP.
The 3-dimensional case 2) was analyzed using Milnor's description of left-invariant metrics on Lie groups [M].In this note we pose the following problems.
1. Find all invariant SNP Weyl connections on unimodular Lie groups of dimension 4.
2. Classify all unimodular solvable Lie groups admitting invariant Weyl connections with SNP property (thus, all solvable Lie groups which can yield SNP Gaussian thermostats (G, g, γE)).
Note that the unimodularity condition is quite natural, because it is a necesary condition for a Lie group G to admit a lattice, and, in particular a co-compact lattice Γ [Ra].If Γ does exist, we get compact homogeneous spaces G/Γ with homogeneous SNP Weyl connections.
In the subsequent sections we solve problems 1) and 2).Note that we don't address here the dynamics of the Gaussian thermostats (which presumably is simple and reduces to the behavior of the integral curves of E).Our aim is to solve the geometric problem of classifying homogeneous spaces admitting SNP Weyl connections.Also, we do not address the more diffiucult problems of finding homogeneous spaces with Weyl connections of nonpositive curvature, which have more interesting dynamics ([W2], Section 10).The main results of this article are Theorem 3 and Theorem 4. These theorems show that the SNP condition on the Gaussian thermostat (G, g, E) imposes structural restrictions on G (Theorem 4).Also, we find all compact 4-dimensional solvmanifolds admitting an SNP Weyl connection (Corollary 1).We find classes of solvable Lie groups admitting and not admitting invariant SNP Weyl connections among solvable extensions of nilpotent Lie groups whose Lie algebras belong to some known classes of nilpotent Lie algebras: metabelian, of Vergne type {2n, 1, 1}, and characteristically nilpotent (Propositions 2, 3, 4, 7).
Finally we want to mention that the general problem of classifying homogeneous spaces with invariant Weyl connections of non-positive curvature seems to be geometrically and algebraically very attractive and challenging.On one hand, one encounters the rigidity phenomena similar to the case of invariant Riemannian metrics on homogeneous spaces.Any simply connected homogeneous space endowed with an invariant Riemannian metric of non-positive curvature admits a simply transitive action of a solvable Lie group S whose Lie algebra satisfies strong structural restrictions (such algebras are called NC-algebras), see [AW].However, imposing the restriction of unimodularity, one necessarily obtains flatness of the metric.On the other hand, the non-positivity of invariant Weyl connections seems to be less rigid and the results of this paper yield some small evidence for this.Note, however, that we impose the solvablity restriction on G from the very beginning.
Our terminology and notation basically follows [TW].We use [GK] as a general reference for the theory of solvable and nilpotent Lie algebras.In the proof of Theorem 3 we use some facts of the theory of lattices in Lie groups which can be found in [Ra].The material related to invariant connections and Riemannian metrics on homogeneous spaces is contained in [KN1], [KN2].The Lie algebras of Lie groups G, H,..., are denoted by g, h,... .The universal cover of a Lie group G is denoted by G.
2 Weyl connections on solvable Lie groups of dimension 4 Our purpose is to fully classify all 4-dimensional unimodular solvable Lie group admitting invariant SNP Weyl connections and also all possible pairs (g, E) on G, that is, all possible invariant Weyl SNP connections on them.To do this, we use a classification of invariant Riemannian metrics of such groups from [T].Let us begin with a short description of this classification which we need to formulate our results.Let G be a simply connected Lie group.A left invariant Riemannian metric on G determines a scalar product on g, and conversely, a scalar product −, − on g determines a left-invariant Riemannian metric on G. Let M be the set of scalar products on G.The group Aut(g) acts on M by the formula A • v, w = A −1 v, A −1 w for any A ∈ Aut(g).It is straightforward to see that to classify the left invariant Riemannian metrics on G is equivalent to classify the scalar products on g, which is the same as to find representatives of the orbits of Aut(g) in M.This is the result of [T].Note that in general this classifcation is finer than just up to isometry, because an isometry of (G, g) need not be a group isomorphism.However, this does not influence our arguments.In this article, we present the list of all simply connected 4-dimensional solvable Lie groups which admit an SNP pair (g, E) and show a representative in the appropriate Aut(g)-orbit.Our notation follows that of [T].Here is the list of all 4-dimensional unimodular solvable Lie groups.

Solvable Lie groups:
where ϕ : R → GL(3, R) is defined as follows: • Nil ×R: Groups Sol 4 l and S 1 ⋊ ϕ Nil are described as semidirect products with the 3-dimensional group of unipotent triangular matrices.The action of ϕ(t) is given as follows: • Sol 4 1 : ϕ(t) acts on the unipotent matrix -action is given by the formula where p(x, y, t) = 1 2 (y 2 − x 2 ) cos t sin t − xy sin 2 t.Note that the Lie algebras corresponding to the Lie groups from the above table are well known and can be easily calculated.The standard Lie bracket structure on R ⊕ ϕ R 3 and on R ⊕ ϕ n 3 is not convenient to analyze the SNP condition which is a condition on the sectional curvature.However, [T] contains a classification of all invariant Riemannian metrics on the given Lie groups G in the following sense: the scalar product −, − on g is determined by expressing the orthonormal base X 1 , ..., X 4 ("Milnor base") through the standard base e 1 , ..., e 4 .We do not reproduce the standard bases here, however, each of them is explicitly written in the course of the proof of Theorem 3.Here we assume that e 1 , e 2 , e 3 span either R 3 or n 3 , where n 3 is the Lie algebra of the triangular (3 × 3)-matrices with zeros on the diagonal (the "3-dimensional Heisenberg algebra").Note again, that in the formulation of Theorem 3 we describe the representatives of the Aut(g)-orbits by writing down the family of Milnor bases.
Theorem 3. A non-abelian solvable unimodular Lie group G admits an SNP Weyl connection if an only if it is one of the following: Any of these groups admits a co-compact lattice Γ, so any of solvmanifolds G/Γ is a compact 4-dimensional manifold with an SNP connection.The following list yields all possible families of pairs (g, E) determining invariant SNP Weyl connections: • Sol 3 ×R, three-parameter family {e 1 , e 2 , be 3 + e 3 , ce 4 , b, c > 0}, E = αe 1 .
Proof.Any invariant Riemannian metric on G is determined by the choice of the scalar product −, − on g.By Theorem 1 if (g, E) is invariant and SNP, then ∇ Y E = 0 for any invariant vector field E. Also, it satisfies (W2), which translates into the identity for the scalar product −, − on the Lie algebra g: for all Y ∈ g orthogonal to E (see the proof of Theorem 7.2 in [TW], or verify directly).
The usual Riemannian geometry formula for for all Y, Z ∈ g.We check these identities in each case separately, writing down the corresponding expressions for (2) and (3) in terms of the Milnor bases.
The case of nilpotent g.
(I) g = Lie(Nil ×R).From [T], Theorem 3.1, every metric is isometric to the one defined by the orthonormal base Calculating the Lie brackets we get Therefore, a 1 = 0 and E = α 2 X 2 .
[T], Theorem 4.2 implies that every metric is defined by an orthonormal base: Calculating the brackets we get for some γ 3 < 0, γ 4 , γ 5 ∈ R We get the following expressions of the Lie brackets [T], Theorem 4.4 shows that every metric is defined by an orthonormal basis We obtain the equalities The second subcase arises when (IIIE) g = Lie(Sol 4 µ ).We get E = 0 in a way similar to (IVC).
Here [e i , e j ] = 0 for i, j = 1, 2, 3, and By [T], Theorem 4.6 there are two forms of the corresponding orthonormal bases.We will write down the first one, the second is analogous.We have: The case of solvable g with a non-abelian nilradical.
[e 1 , e i ] = 0 for all i [T],Theorem 5.1 implies that every metric is defined by an orthonormal basis: We have If we write down the equation 0 = [E, Y ], Y and choose the appropriate values β i we will obtain α 2 = α 3 = α 4 = 0. Therefore E = α 1 X 1 .We obtain [E, Y ] = 0 for all Y ∈ g and ( 3) implies [Z, Y ], E = 0.By choosing the appropriate coefficients, we get This shows that α 1 = 0 and E = 0.
(IVB) g = Lie( Nil ⋊S 1 ).We have • Consider the case Sol 3 ×R.One can easily see that this group can be also presented in the form R × G 1 , where G 1 is a 3-dimensional solvable Lie group of the form where φ is a one-parameter subgroup of the form It known that G 1 admits a co-compact lattice Γ 1 (the full proof can be found in [TO], Theorem 1.9).Thus, Z × Γ 1 is the required co-compact lattice in G.
It is known that in dimension 4 all compact solvmanifolds are of the form G/Γ, where Γ is a co-compact lattice in a four-dimensional solvable Lie group G [GO], Theorem 3.1.This yields the following.
Corollary 1.All 4-dimensional compact solvmanifolds with invariant SNP Weyl connections are exhaused by the list given in Theorem 3.

A structure of solvable Lie groups admitting SNP Weyl connections
Here we determine the structure all solvable Lie algebras such that the corresponding Lie groups admit invariant SNP Weyl connections.
Theorem 4. Unimodular Lie group G admits an invariant SNP Weyl connection determined by a non-central E ∈ g if and only if g has the form g = E ⊕ ϕ s, where: 1. s is a unimodular solvable Lie algebra such that Aut(s) contains a compact torus T of positive dimension, Proof.Choose any scalar product −, − on g and assume that E determines an SNP Weyl connection.By Theorem 2, E ⊥ [g, g] = n and so s :=< E > ⊥ is an ideal of g.Also the action of ad E on n is skew-symmetric.It follows that ad tE ∈ Der(n) corresponds to a subgroup T in Aut(n) such that the closure T of T is a compact torus.Assume that there is no non-trivial compact torus in Aut(n) and so ad E | n = 0. Consider the orthogonal decomposition g = a ⊕ n, E ∈ a.By [TW], Theorem 4.3 we have Thus for any A ∈ a and any Z ∈ n It follows that [E, A] = 0 for any A ∈ a and so E is central in g.On the other hand if ad E | n = 0 and T is a compact torus of positive dimension.If g = E ⊕ ϕ s and satisfies 1)-3), then by Theorem 2 g admits an invariant SNP Weyl connection.

Classes of solvable Lie groups admitting invariant SNP Weyl connections
We simplify the general description given by Theorem 4 as follows.
Proposition 2. Assume that n is a nilpotent Lie algebra with the following property: Aut(N ) contains a compact subgroup S. Let T ⊂ S be a torus.There exists a semidirect product g = a ⊕ ϕ n and a scalar product −, − on g such that for any E ∈ a the pair ( −, − , E) determines an SNP Weyl connection.
Proof.Consider the Lie algebra Der(n) of the Lie group Aut(N ).Let s be the Lie algebra of S ⊂ Aut(N ), and t be the Lie algebra of T .Since t is abelian, for any abelian Lie algebra a and any linear map ϕ : a → t ⊂ s ⊂ Der(n) one can construct a semidirect sum Define a T -invariant scalar product −, − n on n and take any scalar product on a. Declairing a and n orthogonal, we get a scalar product on g which determines an invariant Riemannian metric on g.Then, by the construction, E ⊥ [g, g], and ad E| n = ϕ(E) ∈ t is skew symmetric.
Proposition 2 yields series of examples of solvable Lie groups with SNP Weyl connections in any dimension.

Lie algebras of some particular Vergne's types
Definition 2 ( [V]).The Vergne type {d 1 , ..., d r } of a nilpotent Lie algebra n with descending central series n (i) = [n, n (i−1) ] is defined by In particular, we can characterize a subclass of the class of nilpotent Lie algebras of type {n, 2} as follows.
Definition 3. Nilpotent Lie algebras of type {n, 2})-Heisenberg are nilpotent Lie algebras V ⊕ x, y of dimension n + 2 defined by a pair of alternating forms F 1 and F 2 on the n-dimensional vector space V putting for any v, w ∈ V, [v, w] = F 1 (v, w)x + F 2 (v, w)y.
Proposition 3. Any unimodular solvable Lie group which is a semidirect product A ⋊ ϕ N of an abelian Lie group A and a nilpotent Lie group N whose Lie algebra n has type {n, 2}-Heisenberg, admits an SNP Weyl connection.
Proof.In view of Proposition 2 it is sufficient to prove that Aut(n) has a compact subgroup.This can be seen as follows.Denote by Sp(V, F 1 ) a subgroup of GL(V ) which preserves the alternating form F 1 .This group naturally embeds into Aut(n) by the formula Since Sp(F 1 ) is a closed subgroup of GL(V ), it is a Lie group, containing Sp(2l), where 2l ≤ n is the rank of F 1 .Thus, it contains a non-trivial maximal compact subgroup, as required.
Remark 1.A full classification of nilpotent Lie algebras of type {n, 2} whose automorphism group contains a compact torus is obtained in [FF].We do not reproduce it here, since the description of such Lie algebras is rather complicated.
Proposition 4. Any unimodular semidirect product A ⋊ N of an abelian Lie group and a realification of a complex nilpotent Lie group of type {2n, 1, 1} admits an SNP Weyl connection.
Proof.The proof follows from Proposition 2. It is known that under the assumptions of the Proposition, Aut(N ) contains a compact subgroup.In greater detail, let n be a complex nilpotent Lie algebra of type {2n, 1, 1}.Then, by [BBF] the Lie algebra Der(n) has the form Der(n) = r ⊕ sp(n, C), where r denotes the radical of Der(n).It follows that for the realification (denoted by the same letter), the automorphism group contains a compact torus.
Example 1.A (2n + 1)-dimensional Heisenberg Lie algebra is defined as a a vector space V ⊕ x of dimension 2n + 1 with the only non-zero Lie brackets of the form [v, w] = F (v, w)x, where F is a symplectic form on V .As in the proof of Proposition 3 one can see that Aut(n) contains Sp(n, R).Thus, any semidirect product of an abelian Lie group with the Heisenberg Lie group admits an SNP Weyl connection.

Metabelian Lie algebras
A finite dimensional Lie algebra g is called metabelian, if [g, [g, g]] = 0.The signature of a metabelian Lie algebra is a pair (m, n), where m = dim g/[g, g], n = dim [g, g].Note that this is a particular case of Definition 2. A metabelian Lie algebra structure on g is completely determined by the commutator map Λ 2 U → V , where V = [g, g] and U is a complement in g (different complements determine different structures).Conversely, let g = U ⊕ V be a direct sum of two vector spaces U and V of dimensions m and n.Then each skew symmetric bilinear surjective map f : Λ 2 U → V determines a metabelian Lie algebra structure on g of signature (m, n).The space of maps f is Λ 2 U * ⊗V , and the group GL(U ) × GL(V ) naturally acts on it.Thus, a classification of metabelian Lie algebras can be understood in terms of the orbits of this group.In [GT] the canonical elements f determining the orbits are found.We will call such classification a Galitskii-Timashev classification and the corresponding families of Lie algebras the Galitskii-Timashev types.Note that these types are determined over C and we condier algebraic tori in algebraic groups.
Proposition 5 ([GT], Section 1.1).Let g be a metabealian complex Lie algebra determined by f : Λ where N is a unipotent subgroup, and G(f ) denotes the GL(U ) × GL(V )-stabilizer.
Thus, if the connected component of G(f ) of a metabelian Lie algebra n contains an algebraic torus, any semidirect sum g = a ⊕ ϕ n given by Proposition 2 admits an SNP invariant connection, provided that n is a realification (denoted also as n) .The paper [GT] contains examples of metabealian Lie algebras of both types, containing an algebraic torus and not containing it.In greater detail we can describe the classification as follows.Treat f ∈ Λ 2 U * ⊗ V as a tensor.Choose bases 1 e, ..., m e of U and e 1 , ..., e n of V , and write the base of Λ 2 U * ⊗ V in the form ij e k = e i ∧ e j ⊗ e k .
Note that dual elements are denoted by raising or lowering the indices.The tables in [GT] contain the description of f in the dual form, so the following notation is used:  In total, there are 223 Galitskii-Timashev classes of metabelian Lie algebras and 20 of them do not have an algebraic torus in the automorphism group.
Corollary 2. Let n be a realification of a complex metabelian Lie algebra of signature (m, n), m, n ≤ 5 or m ≤ 6, n ≤ 3 such that its automorphism group contains an algebraic torus according to Proposition 6.Then any solvable Lie group whose Lie algebra is a semidirect extension as in Proposition 2 admits an SNP Weyl connection.
Remark 2. A classfication of metabelian Lie algebras in terms of generators and relations is obtained in [Ga].It would be interesting to find more general result than that of Proposition 6.

Solvable Lie groups which do not admit invariant SNP connections
Recall that a characteristically nilpotent Lie algebra is a nilpotent Lie algebra n such that Der(n) is nilpotent [GK].
Definition 4. W say that a nilpotent Lie algebra n is characteristically nilpotent of Dyer type, if Aut(n) is unipotent.Remark 3.An example of a characteristically nilpotent Lie algebra n with unipotent Aut(N ) was found by Dyer [D].More examples can be found in [AC].Dyer's example is a follows: it is a nine-dimensional Lie algebra spanned by X 1 , ..., X 9 with the Lie brackets

(
abc...ijk) stands for ab e c + • • • + ij e k .Analysing Tables 1-8 in [GT] we get a description of canonical elements f and the connected components of their stabilizers.By what we have said the following holds.Proposition 6.Any metabelian Lie algebra over C of signature (m, n) such that m, n ≤ 5, or m ≤ 6, n ≤ 3 has an automorphism group which contains an algebraic torus with the following exceptions determined by the canonical choice of tensor f : for m, n ≤ 5: