The homogeneous generalized Ricci flow
Elia Fusi Ramiro A. Lafuente James Stanfield



 
 We develop a framework inspired by Lauret's “bracket flow” to study the generalized Ricci flow, as introduced by Streets, on discrete quotients of Lie groups. As a first application, we establish global existence on solvmanifolds in arbitrary dimensions, a result which is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton on exact Courant algebroids that is geometrically meaningful and allows for non-trivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the generalized Ricci flow, provided the initial metrics have “harmonic torsion", and we classify them in low dimensions. Finally, we provide a new formula for the generalized Ricci curvature of invariant generalized metrics in terms of a moment map for the action of a non-reductive real Lie group.
 
 
 
 
 
Real-Time Compressed Sensing for Joint Hyperspectral Image Transmission and Restoration for CubeSat

	
	This study was supported in part by the National Science and Technology Council (NSTC), Taiwan, under grants MOST 112-2221-E-006 -157 -MY3 and 111-2221-E-006 -210; partly by the Higher Education Sprout Project of Ministry of Education (MOE) to the Headquarters of University Advancement at National Cheng Kung University (NCKU). We thank to National Center for High-performance Computing (NCHC) of National Applied Research Laboratories (NARLabs) in Taiwan for providing computational and storage resources.

(Corresponding author: Chih-Chung Hsu.)


		C.-C. Hsu, C.-Y. Jian, C.-M. Lee and G.-L. Chen are with Institute of Data Science and Department of Statistics, Miin Wu School of Computing, National Cheng Kung University, Tainan, Taiwan (R.O.C.), (e-mail:cchsu@gs.ncku.edu.tw, zuw408421476@gmail.com, alright1117@gmail.com)
		E.-S. Tu is with Department of Computer Science, National Cheng Kung University, Tainan, Taiwan (R.O.C.)

    Chih-Chung Hsu, Senior Member, IEEE, Chih-Yu Jian, Eng-Shen Tu, Chia-Ming Lee and Guan-Lin Chen  <https://rcts01.github.io/>
    
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§ INTRODUCTION
  
Motivated by non-Kähler Hermitian geometry and mathematical physics, the generalized Ricci flow is an analogue of the classical Ricci flow in the setting of Hitchin's generalized geometry <cit.>. It is a flow of generalized Riemannian metrics on an exact Courant algebroid. The latter can be described, up to isomorphism, by a smooth manifold M and a choice of cohomology class [H]∈ H^3(M, ), whereas the former can  be thought of as a pair (g,b), with g a Riemannian metric on M, and b ∈Ω^2(M) a 2-form on M. A smooth family (g_t,b_t)_t ∈ I of generalized Riemannian metrics on (M,[H]) solves the generalized Ricci flow if it satisfies the following weakly parabolic PDE:

    ∂∂ t g_t    =  -2 Ric(g_t) + 1/2 (H +  b_t)^2_g_t,
    ∂∂ t b_t    =   - ^*_g_t(H +  b_t),

where, for a 3-form H and Riemannian metric g, H_g^2 ∈Sym^2(T^*M) is defined by H^2_g(X,X) = |ι_XH|_g^2 for all X ∈ TM. The classical Ricci flow is the particular case where H = 0 and b_t ≡ 0.
 
By abuse of notation, we will sometimes call the pair (g,H +  b) a generalized Riemannian metric on M, meaning that the exact Courant algebroid is (M,[H]) and the generalized metric is the pair (g,b). Similarly, we may refer to a family (g_t,H_t := H +  b_t) as a generalized Ricci flow on M.

  

The primary goal of this article is to give a systematic treatment of the generalized Ricci flow in the locally homogeneous case. To simplify the presentation, we focus on exact Courant algebroids over Lie groups (or their discrete quotients), endowed with left-invariant generalized metrics. That is, we consider exact Courant algebroids of the form (𝖦,[H]), where 𝖦 is a Lie group and [H] ∈ H^3(𝖦, )^𝖦 is an invariant cohomology class, and we restrict ourselves to generalized metrics (g,b) that are left-invariant. In this direction, our first main result is that the invariant generalized Ricci flow exhibits long-time existence on solvmanifolds:

 Any invariant solution to the generalized Ricci flow on a solvmanifold exists for all positive times.



 More generally, we characterise the maximal existence time for the invariant generalized Ricci flow on any Lie group 𝖦 in terms of the blow-up behaviour of the generalized scalar curvature (see <Ref>). Here by solvmanifold we mean any (not necessarily compact) locally homogeneous space of the form 𝖦 / Γ, where 𝖦 is a solvable Lie group, and Γ≤𝖦 is a discrete subgroup. It is called a nilmanifold if 𝖦 is nilpotent. An invariant solution is simply a solution to (<ref>) on 𝖦/Γ that lifts to a left-invariant solution on 𝖦. The class of solvmanifolds is known to be very rich. Examples in four dimensions include  Inoue surfaces and primary Kodaira surfaces.

 



Notice that <Ref> is new even in the case of the pluriclosed flow. Recall that when (M,J,g) is Hermitian and H is the torsion of the Bismut connection, (<ref>) is gauge-equivalent to the pluriclosed flow on (M,J) <cit.>. As an immediate application of <Ref> we get 

 
Any invariant solution to the pluriclosed flow on a solvmanifold exists for all positive times.


This was previously known in the case of complex surfaces, for nilmanifolds, for 6-dimensional solvmanifolds with trivial canonical bundle, for almost abelian solvmanifolds, for some almost nilpotent solvmanifolds, and for Oeljeklaus–Toma manifolds <cit.>. Our approach gives a unified treatement of all these results, and applies to SKT solvmanifolds on which the pluriclosed flow has not yet been studied. For instance, the Endo–Pajitnov manifolds <cit.> which were recently shown to admit SKT structures in <cit.>. Moreover, the existence of SKT metrics was characterized  by Freibert–Swann in <cit.> on a large class of 2-step solvmanifolds,   and on solvmanifolds admitting a codimension 2 abelian ideal by Guo–Zheng <cit.> and Cao–Zheng <cit.>.

It is conjectured that the pluriclosed flow on a compact complex manifold (M,J) exists for all time if and only if -c_1^A(M,J) is a nef class (that is, -c_1^A(M,J) lies in the closure of the cone of classes in H_A^1,1(M,J) containing a positive representative), where c_1^A(M,J)∈ H_A^1,1(M,J) is the first Aeppli–Chern class of (M,J) <cit.>. It is well-known that the first Chern class of solvmanifolds is zero, hence the first Aeppli–Chern class vanishes as well. In particular, -c_1^A(𝖦/Γ,J) is a nef class on compact solvmanifolds, and thus <Ref> verifies the conjecture in this setting. 

After having established global existence, a natural problem is to determine the long-time asymptotics of solutions. In this direction, on a nilmanifold M = 𝖦/Γ with harmonic H_0  (i.e.  ^*H_0 = 0), we can characterise the possible limits as  follows: 


 Let  (M,g_t,H_t = H_0 +  b_t)_t ∈ [0,∞) be an invariant solution to the generalized Ricci flow on a nilmanifold M=𝖦/Γ with  H_0 harmonic. Then, after pulling back to the universal cover, the rescaled generalized metrics (1/1+tg_t,1/1+tH_t)_t ∈ [0,∞) on the time-dependent exact Courant algebroids (𝖦, 1/1+t[H_0]) converge, in the generalized Cheeger–Gromov topology, to a left-invariant, expanding generalized Ricci soliton (g_∞,H_∞) on a nilpotent Lie group 𝖦_∞.


By generalized Ricci soliton we mean a solution (g_t, H_t = H +  b_t) to (<ref>) which, up to pull-back by time-dependent diffeomorphisms, evolves only by scaling. That is, there exist c(t) ∈_>0, and φ_t ∈ Diff(M) such that, for all t,  it holds that

    ( φ_t^* g_t,  φ_t^* H_t) = (c(t)   g_0, c(t)  H_0).

Notice that the generalized metrics (c(t)   g_0, c(t)  H_0) live on the time-dependent exact Courant algebroids (M, c(t) [H]). By allowing H to scale, this definition differs from others found in the literature <cit.> (compare however with <cit.>), but agrees with the definition posed in <cit.>. We adpot it here because it allows for non-trivial, non-steady solitons, and because examples arise naturally as asymptotic limits by Theorem <ref>. We refer the reader to Section <ref> for more technical details regarding this notion and the scaling of exact Courant algebroids, and to Section <ref> for the definition of generalized Cheeger-Gromov topology.


Notice that the limit group may be non-isomorphic to  (this occurs already in the classical setting <cit.>).  We conjecture that the assumption H_0 harmonic  is not necessary. This will be the subject of forthcoming work.  However, let us mention that in dimensions up to 4, all  left-invariant generalized Ricci solitons on nilpotent Lie groups do have harmonic H_0. This follows from our complete classification  in <Ref>.





Let us now turn to the methods used to prove our main results. Proceeding more precisely, the exact Courant algebroid over M correponding to the cohomology class [H_0] ∈ H^3(M, ) can be identified with the bundle (TM⊕ T^*M,[·,·]_H_0) endowed with the twisted Dorfmann bracket [·,·]_H_0, a bracket law on TM ⊕ T^*M determined by H_0 and the usual Lie bracket of vector fields [·,·] on TM, see (<ref>). The pair (g,b) then determines a generalized metric on TM ⊕ T^*M, which is an endomorphism 𝒢(g,b) ∈End(TM ⊕ T^*M). The generalized Ricci curvature ℛc(𝒢) ∈End(TM ⊕ T^*M) of 𝒢 is a 𝒢-symmetric endomorphism on TM ⊕ T^*M (see <Ref> for its definition) and the generalized Ricci flow is the following PDE:

    ∂/∂ t𝒢 = -2  𝒢 ℛc(𝒢).


If M = 𝖦 is a Lie group and [H_0] ∈ H^3(𝖦, )^𝖦 is invariant, then the exact Courant algebroid structure (T𝖦⊕ T^*𝖦,[·,·]_H_0) can be identified with the pair ( g⊕ g^*,_H_0), where g is the Lie algebra of 𝖦, and _H_0 is a bracket on g ⊕ g^* determined by H_0 ∈Λ^3 g^* and μ, the Lie bracket of g. The invariant generalized Ricci flow is then equivalent to an ODE on the space of endomorphisms of g ⊕ g^*. We now follow Lauret's bracket flow approach. The non-reductive group 𝖫 = 𝖦𝖫( g)·exp(Λ^2  g^*) ≅𝖦𝖫( g) ⋉Λ^2 g^* acts transitively on the space of invariant generalized metrics. Thus, by fixing a background generalized metric 𝒢 = 𝒢(g,0), we pull-back the generalized Ricci flow solution to an ODE on the space of brackets on g ⊕ g^* via the action of 𝖫. 

 The invariant generalized Ricci flow on (𝖦,[H_0]) is gauge-equivalent to the following ODE of Dorfman brackets on g ⊕ g^*:

    / t(t) = -  Rc_(t)(t) + (t)( Rc_(t)·,·) + (t)(·, Rc_(t)·).


Here, for a twisted Dorfman bracket = _H,μ, Rc_ :=  Rc_ - A_ where ℛc_∈End( g ⊕ g^*) is the generalized Ricci curvature of the infinitesimal exact Courant algebroid structure ( g ⊕ g^*,_H), and A_ := Skew_𝒢(^*H), where we consider the two-form ^*H  g → g^* as a map from g to g^*.

Let us remark that a bracket flow approach to the homogeneous generalized Ricci flow was considered by Paradiso in <cit.> using the action of 𝖦𝖫( g) ≤𝖫. The novelty of our approach, and what makes <Ref> work, is that we consider the bigger group 𝖫 which acts transitively on the space of generalized metrics.

The two approaches agree only when H_0 is harmonic (see <Ref>).

Another important ingredient in our methods is a new formula for the generalized Ricci curvature of an invariant exact Courant algebroid resembling that of the classical Ricci curvature on a Riemannian homogeneous manifold:

    Rc_ - A_ = _ - 12 B -Sym_g(_U).

Here, U ∈ g is the mean curvature vector defined by g(U,X) = (_X), for X ∈ g, and B is the Killing form defined by g(BX,Y) = (_X ∘_Y) for X,Y ∈ g. For F ∈End( g), we define F := F ⊕ -F^* ∈End( g ⊕ g^*). Finally, _∈End( g ⊕ g^*) is the moment map for the natural action of 𝖫 on the space of brackets Λ^2( g ⊕ g^*)^* ⊗ ( g ⊕ g^*)  (see <cit.>).


The identification of a moment map in the structure of the generalized Ricci curvature allows us to use techniques from real geometric invariant theory <cit.>. In particular, for nilmanifolds with H_0 harmonic, it holds that ℛc_ = _. This is the key ingredient in the proof of <Ref> since after normalizing, <Ref> becomes a real analytic gradient flow (see <Ref>). If H_0 is not harmonic, we have ℛc_ = _ which is not symmetric, as 𝖫 is not a reductive Lie group.

There have been many other recent works on generalized Ricci flows with symmetries. The Bismut flat spaces - which are fixed points of the generalized Ricci flow - were classified by <cit.>. They are covered by compact Lie groups with bi-invariant metric, and the form H given by the contraction of the Lie bracket with said metric. By a recent result of Lee <cit.>, they are dynamically stable with respect to the GRF. Furthermore, Barbaro in <cit.> proved  global stability for Bismut-flat metrics on complex  Bismut–flat manifolds with finite fundamental group with respect to the pluriclosed flow using a stability result appeared in  <cit.>. The first examples of homogeneous Bismut–Ricci flat metrics which are non-flat were given by Podestà–Raffero <cit.>. Lauret–Will in <cit.> constructed further examples in this setting. The stability of many of these examples was studied by Gutierrez in <cit.>. In <cit.> Podestà–Raffero construct examples of 𝖲𝖮(3)-invariant steady, gradient generalized solitons on ℝ^3, resembling the structure of the classical Bryant soliton for the Ricci flow, see <cit.>. On the other hand, it was recently shown that all compact shrinking solitons of gradient type are in fact classical solitons <cit.>. Moreover, in <cit.>,  Streets–Ustinovskiy classified  generalized Kähler–Ricci solitons on complex surfaces. Beyond soliton solutions, the generalized Ricci flow was studied on nilmanifolds by Paradiso <cit.> and on cohomogeneity-one manifolds with nilpotent symmetry by Gindi–Streets <cit.>. The toric, Fano case of the generalized Kähler–Ricci flow was treated by Apostolov–Streets and Ustinovskiy in <cit.>.



The rest of the article is structured as follows: In <Ref>, we recall the notion of an exact Courant algebroid and generalized metrics. We also define scaled exact Courant algebroids with the motivation of defining non-trivial expanding or shrinking solitons. We then discuss rigorously the generalized Ricci curvature and generalized Ricci flow. In <Ref>, we define and discuss generalized Ricci solitons, and present in detail some examples on the Heisenberg group. In <Ref> and <Ref>, we treat homogeneous generalized geometry and the generalized bracket flow, then derive <Ref>, and prove <Ref>. In Sections <ref> – <ref> we prove our main theorems <Ref> and <Ref> using the generalized bracket flow and techniques from real geometric invariant theory. We discuss in <Ref> the relationship between the pluriclosed flow and the generalized Ricci flow in our (not necessarily compact) setting, leading to <Ref>. Finally, in <Ref> we classify generalized Ricci solitons on nilmanifolds in dimensions less than or equal to 4.






 §.§ Acknowledgements
 This project originated during the first-named author's visit to the University of Queensland. For this,  he wishes to express his deep gratitude to it and to the second and third author for their cheerful hospitality and for many inspiring conversations. The third-named author would like to thank Christoph Böhm, Alberto Raffero, and Luigi Vezzoni for fruitful conversations and their interest in this work. The first-named author is supported by GNSAGA of INdAM. The second-named author was supported by the ARC project DE190101063. The third-named author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry–Structure and was previously supported by an Australian Government Research Training Program (RTP) Scholarship.





























§ PRELIMINARIES ON GENERALIZED GEOMETRY



In this section we review the basic notions of generalized Geometry and setup our notation. We mainly follow <cit.>. We also introduce a scaling of exact Courant algebroids that will be central in our definition of soliton solutions for the generalized Ricci flow.  

 For additional motivation and details on exact Courant algebroids we refer the reader to <cit.> and the references therein. 






 §.§ Exact Courant algebroids


An exact Courant algebroid (ECA, for short) is a vector bundle E→ M^n of rank 2n together with the following data:

	
  * A non-degenerate, symmetric bilinear form  of signature (n,n), called the neutral inner product;
	
  * A bracket  on Γ(E), called the Dorfman bracket;
	
  * A bundle map π : E → TM, called the anchor map;

subject to the following axioms being satisfied for all a,b,c ∈Γ(E) and f∈ C^∞(M):

	
  * [a,[b,c]] = [[a,b],c] + [b,[a,c]];
	
  * π[a,b] = [π a, π b];
	
  * [a,fb] = f[a,b] + π(a) fb;
	
  * π(a)    b,c   =  [a,b], c  +  b, [a,c];
	
  * [a,b] + [b,a] =  (π^* ∘)  a,b;
	
  * there is an exact sequence of vector bundles 
	
    0 ⟶  T^*M π^*⟶ E π⟶ TM ⟶ 0   .



In the last two axioms  we have abused notation and considered π^*: T^* M → E^* as a map π^* : T^* M → E, after composing with the isomorphism E^* ≃ E given by . Notice also that   in  (ECA5) is the exterior differential in M,  and the bracket in the right-hand-side of (ECA2) is the Lie bracket of vector fields in M. The Dorfman bracket is not necessarily skew-symmetric, but it does satisfy the Jacobi identity, see <cit.> for the proof. Although we will not be making use of it, it is worth mentioning that,   in the literature, the skew-symmetrization of the Dorfman bracket is called Courant bracket and it is defined by:

    [a, b]_c:=1/2([a,b]-[b, a]) ,    a,b ∈Γ(E) .
 
 Despite being skew-symmetric, the Courant bracket does not satisfy the Jacobi identity.  Indeed, one can show that
 
    [a, [b, c]_c]_c+[c, [a,b]_c]_c+[b, [c, a]_c]_c=1/3([a, b]_c, c+[b, c]_c, a+[c, a]_c, b)
 holds for all a, b,c∈Γ(E).
  
The first example of exact Courant algebroid is the generalized tangent bundle. On this, we can consider different structures of exact Courant algebroids depending on the choice of a closed 3-form on M.

 
The generalized tangent bundle TM ⊕ T^* M has a natural structure of exact Courant algebroid, with data given by:

    X+ξ, Y+ η := 12 (ξ(Y) + η(X)),      [X+ξ, Y+η] := [X,Y] +  L_X η - ι_Y ξ,     π(X+ξ) := X,

for X+ ξ, Y+ η∈Γ(TM⊕ T^*M). 

Moreover, for any closed 3-form H∈Ω^3( M), we can consider another ECA structure on , denoted by

    := (TM⊕ T^*M, ,_H,π).

Here,  and π are as in <Ref>, and _H denotes the H-twisted Dorfman bracket defined by:

    [X+ξ, Y+η]_H := [X,Y] +  L_X η - ι_Y ξ + ι_Yι_XH,     X+ξ,Y+η∈Γ(TM⊕ T^*M).






Let  (E_i → M_i,_i,_i,π_i), i = 1,2,   be  two exact Courant algebroids. Recall that a vector bundle isomorphism is a pair (F,f) where f: M_1 → M_2, F: E_1 → E_2 are diffeomorphisms, and,  for each p∈ M_1, F restricts to a linear isomorphism between the fiber over p and the fiber over f(p). We call such a pair an isomorphism of exact Courant algebroids if,  for any a,b∈Γ(E_1),  we have

	
  * F a, F b _2 =  a, b_1;
	
  * [F a, F b]_2 = F [a, b]_1.
 
It follows from  the second condition above that f∘π_1 = π_2 ∘ F, see for instance <cit.> for the detailed proof.

 The next Proposition guarantees that any exact Courant algebroid is isomorphic  to one described in <Ref>.

<cit.>
Let E be an exact Courant algebroid. Any isotropic splitting σ of (<ref>) induces an isomorphism of exact Courant algebroids

    E ≃_σ ()_H  .

Here H ∈Ω^3 (M) is the closed 3-form given by

    H(X,Y,Z) = 2   [σ X,  σ Y],  σ Z ,      X,Y,Z ∈Γ(TM).


Moreover, given another isotropic splitting σ̃, the corresponding isomorphism satisfies 

    E ≃_σ̃ ()_H +  b  ,

for some b∈Ω^2( M).


More generally, Ševera's classification of exact Courant algebroids is the content of the following Theorem.

<cit.> The isomorphism classes of exact Courant algebroids over M are in one-to-one correspondence with H^3(M,) / Diff(M) = H^3(M,)/Γ_M, where Γ_M = Diff(M)/Diff_0(M) is the mapping class group of M.


Given the correspondence in <Ref>, we can characterise automorphisms of any exact Courant algebroid.  Before stating the  result itself, let us introduce two classes of maps which play an important role in generalized Geometry. Any b∈Ω^2 (M) may be viewed as a map b : TM → T^*M and thus it gives rise to an endomorphism of the generalized tangent bundle which in matrix form is represented by

    [ 0 0; b 0 ] : T ⊕ T^* ⟶ T ⊕ T^*.

We then set

    e^b := [ 𝕀 0; b 𝕀 ]∈(T ⊕ T^*),      b∈Ω^2 (M).
 The latter transformations are classically known as B-field transformations which appear naturally  as a generalization of the electromagnetic field in string theory. 

Additionally,  fixed f∈ Diff(M), we can produce a  bundle map  of the generalized tangent bundle covering f, as follows:

    f̅:=[        f        0;        0 (f^-1)^* ]→ .

Then, the group of automorphism of ()_H, and then of any ECA,  consists  of  bundle maps which are composition of  maps introduced above. 





 <cit.> For any closed 3-form H ∈Ω^3 (M), the group of  automorphisms of the exact Courant algebroid  is given by: 

    Aut() = {(f̅ e^b
    , f) : f ∈Diff(M),   b ∈Ω^2(M),   f^*H = H -  b}.



Moreover, the Lie algebra of Aut() is

    aut() = {X + b ∈Γ(TM) ⊕Ω^2(M) : L_XH = - b}.


 We refer the reader to <cit.> for a detailed study of the ILH Lie group structure  on Aut().

To conclude this Section, we observe that there is a natural way of scaling exact Courant algebroids. This will play a key role in the next Sections. 


Let (E,, , π) be an exact Courant algebroid and let c∈ℝ\{0}.  Then, the data (c  , , π) defines a new exact Courant algebroid structure on E. 




Trivially,  c is a non-degenerate symmetric bilinear form of signature (n,n), regardless of the sign of c. Moreover,  all the axioms except for (ECA5) in the definition of ECA clearly remain true after scaling . As regards  (ECA5), we can notice that the π^* in the right-hand-side involves a composition with the isomorphism φ_ : E^* ≃ E induced by the neutral inner product. For c,  this isomorphism is simply 

    φ_c = c^-1φ_,

which gives the claim.



Let (E,,,π) be an exact Courant algebroid and c∈ℝ\{0 }. We define the scaled exact Courant algebroid c · E to be the one associated with the data (E, c  , , π). 
 
We now study the effect of scaling on the  3-form H after choosing an isotropic splitting:


Let E be an ECA with isotropic splitting σ. Then,  the isomorphism E≃_σ, H∈Ω^3 (M) closed, induces an isomorphism

    c · E ≃_σ ()_c H.








This follows immediately from <Ref> and (<ref>).

























 §.§ Generalized metrics


A generalized metric on an ECA  E is an orthogonal endomorphism ∈(E,) such that the bilinear form 

    (a,b) ↦ a, b ,      a,b ∈Γ(E),

is symmetric and positive definite. We call the pair (E, G) a metric Courant algebroid. A Courant algebroid isomorphism (F,f)  (E_1,  G_1) → (E_2,  G_2) is a metric Courant algebroid isometry (or simply isometry) if F ∘ G_1 =  G_2 ∘ F.

The presence of a generalized metric on  E allows us to consider a preferred isotropic splitting. We quickly recall  its construction since it will  be useful in what follows.  We refer the reader to <cit.> for the detailed description.
Since   G^2= Id,  we can consider the eigenbundles  E_± of E associated to ± 1. It is not hard to see that π E_±→ TM is an isomorphism of vector bundles. Then, we can define 

    σ_±:= (π_|E_±)^-1 TM→ E_± ,   τ_±(X, Y)=σ_±X, σ_±Y ,     X, Y∈Γ(TM) .
  
With these ingredients, we can consider the following splitting:

    σ=σ_±-1/2π^*τ_± .
  
It is easy to see that σ is  isotropic and it does not depend on the choices of σ_±, see <cit.> for the proof.  Moreover, we have that 

    τ_+(X, X)= Gσ_+X, σ_+X  >0 ,     X∈Γ(TM) , X 0  ,
 
thus determining a Riemannian metric g on M. 











 <cit.> Let E be an ECA with isotropic splitting σ.  Under the isomorphism E≃_σ, any generalized metric  on E corresponds to a generalized metric on  of the form

    G(g,b) := e^b [    0 g^-1;    g    0 ] e^-b,

where b ∈Ω^2(M) and g is a Riemannian metric on M. Moreover, the bundle map

    (e^-b,𝕀_M)(, G(g,b)) →(()_H +  b,  G(g,0)),

is an isometry.



The isometry in (<ref>) indicates that a generalized metric G on an ECA E with Ševera class α∈ H^3 (M, ℝ) is equivalent to a choice of Riemannian metric g on M and a preferred representative H∈α. This can be also be observed by considering the preferred isotropic splitting induced by G and the corresponding isomorphism E≃ under which G corresponds to G(g,0).


Furthermore, the existence of a preferred isotropic splitting induced by a generalized metric G allows us to characterize generalized isometries on a fixed exact Courant algebroid.

  The group of generalized isometries of the metric  Courant algebroid  (, G(g,0)) is given by

    𝖨𝗌𝗈(, G(g,0)) = {(f̅
    , f): f ∈𝖨𝗌𝗈(M,g),   f^*H = H}⊂Aut().




Let us now examine the effect of scaling on generalized metrics.


Let (E, G) ≃_σ (,G(g,b)) be a metric ECA and c>0. Then, ± G is a generalized metric on ± c · E and the isomorphism ± c· E ≃_σ ()_± cH gives an isometry 


    (± c · E, ±) ≃_σ (()_± cH, (cg, ± cb)).









The first claim is clear, since the  bilinear form ± c ±·, · = c  ·, · is symmetric and positive definite.

As regards the second one,  we note that the scaling on the Riemannian metric follows directly from (<ref>). On the other hand,  considering  σ_1 to be the preferred isotropic splitting induced by G, then (σ_1-σ)(X)∈π= Im(π^*), for any X∈Γ(TM). Hence,  we can write

    b=(π^*)^-1(σ_1-σ) TM→ T^*M ,
 where the fact that b is a 2-form is straightforward to check, using  that both σ_1 and σ are isotropic. Then, the scaling on b follows from (<ref>). 


The isometry in the second claim  in <Ref> can be also explicitly constructed.  Assuming c>0, we can use  the isotropic splitting σ to obtain the isometry (± c· E,  G)≃_σ(± c·()_H,  G(g, b)). Then, one can easily see that

    [     Id      0;      0 ± c Id ](± c·()_H,  G(g, b))→ (()_± cH,  G(cg, ± cb))
 is an isometry. Hence, composing the two isometries above, we obtain the claim.















 §.§ The generalized Ricci curvature




In this short Section, we will briefly discuss the explicit expression of the generalized Ricci curvature in  the case of (()_H,  G(g, 0 )) which will be central in what follows. For a detailed investigation of the general definition, we refer to <cit.>, <cit.>, <cit.>, <cit.>  and <cit.>.  
Before doing that, we recall that if   (M, g) is a Riemannian manifold  and H∈Ω^3(M),  there exist  unique connections ∇^±, called Bismut connections, such that 

    ∇ g=0 ,    gT^± =± H ,
 where the tensor T^± is the torsion of ∇^±. In what follows, we will abuse the notation denoting with 
  
    Ric^B_g, H:= Ric_g-1/4H^2
 the symmetric part of the Ricci tensor of the Bismut connections ∇^± associated to g and H.  The symmetric (0,2)-tensor H^2 is defined by H^2(X, Y)=g(ι_XH, ι_YH),  for all X, Y∈Γ(TM).
 Then, following the notation in <cit.>, the generalized metric (g, 0) determines two eigenbundles  as in <Ref>,  which,  in this particular case,   takes the following form: 
 
    E_±={X± g(X) :  X∈Γ(TM)}  .

   Now, using <cit.>  and <cit.>, it is easy to see that, for any X∈Γ(TM), 

    Rc^+(X-g(X))=     g^-1  Ric^B_g, HX-1/2g^-1_g^*HX+   Ric^B_g, HX-1/2_g^*HX ,
    Rc^-(X+g(X))=     -g^-1  Ric^B_g, HX-1/2g^-1_g^*HX+   Ric^B_g, HX+1/2_g^*HX .
 


    Rc ( G)= Rc^+- Rc^-=[    g^-1  Ric^B_g, H 1/2 g^ -1 ^*_gHg^-1;          -1/2^*_g H   -  Ric^B_g, Hg^-1 ].

 In <cit.>,  the authors prove the Aut(E)-equivariance  of the generalized Ricci curvature which will be extensively used  in the next Sections.  







 §.§ The generalized Ricci flow



 A one-parameter family ((t))_t∈ [0,T) of generalized metrics on an ECA E is a solution to the generalized Ricci flow if it satisfies

    ^-1∂/∂ t =-2   Rc().

If we use the isotropic splitting σ_0 associated to the initial metric (0) to identify E ≃_σ_0 ()_H_0, then (t) corresponds to (g(t),b(t)) for some one-parameter families of Riemannian metrics g(t) and two-forms b(t) on M. Then, Equation (<ref>) is equivalent to the following coupled system: 

    ∂/∂ t g    = - 2   Ric_g, H^B, 
    ∂/∂ t b    = - ^*_g H,

where H(t) = H_0 +   b(t).  Since H_0 is closed, it follows that H(t) evolves accordingly to the classical heat equation: 

    ∂/∂ t H = Δ_g H .


We should emphasize that <Ref> firstly appeared in <cit.>  as the renormalization group flow. We refer to <cit.> for a detailed explanation of the relation between the renormalization group flow and generalized Geometry.

Since the generalized Ricci curvature is Aut(E)-equivariant, the PDE system  (<ref>) defining the generalized Ricci flow is only weakly parabolic.  A suitable application of the DeTurck  trick can be applied to obtain short-time existence and uniqueness, provided M is compact, see <cit.> for details. 

More generally, a one-parameter family ( G(t))_t∈[0,T) of generalized metrics on an ECA E solves the gauge-fixed generalized Ricci flow if there exists a one-parameter family (F_t)_t∈[0,T)⊆ Aut(E) so that 
F_t G(t)F_t^-1 is a solution to the generalized Ricci flow (<ref>).  Using again the isotropic splitting σ_0 induced by G(0) to obtain (E,  G(t))≃_σ_0( ()_H_0,  G(g(t), b(t))) as above,  it can be seen that G(t) is a solution of the gauge-fixed generalized Ricci flow if only if there exists  a one-parameter family of generalized vector fields X_t+B_t∈aut(()_ H_0) such that (g(t), b(t))_t∈ [0, T) is a solution of the following  coupled system:

    ∂/∂ tg=     - 2   Ric^B_g, H+  L_Xg ,
    ∂/∂ tb=     -^*_gH-B+ L_Xb .



It follows that H(t) will evolve by

    ∂/∂ tH=Δ_gH+  L_XH .











§ SOLITONS OF THE GENERALIZED RICCI FLOW
 
 Motivated by  <Ref> and   by the effect of scaling on the 3-form H described in <Ref>, we give the following new definition of solitons for the generalized Ricci flow. 


A solution (E, (t))_t∈ [0,T) to the generalized Ricci flow (<ref>) is called a generalized Ricci soliton if there exists a one-parameter family of scalings c(t) > 0, c(0) = 1,  and a one-parameter family of generalized isometries 

    F_t : (c(t)· E,  (0)) ⟶ (E,  G(t)).



 Differentiating the family F_t at t=0 gives rise to a pair (X,  D)∈Γ(TM)× End(E) satisfying the following properties:
  
  
  * D∈ Der([·, ·]), i.e. D[a, b]=[Da, b]+[a, Db], for any a, b∈Γ(E);
  
  *  X=  ( D+ λ Id)·, ·+ ·,  ( D+ λ Id)·,
 where c'(0)=-2λ;
 
  *  [ Rc( G(0))+ D, G(0) ]=0.
  
  As one can easily see, the dependence of (<ref>) on the scalings is only through c'(0). This motivates the following definition.


 
 Let E be a  ECA and λ∈ℝ. We say that (X, D)∈Γ(TM)× End(E) is a λ-derivation if <Ref> and  <Ref>  are satisfied.  We denote by Der_λ([·, ·]) the set of all λ-derivations. 
 
 It can easily be observed that  0-derivations are precisely the derivations of the Dorfman bracket introduced in <cit.>. Then, λ-derivations has to be interpreted as the generalization of derivations when a scaling process is taken into account. 
 
 We can  give a characterization of both the set of isomorphisms  between c·()_H and ()_H and the set of λ-derivations which will be useful in the next Sections.
  Let H∈Ω^3(M) be  closed and c∈\{0}. Then, 
 
    Aut(c·()_H, ()_H)={f̅_c e^b :  b∈Ω^2(M),     f∈ Diff(M),    f^*H=c(H- b)} ,

  where 
    f̅_c=[         f         0;         0 c(f^-1)^* ] .

Moreover, if λ∈, 
 
    Der_λ([·, ·])={X+ b∈Γ(TM)⊕Ω^2(M) : ℒ_XH =- b -2λ H } .

 
 
 Let (F, f)∈ Aut(c·()_H, ()_H). Since F preserves the Dorfman bracket, we have that  π∘ F= f ∘π. This naturally gives us that if
 
    F=e^B[   f   0;   0 h^* ]e^b
 then B=0.  On the other hand,  the condition F·, F·=c ·,·
 


 
 forces b∈Ω^2(M) and h=c f^-1 .




  Moreover, we have that, for any X+ξ, Y+η∈Γ (),
 
    f̅_c [(f̅_c)^-1(X+ξ),(f̅_c)^-1(Y+η)]=      [ X,  Y]+ℒ_Xη- ι_Yξ+ cι_Yι_X(f^-1)^*H.
 Composing this with the usual  action of  e^b,  we have that  if F∈ Aut( c· ()_H, ()_H) then 
 
    F=f̅_ce^b ,      f^*H=c(H- b) .
 The viceversa is trivial, concluding the proof of the first claim. 
 
Let us consider a one parameter of scalings  c(t)>0 such that c(0)=1 and c'(0)=-2λ and  a one parameter family (F_t, f_t)∈ Aut( c(t)·()_H,  ()_H).  Differentiating (F_t, f_t) at t=0 gives X+b∈Γ(TM)⊕Ω^2(M) such that 
 
    ℒ_XH=- b- 2λ H .
 Viceversa, given X+ b∈Γ(TM)⊕Ω^2(M) such that ℒ_XH=- b- 2λ H,  we can consider f_t the one parameter family of diffeomorphisms generated by X  and a family of scaling c(t)>0 such that c(0)=1 and c'(0)=-2λ. Hence, we can define 
 
    b̅_t=∫_0^tc(s)f_s^*b_sds ,
 where b_s∈Ω^2(M) such that ℒ_X_sH=- b_s+ c'(s)/c(s) H. 

 	Then, we have that 
 
    b̅_t=    
    -∫_0^t1/c(s)f_s^*(-c'(s)/c(s) H + ℒ_X_sH) s 
     =-∫_0^t/ s(1/c(s)f_s^* H) s=-1/c(t)f_t^* H+ H .
 Then, F=f̅_t, c(t)e^b̅_t∈ Aut(c(t)·()_H,   ()_H), as we wanted. 

 <Ref> is the dynamical definition of solitons for the generalized Ricci flow. As in the classical case, we can deduce from that an equivalent and static definition in terms of classical data, as the next Proposition shows.


Let (E,(t)) be a generalized Ricci soliton, and consider the time-independent isometry (E, (t)) ≃_σ_0( ()_H_0,  (g(t), b(t)) ) induced by the isotropic splitting σ_0 associated to (0). Then, g_0 := g(0) and H_0 satisfy

    Ric_g_0, H_0^B     =  λ  g_0   + 12  L_X g_0, 
    Δ_g_0 H_0    =  -2  λ  H_0  -   L_X H_0,

where λ = -12  c'(0) and c(t)>0 are the scalings from <Ref>. Conversely, if ( ()_H_0,  (g_0, 0) ) satisfies (<ref>), then there exists a soliton solution (g(t), b(t)) to (<ref>) on ()_H_0 with (g(0), b(0)) = (g_0,0). 






By <Ref>,  the isomorphism associated to  σ_0 also induces  time-independent isometries 

    (c(t)· E,  (0))    ≃_σ_0(()_c(t) H_0,  (c(t)g_0, 0) ).

It follows that c(t) g_0 and c(t) H_0 must solve the gauged-fixed equations (<ref>) and (<ref>).

Thus,

    c' g_0    = - 2   Ric^B_g_0, H_0 + c L_X g_0, 
    
    		c' H_0    = Δ_g_0 H_0 +  c  L_X H_0.


 Evaluating at t=0 and using c(0) = 1, c'(0) = -2 λ, (<ref>) follows.




When H_0 = 0, <Ref> implies that <Ref> is equivalent to the classical definition of Ricci soliton for the Ricci flow. 

On the other hand, if c(t) ≡ 1 is constant, we recover the definition of soliton solution given in <cit.>, which only allows for steady solitons (i.e. λ = 0).  Recently, in <cit.>, a definition of generalized Ricci soliton, equivalent to <Ref>, was given. 

 Moreover, in the non-steady case, the second equation in (<ref>), together with H_0=0 and the Cartan's formula, forces [H_0]=0. 
 



 In particular,  if M is compact, this implies that  a non-steady generalized Ricci soliton with harmonic torsion is a classical Ricci soliton. 

 To conclude this Section, we give a family of explicit examples of expanding generalized Ricci solitons.


  Let us consider M_0=H_3:= Heis(3, ) the 3-dimensional Heisenberg group.  It is well known that on M_0 we can choose coordinates {x, y, z} so that 
  
    e_1=∂/∂ x ,    e_2=∂/∂ y+ x ∂/∂ z ,    e_3=∂/∂ z
  are left-invariant vector fields, forming a frame of the Lie algebra of M_0,  subjected to the following structure equation:
  
    [e_1, e_2]=e_3 .
 We will denote with {e^1, e^2, e^3} the dual coframe of {e_1, e_2, e_3} which is, in particular,  given by:
  
    e^1= x ,    e^2= y ,    e^3= z-x y .
We next consider the following one parameter family of diffeomorphisms of M_0:
  
    φ_t(x, y, z)=((1+4t)^1/4x, (1+4t)^1/4y, (1+4t)^1/2z) ,   (x, y, z)∈ M_0 ,    t≥ 0  .
 It is easy to see that φ_t is actually an automorphism of M_0 viewed as a Lie group. 
Another easy computation allows us to state that 
  
    φ_t=(1+4t)^1/4e^1+(1+4t)^1/4e^2+ (1+4t)^1/2e^3 ,   t ≥ 0 .
 Now we consider the closed left invariant 3-form  H= x∧ y∧ z  and observe that H is harmonic with respect to any  metric on M_0.  Then, one can easily see that 

    φ_t^*H=(1+4t)H ,    t≥ 0 .

  Then, denoting c(t)=1+4t>0 and  comparing with <Ref>, we are in the position to promote φ_t to 
   
    F_t=φ̅_t, c(t)∈ Aut(c(t)·()_H, ()_H) ,    t≥ 0 .
 Moreover, we easily compute the vector field generated by φ_t:
   
    X=/ t|_t=0φ_t=x∂/∂ x+y∂/∂ y+ 2z∂/∂ z=xe_1+ye_2+(2z-xy)e_3 ,
 which,  in particular,  satisfies the following bracket relations:
   
    [X, e_1]=-e_1 ,    [X, e_2]=-e_2 ,    [X, e_3]=-2e_3 .
 Finally, X∈ Der_-2([·, ·]) satisfies 
 
    ℒ_XH=4H
 which is  precisely the second equation in (<ref>) with -2λ=c'(0)=4, since H is harmonic with respect to any Riemannian metric on M_0. Now, we consider the following Riemannian metric:
   
    g=e^1⊙ e^1+ e^2⊙ e^2+ e^3⊙ e^3=  x⊙ x+ (1+x^2) y ⊙ y +  z ⊙ z - 2x y ⊙ z ,
 which is the standard left-invariant Riemannian metric on M_0 with respect to the frame {e_1, e_2, e_3}.  It is easy to see that 
   
    Ric^B_g, H=-(e^1⊙ e^1+ e^2⊙ e^2)  ℒ_Xg=2(e^1⊙ e^1+ e^2⊙ e^2+2e^3⊙ e^3) .
 Then, 
   
    Ric^B_g, H=-2g+1/2ℒ_X g
  giving precisely the first equation in (<ref>). Now, we can use <Ref> to infer that (()_H,  (g, 0)) is an expanding soliton for the generalized Ricci flow. Moreover, we can also specify what actually is the one parameter family of generalized isometries in <Ref>. Indeed,  considering F_t as in (<ref>), we have
   
    F_t(g, 0)F_t^-1=((1+4t)(φ_t^-1)^*g, 0)
 but
   
    g(t)=(1+4t)(φ_t^-1)^*g=(1+4t)^1/2e^1⊙ e^1+(1+4t)^1/2e^2⊙ e^2+e^3⊙ e^3
  is precisely the solution of the generalized Ricci flow starting from g, see <cit.>.  
   
   In the same fashion and with    a bit more effort, we can construct examples of expanding solitons  for the generalized Ricci flow on M_k:=H_3×^k, for any k≥ 1.
    On M_k, we consider the following  left-invariant frame {e_1, e_2, e_3, e_4, …, e_k+3} of the Lie algebra of H_3×^k  where e_1, e_2, e_3 are the same defined above while e_3+j=∂/∂ x_3+j where x_3+j is the standard coordinate on the j-th copy of  in  the abelian factor ^k. 

     Then, it is sufficient to consider c(t)=1+4t as scaling, 
     
    φ_t(x, y, z, x_4, …, x_3+k)=((1+4t)^1/4x, (1+4t)^1/4y, (1+4t)^1/2z, (1+4t)^1/2x_4, …, (1+4t)^1/2x_3+k) ,
 for all (x,y,z,x_4, …, x_3+k)∈ M_k, 
   as the one parameter family of diffeomorphisms of M_k , H=e^1∧ e^2∧ e^3 and finally the standard left-invariant metric
   
    g=∑_i=1^k+3e^i⊙ e^i ,
 with respect to the coframe {e^1, …, e^3+k} dual with respect to {e_1, …, e_k+3} and obtain an expanding soliton of the generalized Ricci flow on M_k. 










§ HOMOGENEOUS GENERALIZED GEOMETRY
 
In this Section we specialize our study  to the case where the underlying manifold M = is a simply-connected Lie group. We will denote by e the identity of  and g = T_e its Lie algebra. We also ask the ECA to be compatible with the algebraic structure in the following sense:

We say that a metric ECA (E →, G) is left-invariant if the action of  on itself by left-translations lifts to an action on (E, G) by isometries (as defined in <ref>).

The action of 𝖦 on E in <Ref> is a particular instance of a wider class of Lie group actions on ECAs called lifted or extended actions, see <cit.> and <cit.>.

 The action  of 𝖦 on a left-invariant  metric ECA allows  to consider 𝖦-equivariant isotropic splittings.  The choice of  such isotropic splittings, using <Ref>, gives rise to left-invariant classical data as the following Proposition highlights.

Let (E→, G) be a  left-invariant metric ECA. Then,  any -equivariant isotropic splitting σ T𝖦→ E of (<ref>) induces a -equivariant isometry 

    (E,  G) ≃_σ (, G(g,b)),

where the tensors g, H, b on M are left-invariant and H depends only on σ and not on .


 
If σ is the isotropic splitting given by , firstly,  we note that, by <Ref>, (E, G) is isometric to (, G(g,0)),  for some closed 3-form H and Riemannian metric g. Using this isomorphism, we can define an action of   on , and by definition this action will be  by isometries of G(g,0). By <Ref>,  H and g must be left-invariant.  It remains to prove the invariance of b.

As in (<ref>), the 2-form b can be expressed as 

    b=(π^*)^-1(σ_1-σ) ,
 where σ_1 is the isotropic splitting induced by G,  which is -equivariant. Then, the claim follows directly from the fact that, for any (F, f)∈ Aut(E), we have that F∘π^*=π^*∘(f^-1)^* and  from  the  expression of b above.






As said, the preferred isotropic splitting induced by G is  -equivariant. Thus, any homogeneous metric ECA is isometric to (, G (g,0)) for a left-invariant Riemannian metric g, and a closed left-invariant 3-form H ∈Ω^3(M)^. 




Notice that as a vector bundle any left-invariant ECA is trivial, since after choosing an isotropic splitting we get an isomorphism with , where 

    T ⊕ T^* ≃ () ×

because Lie groups are parallelizable by using left-invariant vector fields. By abuse of notation, we will simply denote 

    ()_H := ,

where H∈Ω^3()^≃Λ^3^* is a closed, left-invariant 3-form on 𝖦.











 §.§ The space of left-invariant generalized metrics as a homogeneous space


Let E → be a left-invariant ECA, and let us fix a background left-invariant generalized metric   on E. The preferred isotropic splitting σ̅: T→ E induced by  yields an isometry

    (E →, ) ≃_σ̅ ( ()_H̅,  G(g̅, 0)),
 
for some left-invariant metric g̅ and closed 3-form H̅ on . By abuse of notation we write =  G(g̅, 0).
Let us set

    ^   := { :  is a left-invariant generalized metric on  ()_H̅}
       ≃{(g,b) : g  left-invariant metric on , b∈Ω^2()^}

 where the  last identification is due to <Ref>.   Next, we consider the following Lie subgroup of (,):

    𝖫 := { e^b ·[        h        0;        0 (h^-1)^* ] : h∈𝖦𝖫( g) ,   b∈Λ^2 ^* },       e^b := [ 𝕀 0; b 𝕀 ].

As an abstract Lie group,  is isomorphic to the semi-direct product Λ^2 ^* ⋊𝖦𝖫(). Moreover, its Lie algebra is 

    := {[    A    0;    α -A^* ] : A∈gl(g) ,  α∈Λ^2 ^* }.



The standard representation of 𝖮( g ⊕ g^*,··) on  gives rise to a representation of  on , and this extends in a standard way to an action of  on all tensor powers of . In particular,  acts on generalized metrics and Dorfman brackets via the following formulas:

    ℓ·  := ℓ∘∘ℓ^-1,     ℓ· := ℓ [ℓ^-1 ·, ℓ^-1· ].

Recall that we also have the `change of basis' actions of 𝖦𝖫() on the spaces of brackets and 3-forms: 

    h·μ (·, ·) = h μ(h^-1· ,h^-1·),      h·ω (·,·,·) = ω(h^-1· ,h^-1· ,h^-1· ),       h∈𝖦𝖫(),

for μ∈Λ^2 ^* ⊗ and ω∈Λ^3 ^*.

The next Proposition allows to present the space of left-invariant metric on E as a homogeneous space.

The action of   on the space of left-invariant generalized metrics ^  is transitive.
 



For any ℓ≃ (b, h) ∈, its action on the background metric is given by

    ℓ·   = e^b   [        h        0;        0 (h^-1)^* ][     0 g̅^-1;    g̅     0 ][ h^-1    0;    0  h^* ]    e^-b = e^b  [    0 g^-1;    g    0 ]  e^-b,

where 

    g = (h^-1)^* g̅ h^-1 = h ·g̅ (· ,·).

Since 𝖦𝖫() acts transitively on left-invariant metrics, the claim now follows from <Ref>.























 §.§ Moving generalized metrics is equivalent to moving  Dorfman brackets
  In this Subsection, we will define and study the main properties of the space of left-invariant Dorfman brackets. A similar treatment of them can also be found in <cit.>. Then, we will focus on describing  the effect of the -action and its infinitesimal version  on Dorfman brackets.


From the axiom (ECA5), the Dorfman bracket associated with a left-invariant ECA is skew-symmetric.
Thus, we consider the vector space Λ^2 ()^* ⊗ () of all skew-symmetric bilinear maps 

    : () × () → ().

 Using <Ref>, we know that choosing an isotropic splitting σ gives rise to  an isomorphism E≃_σ ()_ H. Then,  from left-invariance and (<ref>), it follows that the Dorfman bracket of E gets identified with _H ∈Λ^2 ()^* ⊗ () given by

    [X + ξ, Y + η]_H = [X,Y]_ - η∘_(X) + ξ∘_(Y) + ι_Yι_X H,

where _(X) = [X,·]_ : → and H∈Λ^3 ^* is closed. It is not hard to see that _H ,· is totally skew-symmetric. This motivates the following. 

The space of Dorfman brackets is the algebraic subset of Λ^2 ()^* ⊗ () given by

    D := {∈Λ^2 ()^* ⊗ ()     :   (·, ·), ·∈Λ^3()^* ,   (^*, ^*) = 0,    J() = 0}.

Here,  J () (a,b,c) := ((a,b),c) + ((b,c),a) + ((c,a),b) = 0 is the Jacobi identity.

 We can show that all elements in 𝒟 can be written as in (<ref>) for suitable choices of the 3-form H and the Lie bracket on .

We have that ∈ D if and only if there exists a Lie bracket μ on  and a closed 3-form H∈Λ^3 ^* such that, for any X+ξ, Y+η∈, 

    (X + ξ, Y + η) = μ(X,Y) - η∘μ_X + ξ∘μ_Y + ι_Yι_X H,     μ_X := μ(X,·).
 





Sufficiency was observed above. Regarding necessity, for each X,Y, Z ∈,  we set 

    μ(X,Y) := π∘(X,Y),      H(X,Y,Z) := 2  (X,Y), Z .

The linear conditions on   imply that,  for every ξ∈^*,  we have  (ξ,Y) ∈^*. Thus, 

    (ξ, Y)(X)= 2 (ξ, Y), X = 2 (Y, X), ξ=  ξμ(Y, X) =   ξ∘μ_Y (X).

By skew-symmetry, (X,η) = -η∘μ_X, and from this (<ref>) follows. A simple computation shows that the Jacobi identity for  implies the Jacobi identity for μ and that _μ H = 0.




















  Moreover, one can notice that the following conditions hold for brackets in D: 

    (,) ⊂,      (,^*) ⊂^*,     (^*,^*) = 0.

That is, (,) is a Lie algebra which is a central extension of (,μ). We thus have that the space of Dorfman brackets is contained in the linear subspace:

    V_ D := {∈Λ^2 ()^* ⊗ ()   :  satisfies (<ref>) and (·, ·), ·∈Λ^3()^*  } .



  


We  then have two projections (·)_ :  D →Λ^2 ^* ⊗ and (·)_Λ^3 :  D →Λ^3 ^*,

    ↦_ = μ∈Λ^2 ^* ⊗,     ↦_Λ^3 = H ∈Λ^3 ^*,

where here μ is simply the Lie bracket of .  Furthermore, we notice that, by (<ref>), these projections completely determine a Dorfman bracket ∈ D. 

Let us first study how the action of  relates to the projections in (<ref>):



Let ∈ D with μ := _, H := _Λ^3. Then, for any ℓ=  h̅ e^α∈, we have

    (ℓ·)_ = h·μ,      (ℓ·)_Λ^3 = h · (H - _μα).



First, we compute for X+ξ,Y + η∈ g⊕ g^*:

    (e^α·)(X + ξ,Y + η)    = e^α(X  + ξ - i_X α ,    Y + η - i_Y α)
    
       = e^α( (X+ξ, Y + η) + α(Y, μ(X,·)) - α(X, μ(Y,·)))  
    
       = (X+ξ, Y + η)  -  i_Y i_X (_μα).

Similarly,

    ([        h        0;        0 (h^-1)^* ]·)(X + ξ,Y + η)    = [        h        0;        0 (h^-1)^* ](h^-1X + ξ∘ h, h^-1Y + η∘ h)
    
    
       = (h·μ)(X,Y) - η∘ (h·μ)_X  + ξ∘ (h·μ) + i_Y i_X (h· H).

The result follows from composing the two actions.



Given the actions of  on Dorfman brackets on , and of 𝖦𝖫() on brackets and 3-forms on , we denote the corresponding Lie algebra representations (or `infinitesimal actions') by

    Θ : →(Λ^2 ()^* ⊗ ()),     θ : gl(g)→(Λ^2 ^* ⊗),     ρ : gl(g)→(Λ^3 ^*).
 
More precisely, they are given by

    Θ(A)  (· ,·)    = A  (·, ·) - (A ·, ·)  - (·, A ·),      A∈gl(), 
    θ(A) μ (· ,·)    = A μ (·, ·) - μ(A ·, ·)  - μ(·, A ·),       A∈gl(), 
    ρ(A) ω (·, ·, ·)    = -ω(A ·, ·, ·) -ω(·, A ·, ·) -ω(·,  ·, A ·),       A∈gl().

From <Ref> we immediately deduce: 

For any L ≃ (α, A) ∈≃Λ^2 ^* ⋊gl(g),  we have

    (Θ(L))_  = θ(A) μ,      (Θ(L) )_Λ^3 = ρ(A) H - _μα.







Since we will be moving the Dorfman and Lie brackets on  and  respectively, it is convenient to introduce the following notation: given  a Dorfman bracket as in (<ref>), we denote by

    ()_ := (, , , π)

the left-invariant ECA structure given by the data (, , π), where  and π are defined in (<ref>). 
Notice that in order for (ECA2) to be satisfied, we must necessarily have that the Lie bracket on  is μ := _.

  Let ℓ =  h̅  e^α∈, for h ∈𝖦𝖫() and α∈Λ^2 ^*. Then, the bundle map 

    (ℓ, h) : (()_H, (ℓ^-1) ·) →( ()_ℓ·, )
 is an isometry. Here _ = _ is the original Lie bracket of  and _Λ^3 = H.



We first prove the claim for ℓ = h̅ covering h ∈𝖦𝖫(). By definition of the action, the map h : (, _) → (, h·_) is a Lie algebra isomorphism. Set μ := h·_, let ,_μ be respectively the simply-connected Lie groups with Lie algebras (,_) and (, μ). Then, there exists a Lie group isomorphism φ : →_μ for which _eφ = h, where e∈ is the identity. The corresponding bundle map φ̅: T⊕ T^*→  T_μ⊕ T^*_μ covering φ clearly maps left-invariant sections to left-invariant sections, and since the trivialization (<ref>) is done via left-invariant sections, it follows that φ̅ may be represented by an element of 𝖦𝖫(). The latter is precisely h̅, and the pair (h̅, h) is the infinitesimal data corresponding to the ECA isomorphism 

    φ̅: ()_H → ()_h̅·.

Indeed, h̅ preserves the Dorfman bracket by definition of the action, and it preserves the neutral inner product because h̅∈(,). Finally, 

    h̅·  (h̅^-1·)  = ,

thus we have an isometry. 

The proof for ℓ = e^α covering 𝕀∈𝖦𝖫() is similar (in this case, = _μ). 


Let us now consider the effect of scaling when  Dorfman brackets are moving. Recall that, by <Ref>,  a choice of isotropic splitting σ induces an isometry (c· E, ) ≃_σ (()_cH, (cg, cb)), for any c>0. 


 
Let c > 0

 and let  be the Dorfman bracket determined by _ = _ and _Λ^3 = H. Then, the following bundle map is an isometry:

    (±c^1/2𝕀 ,  ± c^1/2𝕀) : (()_± cH, (cg, ± cb)) ⟶  (()_± c^-1/2, (g,b)).



First, we see that it is an ECA isomorphism. Indeed, we have that ±c^1/2𝕀∈(,), 

and 

    ±c^1/2𝕀[(±c^1/2𝕀)^-1 · , (±c^1/2𝕀)^-1 · ]_cH = ± c^-1/2(·, ·)
 
can be easily verified using  (<ref>) and (<ref>) (with μ = _). Regarding the isometry claim, we directly compute:

    (c^1/2𝕀)  (cg,cb)  (c^1/2𝕀)^-1   =
    	 [  c^1/2𝕀       0;       0 c^-1/2𝕀 ]  e^cb 
    	 [        0 c^-1g^-1;      c g        0 ]  e^-cb 
    	 [ c^-1/2𝕀       0;       0  c^1/2𝕀 ]
        = [      -g^-1 b         g^-1; g - b g^-1 b       b g^-1 ]  
    	  = (g,b).










 §.§ The moment map
 

In this section we define a moment map for the action of  on Dorfman brackets, in the sense of real GIT (see <cit.>). In <Ref> we show that this moment map encodes the most complicated part of the generalized Ricci curvature of left-invariant generalized metrics.

As before, we fix a background generalized metric  on a left-invariant ECA, yielding an isomorphism with (()_H̅, (g̅, 0)). The metric g̅ gives rise to an inner product on Λ^2 ()^* ⊗ (),  also denoted by g̅. To define it, we fix a g̅-orthonormal basis { e_i} of   with dual basis {e^i} and set

    g̅(,) := 2  ∑_i,j(e_i,e_j), (e_i,e_j) + 4  ∑_i,j(e_i,e^j), (e_i,e^j) + 2  ∑_i,j(e^i,e^j), (e^i,e^j).
 
The overall factor of 2 is due to the 12 factor in the definition of . Since we are only interested in Dorfman brackets and their infinitesimal variations, we will mostly work with brackets in V_ D (see (<ref>)). Then, the only non-vanishing structure coefficients  are: 

    (e_i, e_j) = _ij^k e_k + _ijk e^k,     (e_i, e^j) = -_ik^j e^k = -(e^j, e_i),

using Einstein's summation convention (summing over all i,j,k and not just i<j). The fact that the coefficient of e^k in (e_i, e^j) is related to that of e_j in (e_i,e_k) follows from (·,·), ·∈Λ^3()^*. We then have

    g̅(, )	:= ∑_i,j,k( 3  _ij^k _ij^k  + _ijk_ijk) = 3  g̅(μ, ν) + g̅(H, H̃),     , ∈ V_ D,

where μ = _, ν = _,  H = _Λ^3,  H̃ = _Λ^3. Here we have used the extension of g̅ to a positive-definite inner product on Λ^2 ^* ⊗, and on  Λ^k ^*, given respectively by

    g̅(μ,ν) :=  ∑_i,j,kμ_ij^k ν_ij^k,     μ,ν∈Λ^2 ^* ⊗,     μ_ij^k := g̅(μ(e_i,e_j), e_k).

and

    g̅(α, β) := ∑_i_1, …, i_kα_i_1 ⋯ i_kβ_i_1 ⋯ i_k,     α,β∈Λ^k ^*,     α_i_1⋯ i_k := α(e_i_1, …, e_i_k).













The metric g̅ also determines  a  maximal compact subgroup (, g̅) ≤. We fix on  the ((,g̅))-invariant, positive-definite inner product g̅_ given by 

    g̅_([    A    0;    α -A^* ], [    B    0;    β -B^* ]) :=  2  A B^T + 1/6β^* α = 2  ∑_i,j A_ij B_ij + 1/6∑_i,jα_ijβ_ij,

where A_ij := g(A e_i, e_j), α_ij = α(e_i, e_j), etc, for a g-orthonormal basis { e_i }. While the  1/6-factor looks arbitrary, it will play a key role in the forthcoming computations.






The moment map for the action of  on (Λ^2 ()^* ⊗ (), g̅) is the map 

    : Λ^2 ()^* ⊗ () →,     ↦_,

 implicitly defined by

    g̅_(_, L) =  1/6 g̅(Θ(L), ),        L∈.





The moment map is (,g̅)-equivariant. Moreover,  for each ∈Λ^2 ()^* ⊗ () and L∈,  it satisfies

    g̅_(_, L) = 1/12 / t|_t=0|exp(t L)·|^2_g .





 Let K∈(,g̅), for every L∈,  we have that 

    g̅_(_K·, L)=
    
    1/6g̅(K ^-1·Θ(L)(K·), )=1/6g̅(Θ(K ^-1LK ), )
    = g̅_(_, K ^-1LK )=g̅_(K_K^-1,L )

yielding the first claim. The second one is trivial computing the derivative on the right hand side. 



We now define Rc_∈() to be the generalized Ricci curvature of (()_, ) (at the identity). We also denote by Ric_μ∈() the (1,1)-Ricci tensor of (, μ,g̅), and by _μ∈() the moment map for the action of 𝖦𝖫() on Λ^2 ^* ⊗:

    g̅(_μ, A) = 1/4g̅(θ(A) μ, μ),      A∈gl(g),     μ∈Λ^2 ^* ⊗.

The factor 14 makes this consistent with the standard notation in the homogeneous Riemannian setting, see for instance <cit.>. In this notation, for ∈ D with _ = μ, _Λ^3 = H, we have

    Rc_ = 
    	[                   _μ,H 1/2g̅^-1(_μ^*H)  g̅^-1;             -1/2^*_μ H             - (_μ,H)^* ],     _μ,H :=  Ric_μ - 1/4 g̅^-1 H^2.





The generalized Ricci curvature and the moment map for the action of  on Dorfman brackets are related by 

    Rc_ =  _ +   Ric_μ - _μ + A_ ,

for some A_∈so(,) ∩so(,).








Let us set

    A_ := 
    		[                      0 1/2g̅^-1(_μ^*H)  g̅^-1;              1/2^*_μ H                      0 ].

It is not hard to see that indeed A_∈so(,) ∩so(,). We are thus left with

    Rc_ - A_ = 
    		[       _μ,H          0;   - ^*_μ H - (_μ,H)^* ]∈.

Furthermore, one can easily see that (<ref>) defines the unique element in so(,) ∩so(,) such that Rc_ - A_∈.

 Moreover, for any L ≃(  B, β)∈, we compute using the definition of g̅_:


    g̅_( Rc_ - A_μ, L)     = 
    		2tr(Ric_μ B)  - 1/2 (H^2 B) - 1/6g̅(^*_μ H,β).

Furthermore, we notice that

    g̅(ρ(B)H,H)    = ∑_i,j,k (ρ(B)H)(e_i,e_j,e_k)H(e_i,e_j,e_k) = -3 ∑_i,j,k H(Be_i,e_j,e_k)H(e_i,e_j,e_k)
       = -3 ∑_i g̅(ι_e_iH,ι_Be_iH) = -3 g̅(H^2e_i,Be_i) = -3(H^2B),

from which we get

    g̅_(  Rc_ - A_  , L)  = 
    		g̅_( Ric_μ, L ) +  1/6 g̅(ρ(B)H - _μβ,H).

On the other hand, by <Ref> and (<ref>) the moment map satisfies:

    g̅_( _, L ) = 1/6 g̅(Θ(L) , )    = 1/2 g̅(θ(B) μ,μ) + 1/6 g̅( ρ(B) H - _μβ, H) 
       = 2  g̅(_μ, B) + 1/6 g̅( ρ(B) H - _μβ, H) 
        = g̅_( _μ, L ) + 1/6 g̅( ρ(B) H - _μβ, H),

and the Proposition follows.




If the Lie algebra (,μ) is nilpotent, then 

    Rc_  - A_ =  _,      A_∈so(,) ∩so(,).



Lauret's formula for the Ricci curvature of left-invariant metrics in terms of the moment map on the variety of Lie algebras <cit.> yields

    Ric_μ = _μ - 1/2  B_μ - 1/2 (_μ U + (_μ U)^t ),
 
where g̅(B_μ·, ·) is the Killing form of (,μ), and U∈ denotes the mean curvature vector of the metric Lie algebra (,μ,g̅), which vanishes if and only if (,μ) is unimodular. In particular, for nilpotent Lie algebras –for which it is known that the Killing form vanishes– this gives Ric_μ = _μ, and the stated formula follows from <Ref>.








§ A  FLOW OF DORFMAN BRACKETS
 
	

The fact that the Lie group  (see (<ref>)) acts transitively on the space of generalized left-invariant metrics ^ on ()_H (<Ref>) and the equivalence between acting on generalized metrics and acting on Dorfman brackets (<Ref>) lead us to consider what would be a natural counterpart to the generalized Ricci flow of left-invariant metrics, on the space of Dorfman brackets D ⊂Λ^2()^* ⊗ ().


As in previous Sections, we fix a left-invariant ECA ()_H̅ and a background left-invariant generalized metric = (g̅, 0) on it.


The generalized bracket flow  with background ( ()_H, ) is the initial value problem for a curve of brackets (t) in Λ^2 ()^* ⊗ ():

    / t = - Θ( Rc_ - A_) ,     (0) = _0,

where Rc_ and A_ are respectively defined in (<ref>) and (<ref>). 
 

Since a Dorfman bracket ∈ D  (see <Ref>) is determined by its two projections _, _Λ^3, it is natural to expect the generalized bracket flow to be equivalent to a coupled flow of Lie brackets on  and of closed 3-forms. Indeed:


The generalized bracket flow (<ref>) with initial condition a Dorfman bracket _0 ∈ D is equivalent to the following coupled flow of brackets μ(t) and 3-forms H(t):


    / tμ = - θ(_μ,H) μ ,       μ(0) = μ_0 := (_0)_,  
    / t H =  Δ_μ H - ρ(_μ,H) H,        H(0) = H_0 := (_0)_Λ^3.






 Let (μ_t, H_t)_t∈ [0,T) be a solution of (<ref>).  Then, we  define 
 
    _t(X+ξ, Y+η)=μ_t(X, Y)-η∘ (μ_t)_X+ ξ∘ (μ_t)_Y + ι_Yι_X H_t ,    X, Y∈ ,  ξ, η∈^*,   t ∈[0,T).
 
 Using <Ref>, we have that _t∈ D , for all t∈ [0,T). Now, using (<ref>) and <Ref>, we have that 
 
    / t_t=-Θ( Rc__t- A__t)_t ,   (0)=_0 .

 Then,  since they solve the same ODE with the same initial datum, (_t)_t∈[0,T) coincides with the solution of the generalized bracket flow starting from _0.

Viceversa, let (_t)_t∈ [0,T') be  a solution of the generalized bracket flow starting from _0.  For the sake of simplicity, we will denote μ̃_t:=(_t)_ and H̃_t:=(_t)_Λ^3.  By definition, _t belongs to the 𝖫-orbit of _0. Using the fact that  the 𝖫-action does not change π and <Ref>, we obtain  that

    / tμ̃=π/ t=-πΘ( Rc_- A_)=-θ( Ric_μ̃, H̃^B)μ̃ .

Moreover, using again the fact that the 𝖫-action acts trivially on , we have, for all X, Y, Z∈, 

    / t H̃(X, Y, Z)=    2/ t( X,  Y),  Z=-2Θ ( Rc_- A_)( X,  Y),  Z
    
    =     -(Θ ( Rc_- A_))_Λ^3(X, Y, Z)=(Δ_μ̃H̃-ρ( Ric_μ̃, H̃^B)H̃)(X, Y, Z) .

where the last equality is due again to <Ref>. <Ref> and <Ref> readily guarantees that (μ̃_̃t̃, H̃_t)_t∈[0,T') coincides with the solution of (<ref>), concluding the proof. 



 Before going deeply into the discussion on how the generalized bracket flow is connected to the generalized Ricci flow, we state the following Lemma which will be useful in  what follows.
 
  Let (()_(t), G̅)_t∈ [0, T) be a solution of the generalized bracket flow starting from . Then, 
  
    / t|_μ^* H|^2= 2(g̅(ρ( Ric_μ, H^B)_μ^* H,_μ^* H) -2g̅(ρ( Ric^B_μ, H) H, _μ^*_μ H )-|Δ_μ H|^2) .
 In particular, if _μ(0)^*H(0)=0, then _μ^*H=0, for any t∈[0,T). 
 

First of all, we  easily see that, for all  A ∈gl( g) and brackets μ, one has
   
    _θ(A)μ^* = [_μ^*,ρ(A^t)].

Then, we can use (<ref>) to infer that 
   
    / t^*_μ H
       =     [ρ( Ric_μ, H^B), ^*_μ] H-^*_μρ( Ric_μ, H^B)H+Δ_μ^*_μ H
    
       =  ρ( Ric_μ, H^B)^*_μ H - 2^*_μρ( Ric_μ, H^B) H+ Δ_μ^*_μ H .
 
     Then, we easily obtain that 
   
    / t|_μ^* H|^2  =     2(g̅(ρ( Ric_μ, H^B)_μ^* H,_μ^* H) -2g̅(ρ( Ric^B_μ, H) H, _μ^*_μ H )-|Δ_μ H|^2) ,
 as claimed. The second part follows trivially from the first one. 
  

 

A similar approach was firstly introduced by Paradiso in <cit.>. As a main difference from our setting, the author considered the action of 𝖦𝖫()⊂ on D. This  choice yields  a coupled flow as follows:

    / tμ = - θ(_μ,H) μ ,       μ(0) = μ_0 := (_0)_,  
    / t H =  - ρ(_μ,H) H,        H(0) = H_0 := (_0)_Λ^3

 which, in view of <Ref>,  is equivalent to (<ref>) only in the particular case in which H_0 is harmonic. 


The following is the main result of this Section. It shows that from a left-invariant  solution of the generalized Ricci flow one can construct a generalized bracket flow solution whose generalized geometry is gauge-equivalent to the original solution, and that the converse is also true. This in particular implies that the maximal existence times for both flows coincide, and that any generalized geometry question can be studied by means of the generalized bracket flow.


Let ( ()_H_0, _t)_t∈ [0,T) be a left-invariant solution to the generalized Ricci flow equation (<ref>) and let ((t))_t∈ [0,T') be the  solution to the generalized bracket flow (<ref>) with background  ( ()_H_0, _0) and initial condition (0) defined by (0)_ = _, (0)_Λ^3 = H_0, with both solutions defined on a maximal interval of time. Then, T = T' and there exists a one-parameter family of generalized isometries

    F_t : ( ()_H_0, _t) ⟶( ()_(t), _0 ),      t∈ [0,T).






First, given any metric ECA (( g ⊕ g^*)_, 𝒢) , we define the endomorphism

    A_,𝒢 := Skew_𝒢(^*_g_μ(H + _μ b)) ∈o( g ⊕ g^*,𝒢··),

where μ = _ g, H := _Λ^3, and g ∈Sym^2_+( g), b ∈Λ^2 g^* satisfy

    𝒢 = e^b [    0 g^-1;    g    0 ]e^-b.

One readily checks that the map ( , 𝒢) ↦ A_ , 𝒢
is 𝖫-equivariant: A_F·, F ·𝒢 = F · A_ , 𝒢. Then, since the generalized Ricci curvature is similarly 𝖫-equivariant, it follows from <Ref> that

    Rc(,𝒢) - A_,𝒢∈ l,

where Rc(,𝒢) is the generalized Ricci curvature of (( g⊕ g^*)_,𝒢).


Now, given a solution to the generalized bracket flow ((t))_[0,T') (<ref>), we define  the family of left-invariant endomorphisms  ( G_(t))_t∈[0, T')  as follows:

    G_(t)(x)= L_(t)(x)· G_0(e) ,     x∈𝖦,
 where L_(t)(x) is the lift  on  of the left-translation in 𝖦_μ(t) by x  while  G_0(e) and  G_(t)(x) are, respectively, the endomorphisms on the fibers in e and x.  We easily see that 
G_(t) are left-invariant generalized metric on . 
Let us then consider

    / tF_t=F_t( Rc_(t)- A_(t)) ,    F_0= Id ,
 where A_(t)∈so(,_0) ∩so(,) is defined as in (<ref>).   By a standard result in ODE theory, the family F_t is defined on [0,T') and (F_t)_t∈[0,T')⊆. We then define λ(t):=F_t^-1·(0) yielding that 

    / tλ=-Θ(F_t^-1F_t')λ=-Θ( Rc_(t)- A_(t))λ .

 Thus, λ(t) and (t) satisfy the same ODE with equal initial datum  concluding that (t)=F_t^-1·(0). From this,  we can deduce that 

    F_t(()_(t),  G_(t))→ (()_H_0, F_t· G_(t)) ,
  is an isometry. So,  denoting 
G̃_t:=F_t· G_ (t) we  obtain that,  in e, 

    G̃_t^-1/ tG̃_t
    
    =      F_t G^-1_ (t)( Rc_(t)- A_(t))  G_ (t)F_t^-1- F_t( Rc_(t)- A_(t))F_t^-1
    = -2 Rc(G̃_t) ,
 where the last equality follows from  Rc( G) G =- G Rc( G) and from [A_(t),  G_ (t)]=0.   This allows us to conclude  that G_t=G̃_̃t̃,  since they solve the same ODE with the same  initial condition.

Conversely, we define

    / tF_t=( Rc( G_t)- A(𝒢_t))F_t ,    F(0)= Id,
 where A(𝒢_t) := A_,𝒢_t. As before,  we consider
G̃_̃t̃:=F_t· G_0  and easily note that

    G̃_̃t̃^-1/ tG̃_̃t̃=G̃_̃t̃^-1( Rc( G_t)- A(𝒢_t))G̃_̃t̃- ( Rc( G_t)- A(𝒢_t)) .
On the other hand, 

    G_t^-1/ t G_t=-2 Rc( G_t)
    
    = G_t^-1( Rc( G_t)-A(𝒢_t))  G_t- ( Rc( G_t)-A(𝒢_t)),
  since, again,   Rc( G_t) G_t =- G_t Rc( G_t) and   [A( G_t),  G_t]=0.  So, in particular, we have that 

    / tG̃_̃t̃=[ Rc( G_t)- A(𝒢_t), G̃_̃t̃] ,   / t G_t=[ Rc( G_t)- A(𝒢_t),  G_t] ,
 Then, G̃_̃t̃ and G_t satisfy the same ODE with the same initial condition yielding that G_t=F_t· G_0. 
Therefore,  if we let  λ(t)=F_t^-1·(0), 
 
    F_t(()_λ(t),  G_λ(t))→ (()_H_0,  G_t)

 is an isometry. In particular,  this implies that 

    F_t Rc _λ(t)F_t^-1= Rc( G_t) ,     F_tA_λ(t)F_t^-1=A( G_t),
 using the  Aut(E)-equivariance of the generalized Ricci curvature and the uniqueness of A( G)∈so(,) ∩so(,) such that Rc( G)- A( G)∈𝔩.  Then we have that 

    / tF_t=F_t( Rc_λ(t)- A_λ(t)) ,    F_0= Id
 and repeating the computations in (<ref>), we infer that (t)=λ(t), concluding the proof. 





We do not necessarily need to use _0 as background metric, but this simplifies the formulas.





§ GLOBAL EXISTENCE ON SOLVMANIFOLDS
 


As a first application of <Ref>, we prove in this Section the long-time existence of invariant solutions on any solvmanifold. Recall from <cit.> that the generalized scalar curvature of a metric Courant algebroid (, G(g,0)) is given by

    S_g, H := _g - 112|H|^2,

where _g denotes the Riemannian scalar curvature of the metric g. Let us remark that this is not the trace of the generalized Ricci curvature (which is in fact traceless), but it is obtained instead by studying Lichnerowicz formulae  for certain Dirac operators <cit.>.
 Within the moving Dorfman bracket framework, a  direct consequence of  <cit.> allows us to infer  that on a nilpotent Lie group

    S_=-1/12(3|μ|^2+| H |^2)=-1/12||^2 .

Before stating the main Theorem of this Section, we will need a preliminary Lemma. 

Let ((t))_t∈  (ε_-,ε_+) be a solution of the generalized bracket flow defined on its maximal time interval. Then, there exists a uniform constant C>0 such that:


  * if ε_+<∞, then

    |(t)|≥C/(ε_+-t)^1/2 ,    t∈[0,ε_+) ;


  * if ε_-> -∞, then

    |(t)|≥C/(t-ε_-)^1/2 ,    t∈(ε_-, 0] .



 For the ease of notation, we will always denote with C a uniform  and positive constant which may change from line to line.  
 First of all, we observe that,  
 as in <cit.>, we have | Ric_μ|^2≤ C |μ|^4. Moreover, one can easily show that | H^2|^2≤ C | H|^4. Finally,  we  obtain that |_μ^*H|^2≤ C|μ|^2| H|^2≤ C||^4, using that *_μ is an isometry and that |_μα|^2≤ C |μ|^2|α|^2, for any form α and any Lie bracket μ. Then, one can deduce 
 
    | Rc_- A_|^2=2| Ric^B_μ, H|^2+1/6|_μ^*H|^2≤ C||^4 .

 Now,  we fix t_0∈[0,ε_+) and we can use (<ref>) and the linearity of the infinitesimal 𝖫-action Θ to infer that 
 
    / t||^2=2g̅(, / t)≤ C||^4 ,    t∈ [t_0, ε_+) .
 This implies, by comparison, that 
 
    ||^2≤1/-C(t-t_0)+ |(t_0)|^-2 ,     t∈ [t_0, ε_+) .

  On the other hand, (<ref>) readily guarantees   that ε_+≥ t_0+1/C|(t_0)|^-2, giving us the  first claim.  The second claim can be obtained  by reversing the time variable.

We can now state  the following blow-up result:

 Let (E, _t)_t ∈ (ε_-,ε_+) be a left-invariant solution of the generalized Ricci flow over a simply-connected Lie group 𝖦 defined in its maximal time interval. Then,  


  * if ε_+<∞, then S_t→∞, as t→ε_+ ;

  * if ε_->-∞, then S_t→ -∞, as t→ε_- .




Fixing the  preferred isotropic splitting σ_0 associate with _0, we have the time-varying isometry (E, G_t)_t ∈ (ε_-,ε_+)≃_σ (( g ⊕ g^*)_H_t, G(g_t,0))_t ∈ (ε_-,ε_+). By <cit.>, if ϕ× (ε_-,ε_+) → satisfies

    (∂/∂ t - Δ_g_t)ϕ_t =  16 |H_t|^2_g_t;     t∈ (ε_-,ε_+),

then, 

    (∂/∂ t - Δ_g_t)( S_g_t, H_t + 2Δ_g_tϕ_t - |∇_g_tϕ_t|^2) = 2 |_g_t, H_t + ∇^2_g_tϕ_t -  12 (^*_g_tH_t + ι_∇_g_tϕ_tH_t)|^2;     t ∈  (ε_-,ε_+).

In this case, we set ϕ_t := ∫_0^t  16 |H_s|^2 s, which is constant in space and hence satisfies <Ref>. Thus, by left-invariance, we have

    / t S_g_t, H_t = 2 |_g_t,H_t -  12 ^*_g_tH_t|^2 = | Rc( G_t)|^2.

The remainder of the proof follows <cit.> for the classical Ricci flow. By <Ref>, (( g ⊕ g^*)_H_0, G_t)_t ∈ (ε_-,ε_+) is isometric to ( ()_(t), (g_0,0) )_t∈ (ε_-,ε_+), where  satisfies <Ref>. Let S_(t) =  S_g_t denote the generalized scalar curvature of this latter metric ECA and note that by <Ref> it satisfies t S_(t) = | Rc_(t)|^2. Also observe that

    t||^2 = - 2 Θ( Rc_ - A_)≤ C | Rc_ - A_|||^2,

for some constant C > 0. Hence, observing that | Rc_ - A_|^2 ≤ 2 | Rc_|^2, we have,  for t ∈ (ε_-,ε_+),

    log |(t)|^2 - log |(0)|^2    = ∫_0^t  slog |(s)|^2  s ≤ C ∫_0^t | Rc_(s) - A_(s)| s ≤ C ∫_0^t | Rc_(s)|  s
       ≤ C ∫_0^t 1 + | Rc_(s) - A_(s)|^2  s = C(t +  S_(t) -  S_(0)).

On the other hand, by <Ref>, we have that |(t)|^2 →∞,  as t →ε_+,  forcing S_(t)→∞,  as t →ε_+. A similar argument shows the statement for t →ε_-.


Recall that left-invariant metrics on solvable Lie groups have non-positive scalar curvature, and zero scalar curvature if and only if they are flat <cit.>. Hence, we yield the immediate Corollary:

 
Any left-invariant  solution  of the generalized Ricci flow on a solvable Lie group exists for all positive times.


In the case of a left-invariant solution on a nilpotent group, we are able to say much more thanks to <Ref>. In particular, we can describe the precise asymptotic behaviour of the generalized scalar curvature:
 For any left-invariant  solution (g_t,H_t) of the generalized Ricci flow on a nilpotent Lie group , the generalized scalar curvature satisfies 𝒮_g_t,H_t∼ -1/t, as t →∞.

 Let (g_t,H_t) be a solution of the generalized Ricci flow  on  and (t) the corresponding generalized bracket flow. Then by <Ref>, <Ref>  and <Ref>, we obtain

    / t|(t)|^2 = -2Θ(_(t))(t)(t) = -6g_ l(_(t),_(t)) = -6|M_(t)|^2.

We now claim that |_(t)|^2 ≥ c||^4, for some small constant c > 0. To see this, first note that the map

    𝕊(Λ^2( g ⊕ g^*) ⊗ ( g ⊕ g^*)) ∋↦ |_|^2,

is never zero on the unit sphere. Indeed _ = 0 implies that 0 = (_) = -1/6||^2 = -1/6, which is a contradiction. The claim therefore follows from compactness of the unit sphere. Thus,

    -C|(t)|^4 ≤/ t|(t)|^2 ≤ -c |(t)|^4,

so |(t)|^2 ∼1/t by ODE comparison. The result now follows from <Ref>.








§ GENERALIZED NILSOLITONS


	
	In this Section,  following the approach by Lauret in <cit.> and <cit.>, we give the definition of algebraic generalized solitons, proving their main  properties  in the nilpotent case. 	
	 First of all, we  introduce the normalized generalized bracket flow. 
	 
	  The ℓ-normalized generalized bracket flow with background data (()_H,  G ) is the  initial value problem for a curve of brackets (t) and ℓ=ℓ(t):
	
    / t=-Θ( Rc_ - A_) + ℓ ,     (0)=_0  .

	 
	 The multiplication of  in (<ref>) by the factor ℓ has to be interpreted using <Ref> and <Ref>.  On the other hand, given a function ℓ,  one can  obtain a solution of the ℓ-normalized generalized bracket flow from a solution of the generalized bracket flow (<ref>)  via a time reparametrization and a scaling. 
	 	   
	    Let  (()_H, ) be a left-invariant metric ECA over a Lie group 𝖦. Let  ^ℓ(t) and (t), respectively, be  the solution of the ℓ-normalized generalized bracket flow and the solution of the generalized bracket flow  starting from _0.  Then, there exist  a family of scaling c(t)>0 such that c(0)=1 and a time reparametrization τ=τ(t) so that 
	    
    ^ℓ(t)=c(t)(τ) .

	   
	   
	 Following  <cit.>, we choose 
	
    c'=ℓ c ,    c(0)=1    τ'=c^2 ,   τ(0)=0.
  Then,  we have  
	
    / t(c(t)(τ))=-c^3Θ( Rc_(τ) - A_(τ)) (τ)+ℓ c(τ).
  
	Using that Rc_c- A_c=c^2( Rc_ - A_), we obtain the claim. 
	
	
	







	   
	 The case in which we will be interested the most is when  we choose c=1/||. Applying  <Ref>, we have that :=/|| is a solution of the ℓ-normalized generalized bracket flow with 
	 
    ℓ_=g̅(Θ( Rc_ - A_) , )/||^4
    	 =6g̅( Rc_ - A_,_ ) .
  With this choice of the normalization, we will refer to  the corresponding normalized generalized bracket flow as scalar-normalized generalized bracket flow.  In this case,  the norm of the solution of the scalar-normalized generalized bracket flow will remain constantly 1, provided the initial datum has unit norm. Indeed, 
	 
    / t||^2= 2g̅(Θ( Rc_ - A_) , )(-1+||^2) ,
 then ||^2=1, since |_0|^2=1.  Recalling (<ref>), this  also implies that S_= S__0=-1/12 along the scalar-normalized generalized bracket flow.
	 
	 With this in mind, we can give the definition of generalized algebraic soliton.
	 
	 Let ∈ D. We say that  is an algebraic soliton for the generalized Ricci flow if it is  a fixed point of the scalar-normalized generalized bracket flow,  i.e.  the following is satisfied
	 
    Θ( Rc_ - A_) = ℓ_ .

	 We say that  is expanding, steady or shrinking if, respectively, ℓ_>0, ℓ_=0 or ℓ_<0. 
	 
	  Building from this definition,  we can derive equivalent conditions for a Dorfman bracket to be an algebraic soliton for the generalized Ricci flow. 
	 
	 Let ∈ D. Then, the following are equivalent:
	 
	 
  *  is an algebraic soliton;
	 
  *  the generalized bracket flow starting at    evolves only by scaling;
	 
  *  Rc_ - A_=λ Id+ D with D∈ Der_λ() and λ∈ℝ;
	 
  * there exist  D∈ Der(μ) and λ∈ such that 
	 
    Ric_μ, H^B=λ Id+ D ,  
    
    	 Δ_μH=λ H+ρ( Ric_μ, H^B)H .

	 
	  






	  
	  The equivalence between <Ref> and <Ref> is trivial using (<ref>) and <Ref>.  On the other hand, if  is an algebraic soliton,   the fact that Θ( Id)=-, for all ∈𝒟,  implies Rc_ - A_=λ Id+ D with  D∈ Der(). Moreover,  Rc_ - A_∈𝔰𝔬(, ), then,  D has to satisfy
	  
    2λ+  D·, ·+ ·,   D·=0 ,
 which gives <Ref>.
	  
	  
	  
	 
	  Viceversa, it is easy to show that if <Ref> holds, then (<ref>) is satisfied with ℓ_=-λ. 
	     Furthermore, <Ref> guarantees the equivalence between  (<ref>) and 
	  
    θ( Ric^B_μ, H+ℓ_ Id)μ=0 , 
    ρ( Ric_μ, H^B)H-Δ_μH=ℓ_ H
 which is equivalent to (<ref>) with λ=-ℓ_, concluding the proof. 
	  
	  	  
	 <Ref>  and <Ref> in <Ref> can be considered as the static definition,  in terms of  both generalized  and classical  objects, of  algebraic solitons   and so <Ref> has to be regarded as the analogue of <Ref> in the varying  Dorfman bracket framework.


Moreover, <Ref> allows us to construct  from algebraic solitons for the generalized Ricci flow  generalized metrics which are  generalized Ricci  solitons as in <Ref>.

Let  be an algebraic soliton for the generalized Ricci flow. Then, G_, defined as in (<ref>),  is a generalized Ricci soliton. 


	 
	  The first equation in (<ref>) can be equivalently written, in terms of symmetric (0,2)-tensors, as  
	  
    Ric_g_μ, H^B=λ g_μ+1/2(g_μ(D·, ·)+ g_μ(·, D·)=λ g_μ-1/2 L_X_Dg_μ ,
 where X_D=/ t|_t=0φ_t  with φ_t ∈ Aut(𝖦_μ) such that _eφ_t=e^tD. Moreover,  plugging   the first equation of (<ref>) into the second one, we have 
	  
    Δ_μ H=λ H+ρ(λ Id+ D)H=-2λ H +ρ(D)H =-2λ H + L_X_DH .
 The claim follows  using <Ref>.
	  
	   Before  going into the discussion of algebraic solitons in the nilpotent case, we will need  the following preliminary Lemma about λ-derivations, which is nothing but an adaptation of <Ref> to the invariant case. 
	 
	Let  ()_ be a   left-invariant  ECA  over a Lie group 𝖦. Then,  we have that, for any c∈ℝ\{0} and λ∈, 
	 
    Aut(c·()_, ()_)={A̅_ce^b  :  b∈Λ^2^* ,   A∈ Aut(μ) ,   A^-1· H=c( H- b )}

and 	 
    Der_λ()={[          D          0;          α -2λ Id-D^* ] : D∈ Der(μ),   ρ(D)H-_μα -2 λ H=0} .

	 
	  The first assertion follows directly from the discussion in the non-invariant case.  <Ref> guarantees that  a λ-derivation  has to be of the form 
	 
    D= [          D          0;          α -2λ Id-D^* ] ,
 where D∈ Der(μ) and ρ(D)H-α -2λ H =0 . 








	 

	   
	The moment map formulation in the nilpotent case   allows us to derive many other properties concerning algebraic solitons, which, in this particular case, will be called generalized nilsolitons, mimicking the classical nomenclature, see <cit.>.

Let ∈ D be a generalized nilsoliton.  Then, it is expanding.  Moreover, the derivation D satisfies the following property:

    tr((D+1/4H^2)D')=-λ tr(D') ,
  for all D'∈ Der( μ)  defining D'=(D', α)∈ Der_λ'(), for some λ'∈. Finally,  g̅(ι_XH, _μ^*H) = 0,  for all X ∈ g.



   We have that
   
    ℓ_=6g̅( Rc_ - A_,_ )=6| Rc_ - A_|^2≥ 0 ,

   where the last equality follows from <Ref>. On the other hand, 
   | Rc_ - A_|^2= 0 would, in  particular,  imply that
   
    Ric_μ=1/4 H^2≥ 0  ,
  which cannot happen in the nilpotent case, see <cit.>, giving us the first  claim. 
   
   As regards the second,  fixed  D'=(D', α)∈ Der_λ'(), we have that 
   
    g̅_𝔩(λ Id+D, λ' Id+D')=1/6g̅(Θ(λ' Id), )
       =-λ'/6||^2=λ'(2 R_μ-1/6| H|^2) .
  On the other hand,  we can use <Ref> to deduce  that D=(D, -_μ^*H) and then obtain 
   
    g̅_𝔩(λ Id+D, λ' Id+D')=2 tr((λ Id+D)(λ' Id+D'))- 1/6g̅(_μ^*H, α) .
 The fact that D'∈ Der_λ'() implies, thanks to <Ref>, that 



    
     _μα=ρ(D')H-2λ'H. This can be used to infer that
       
    g̅_𝔩(λ Id+D, λ' Id+ D')=      2λ' tr( Ric_μ, H^B)+ 2λ tr(D')+ 2 tr(DD')-1/6g̅(α, _μ^*H)
    
       =     2λ'( R_μ-1/4| H|^2)+ 2λ tr(D')+ 2 tr(DD')+1/2 tr(H^2D')+ 1/3λ'| H|^2
    
       =    λ'(2 R_μ-1/6| H|^2)+ 2λ tr(D')+ 2 tr(DD')+1/2 tr(H^2D') .
 Now, it is sufficient to use (<ref>) in (<ref>) to obtain the claim.
      Finally, recall that for metric nilpotent Lie algebras, any symmetric derivation D satisfies (_XD) = 0, for all X ∈ g. Then, the soliton equation in μ  implies
   
    0 = tr(_XH^2) = -1/3g̅ (ρ(_X)H, H) = -1/3g̅(ℒ_XH, H) = -1/3g̅ (_μι_XH, H) = -1/3g̅ (ι_XH, _μ^*H),

   as claimed.

   
   	

Motivated by the wide range of results concerning  classical nilsolitons proved throughout the years, see for instance <cit.>, many questions about generalized nilsolitons can be raised.  First of all, as usual,  let (E, ) be a left-invariant  metric ECA over a nilpotent Lie group 𝖦 and assume that  is  a generalized nilsoliton.  Using the preferred isotropic splitting induced by , we can identify (E,)≃(()_H,  G(g̅, 0 )), for some  closed H∈Λ^3^*.  So, in this setting, one can wonder if  is unique, up to scaling  and up to the action of Aut([·, ·]), among left-invariant generalized metrics defining a preferred representative H'∈ [H]∈ H^3(, ).

Considering [H]∈ H^3(𝖦,  ), the uniqueness is not true in general.  Indeed,  on 𝔫_3= Lie( Heis(3, )), the following 

    μ(e_1, e_2)= e_3 ,    H=e^123 ,   λ=-2 ,   D= diag(1,1,2)
 defines a generalized nilsoliton, see <Ref> for the proof,  with [H]=0∈ H^3( Heis(3, ), ). One can easily observe that the above generalized nilsoliton defines exactly the expanding soliton for the generalized Ricci flow described in <Ref>.  On the other hand, μ(e_1, e_2)= e_3 defines  also the classical nilsoliton, see <cit.>, i.e. H=0.


In order to address the uniqueness question, we need to understand better the geometric properties of the space of left-invariant generalized metrics.

   As we saw in <Ref>, 
    fixed E a left-invariant ECA over a Lie group 𝖦, not necessarily nilpotent,  the space of left-invariant generalized metric M ^𝖦 can be presented as a homogeneous space where 𝖫 is acting transitively by conjugation. 
   
   Of course, <Ref> gives also   that the action of 𝖲𝖮(⟨·,·⟩) is transitive on ℳ^. This allows us  to  present  M^𝖦 as a homogeneous space in a different manner. 
   
 Clearly, the isotropy  of 𝖲𝖮(⟨·,·⟩) in ∈ M^𝖦 is 
    
    _𝖲𝖮(⟨·,·⟩)={F∈𝖲𝖮(⟨·,·⟩) :  F F^-1=}=: Isom(, 𝖲𝖮) .

     On the other hand, we observe that
     
    θ𝖲𝖮(⟨·,·⟩)→𝖲𝖮(⟨·,·⟩) ,   σ(F)= F ,
 is an automorphism of 𝖲𝖮(⟨·,·⟩) which is  clearly involutive, 
and the set of fixed points of θ  precisely coincides with  
     Isom(, 𝖲𝖮). 
     Thus, we obtain that (𝖲𝖮(⟨·,·⟩),  Isom(,  𝖲𝖮)) is a Riemannian  symmetric pair, as defined in <cit.>. As a consequence,  we obtain a reductive  decomposition of 𝔰𝔬(⟨·, ·⟩)=𝔥⊕𝔪  such that [𝔪 , 𝔪 ]⊂𝔥,  where, using (E, )≃ (()_H,  G(g̅, 0)), 
    
    𝔥:= Lie( Isom(, 𝖲𝖮))=so(, )=    {[           A g̅^-1αg̅^-1;           α        -A^* ] : A∈𝔰𝔬(n) ,  α∈Λ^2𝔤^*}

     and
     
    𝔪:=(θ +  Id) ={[            A -g̅^-1αg̅^-1;            α         -A^* ]: A∈ Sym(n) ,  α∈Λ^2𝔤^*} .
As usual,  we will be  identifying 𝔪 with T_ℳ ^𝖦. Now, using again the  identification (E, )≃(()_H,  G(g̅ , 0)),  we can define   a non degenerate bilinear form g̅ on 𝔰𝔬()  such that, for all (A, α), (A', α')∈𝔪 and (B, β), (B', β')∈𝔥, 
         
    g̅((A, α), (A', α'))=2 tr(AA')+ 2g̅(α, α') ,   g̅((B, β), (B', β'))=-3/4( tr(BB')+ g̅(β, β')) .
 It is not hard to see that g̅ is an  Ad( Isom(, 𝖲𝖮))-invariant  bilinear form which,  when restricted to 𝔩,  coincides with the inner product g̅_𝔩 defined in (<ref>).  Moreover, using <cit.>, the 𝖲𝖮(⟨·, ·⟩)-invariant Riemannian metric induced by it  on ℳ^𝖦 will be naturally reductive. This readily  guarantees  that all the geodesics emanating from  are of the form exp(tX)·, for some X∈𝔪, see <cit.>. Finally,  <cit.> guarantees that (ℳ^, g̅) has all non-positive  sectional curvatures.

        

Keeping in mind this and  building from <cit.>, we can define the generalized scalar functional

    𝒮ℳ^→ℝ ,
 which associates to   a given  left-invariant generalized metric its  generalized scalar curvature as in (<ref>). We will study some analytic properties of this functional. 
       Let  E be  a left-invariant ECA over a nilpotent Lie group  𝖦. Then, we have:
      
  * the gradient of S at any ∈ M^𝖦 is given by 
      
    ∇_ S=():=[          Ric^B_g̅, H 1/6g̅^-1_g̅ ^*Hg̅^-1;           -1/6_g̅^*H    -( Ric_g̅, H^B)^* ]
 after using (E, )≃_σ(()_H,  G(g̅, 0 ));
      
  * S is concave along  geodesics, i.e. if γ is a geodesic emanating from ∈ M^𝖦, then ( S∘γ)^”(0 )≤ 0;
      
  * ( S∘γ)^”(0 )= 0 if and only if γ(t)=exp(tX)· where X∈ Der_0([·, ·])∩𝔪. In this case, S∘γ is constant. 
      
            
 We fix ∈ M^𝖦 and,  as usual, we identify (E,)≃_σ(()_H,  G(g̅, 0)).  We will make use of the moving Dorfman brackets approach. First of all, we observe that, if α∈Λ^2𝔤^* and ∈𝒟, we have that 
         
    2⟨Θ[            0 -g̅^-1αg̅^-1;            0            0 ](e_i, e_j), (e_i, e_j)⟩=-_ijk_ij^sα_ks ,
 while 
         
    4⟨Θ[            0 -g̅^-1αg̅^-1;            0            0 ](e_i, e^j), (e_i, e^j)⟩=-2_is^j_iksα_jk
 which gives us that 
         
    g̅(Θ[            0 -g̅^-1αg̅^-1;            0            0 ], )=3_ijk_ij^sα_sk=-g̅(_μα, H) .
 Then, combining  (<ref>) with  (<ref>), we obtain that, for X=(A, α)∈𝔪, 
         
    g̅(Θ(X), )=    g̅(Θ[    A    0;    α -A^* ], )+g̅(Θ[            0 -g̅^-1αg̅^-1;            0            0 ], )
             
             = 12 tr( Ric_μ, H^BA)-2g̅(_μα, H) .
 On the other hand, considering 
    _=[         Ric^B_μ, H 1/6g̅^-1^*_μHg̅^-1;          -1/6^*_μH   -( Ric_μ, H^B)^* ]

          we then obtain  that 
         
    g̅(_, X)=2 tr( Ric_μ, H^BA)-1/3g̅(α,_μ^*H)=1/6g̅(Θ(X), )=1/12/ t|_t=0|exp(tX)·|^2 .

         




   We now compute the gradient of 𝒮 at . If ∈𝒟 is the given Dorfman bracket on E,  we know that, for any t, the generalized metric exp(tX)· is isometric   to 𝒢_exp(-tX)·. So,   given X∈𝔪, we have that 
      
    _𝒮(X) =    / t|_t=0𝒮 (exp(tX)·)=-1/12/ t|_t=0|exp(-tX)·|^2=1/6g̅(Θ(X), )
          = g̅(_, X)=g̅((), X) ,
   giving us the first claim. 
          
          
           Let now  γ(t)=exp(tX)·, for some  X∈𝔪,  be a geodesic emanating from ,   we have          
            
    (𝒮∘γ)^”(0)= -1/12^2/ t^2|_t=0|exp(-tX)·|^2
              
              = -1/6(g̅(Θ(X)Θ(X), )+ |Θ (X)|^2)=-1/3|Θ(X)|^2≤ 0  ,
giving us the second claim. 
          
          Finally,  clearly, if X∈ Der_0()∩𝔪, exp(tX)∈ Aut() which guarantees that 𝒮∘γ is constant for all t, the viceversa holds too thanks to (<ref>).


The element _∈𝔪 can also be viewed as the moment map of the 𝖲𝖮()-action on the space of Dorfman brackets D.  Indeed, it precisely satisfies the relation:


    g̅(_, X) =  1/6 g̅(Θ(X), )  ,      X∈𝔪  .
 Unfortunately, _ coincides with the generalized Ricci curvature only on the subspace of Dorfman bracket with harmonic torsion.

<Ref> can be considered as the analogue, in the generalized setting,  of <cit.> specialized in the nilpotent case.  So, following the steps in  <cit.>,   we can give a partial answer to the uniqueness question, proving that generalized nilsolitons with harmonic torsion are unique.

 Let E be a left-invariant ECA over a nilpotent Lie group 𝖦 and  let  be a generalized nilsoliton with harmonic torsion. Then, up to scaling and up to the action of Aut(E),  is unique.


Let us now consider the following closed Lie subgroup of 𝖲𝖮():
            
    𝖪:=±∏_c∈^+ c^-1/2 Aut(c· E, E) .

 First of all, we  need some clarifications  about 𝖪. The action of an element in 𝖪 on ∈ M^𝖦 is defined as follows: an element c^-1/2F∈𝖪 acts on  as the following composition:
            
    c^-1/2F (E, ,  c^-1/2,)→ (c· E,  G)→ (E, F·)

            where the first arrow is defined, using the isometry induced by the preferred isotropic splitting of , by the transformation
            
    c^-1/2 Id(()_ c^-1/2,  G(g̅, 0 ))→ (()_cH,  G(cg̅, 0))
 which is an isometry of ECA's thanks to <Ref>. Furthermore,  we can use the explicit isometry in  <Ref> to identify           
             
    (()_cH,  G(cg̅, 0))≃(c· E,) .
In conclusion, the first arrow in (<ref>) is the composition of the above isometries and it is then induced by the transformation
            c^-1/2 Id.
             Finally, the last arrow in (<ref>) is just the usual action of the  element F∈  Aut(c· E, E). 
             
              Moreover,   by <Ref>, any  element c^-1/2F can be represented with
                           
    c^-1/2 Id∘A̅_ce^b=[            c^-1/2A                  0; ((c^-1/2A)^-1)^* b   ((c^-1/2A)^-1)^* ]=c^-1/2Ae^b∈𝖲𝖮() ,

              where b∈Λ^2^* , and A∈ Aut(μ)  such that A^-1· H=c(H- b). Finally, -c^1/2F is given by the composition of 
    - Id (()_ -c^-1/2,  G(g̅, 0 ))→ (()_ c^-1/2,  G(g̅, 0 ))
 with the previously defined action.
             Now,  we can prove that  𝖪 is actually a  Lie subgroup of 𝖲𝖮(). Clearly, it is sufficient to prove that, if  c^-1/2F∈ c^-1/2 Aut(c· E, E)  and γ^-1/2G∈γ^-1/2 Aut(γ· E, E), then 
             c^-1/2F(γ ^-1/2G)^-1∈𝖪. 
             
             
             
             
             It is not hard to see that 








             
    c^-1/2 A_ce^b(γ^-1/2B_γe^b')^-1=(c/γ)^-1/2(AB^-1)_c/γe^β ,   β:=γ B·(b-b') .
 Then,  we observe that B· H=1/γH+ B· b'.  This implies that  
             
    BA^-1· H=c(B· H-B· b )=c/γ(H-β) ,
giving us the claim.
             Moreover,   𝖪 is closed in 𝖲𝖮().  Thanks to the structure of the subgroup 𝖪, the claim is equivalent to prove that if 
    c_i^-1/2 Aut(c_i· E, E)∋  c_i^-1/2F_i→ F_∞∈𝖲𝖮() ,    i →∞ ,
 then F_∞∈𝖪. Following <cit.>,   let us fix a generalized metric  ∈ M^𝖦, then 
             
    c_i^-1/2F_i·→G̃∈ M^𝖦.

                         But, as a consequence of (<ref>),  the effect of scaling on the scalar curvature and on | H|^2, recalling that F_i preserves the Dorfman bracket, for all i,  we have that,
             
    S( c_i^-1/2F_i·G̅)=c_i S(F_i·G̅)=c_i S(G̅)→ S(G̃) ,    i→∞ .

              If  S(G̃) 0, then c_i→ c, as i→∞. This,  in particular,  implies that F_i →F̃∈𝖲𝖮() and so F_∞=c^-1/2F̃. We just need to prove that F̃∈ Aut(c · E, E). This is guaranteed by the fact that  
              F_i=  (A_i)_c_ie^b_i∈ Aut(c_i· E, E ) if and only if 
    A_i^-1· H=c_i(H- b_i)    A_i∈ Aut(μ)
 and passing to the limit in the last equality gives us the claim.  Finally,  if  S(G̃)=0, then, automatically, the Dorfman bracket on E is abelian, i.e. H=0 and μ is abelian.  In this case, 
              
    Aut(c· E, E)={A̅_ce^b:  A ∈ 𝖦𝖫() ,     b∈Λ^2^*} .
 Then,  𝖪= Aut(E), which is clearly closed. 
            
            
            Easily, we see that 
          
    T_(   𝖪·)=⊕_λ∈(λ Id+( Der_λ()∩𝔪)) .
  Moreover, we  note that, using again <Ref> 
                    
    λ Id+( Der_λ()∩𝔪)={[    λ Id+ D          0;          0 -λ Id- D^* ]: D∈ Der(μ)∩ Sym(n),  ρ(D)H-2λ H=0} .





Now, suppose that  and ' are two  generalized nilsolitons with harmonic torsion which do not belong to the same 𝖪-orbit.
           Since 𝒪=  𝖪· and 𝒪'=  𝖪·'  consist in generalized nilsolitons,  we can assume that d(, ')=d(𝒪, 𝒪') and that the unique geodesic γ[0,1]→ℳ^ such that γ(0)= and γ(1)=' meets 𝒪 and 𝒪' orthogonally.
            Thanks to <Ref>, we know   ()=ℛc()= λ Id+  D , D∈ Der_λ()∩𝔪 and using <Ref> (1),  we have that 0=g(γ'(0), ∇_𝒮)=g(γ'(0), ℛc()) and 0=g(γ'(1), ∇_'𝒮)=g(γ'(1), ℛc(')) which gives us that 
           
    (𝒮∘γ)'(0)=(𝒮∘γ)'(1)=0 .
 On the other hand, using <Ref> (2),  𝒮 is concave along the geodesics, so 
       𝒮∘γ is constant, which, thanks to <Ref> (3),  happens if and only if γ=exp(tX)· where X∈ Der_0()∩𝔪, giving us  a contradiction.

A first consequence of <Ref> is the uniqueness among generalized nilsolitons within 0∈ H^3(, )  of the classical nilsoliton, if it exists. 

Unfortunately, the proof of <Ref> is exclusive of the  harmonic torsion case. In view of this and of <Ref> itself, we can  weaken the uniqueness question as follows.

Let (E, ) a left-invariant metric  ECA over a nilpotent Lie group  and assume  is  a generalized nilsoliton. Using (E, G̅)≃ (( )_H,  G(g̅,0)), is it H g̅-harmonic?

 With last objective of addressing <Ref>, we can find a equivalent condition for it to be true. 
  Let ∈ D be an algebraic soliton. Then, _μ^*H=0 if and only if    ((_μ^* H)^2 Ric^B_μ, H) ≤  0.
   
   
  




 
     Using the fact that (<ref>) completely determines ∈ D and (<ref>), we have that  can be viewed as a pair (μ,H) solving
   
    θ(Ric^B_μ, H)μ = -λμ,
    ρ(Ric^B_μ, H)H + _μ_μ^*H = -λ H,

   for some λ∈. 
   
   Applying ^* to the second equation yields,  using (<ref>),
   
    0 = λ_μ^*H + _μ^*ρ(Ric^B_μ, H)H + _μ^*_μ_μ^*H
       		
       		= ρ(Ric^B_μ, H)_μ^*H + _μ^*_μ_μ^*H,

   where in the last equality we used θ(Ric^B_μ, H)μ = -λμ. Now, just as in the proof of <Ref>, we have that g̅(ρ(A)α, α) = -2 (α^2 A),  for all 2-forms α∈Λ^2^* and A∈gl( g). Now taking an inner product of (<ref>) with _μ^*H gives
   
    0 = g̅(ρ(Ric^B_μ, H)_μ^*H, _μ^*H) + |_μ_μ^* H|^2 = - 2((_μ^* H)^2 Ric^B_μ, H) + |_μ_μ^* H|^2.

	Since  -((_μ^* H)^2 Ric^B_μ, H) ≥ 0, then, we yield _μ^*H = 0 as required.
   
   Besides <Ref>, one can also  pose a finiteness question on generalized nilsolitons, namely:
   
   Let  be a nilpotent Lie group. Is the set of generalized nilsolitons on  finite?
   
   In <Ref> we will provide, within the full classification of generalized nilsolitons up to dimension 4, a counterexample to <Ref>. 
   
 Finally, it is well known, see for instance <cit.>,  that there exist nilpotent Lie algebras which do not admit any classical nilsoliton.  So,  one may hope to produce generalized nilsolitons even on such Lie algebras. This cannot be done in general. Indeed, on characteristically nilpotent Lie algebras, i.e. such that the Lie algebra of derivations is nilpotent, we cannot find neither classical nor generalized nilsolitons,  due to  the fact that any derivation is nilpotent, see <cit.>.



§ LONG-TIME BEHAVIOR ON NILMANIFOLDS
 
In this Section, we will recall a definition of Cheeger-Gromov convergence for exact Courant algebroids introduced in <cit.>. Then, we will discuss the long-time behavior of the Generalized Ricci flow on nilpotent Lie groups in the particular case in which the starting ECA has harmonic torsion, showing that, in this case, the asympotics are precisely the generalized nilsolitons defined in the previous Section. 

Given a subset V ⊂ M of a manifold,  we will denote by i_V  V → M the inclusion map.

A sequence of pointed metric ECAs (E_i → M_i, G_i,p_i)_i≥ 1 converges to ( E → M ,  G, p) in the generalized Cheeger–Gromov topology if there exists a  sequence  {U_i}_i≥ 1  of neighbourhoods of p exhausting M and Courant algebroid isomorphisms 

    (F_i,f_i)  (i_U_i^* E → U_i) → (F_i(i_U_i^* E) → f_i(U_i))⊂ (E_i → M_i),

satisfying:


  * f_i^-1(p_i) → p, as i →∞;

  * F_i^-1∘ G_i ∘ F_i → G in C^∞_loc( E), as i →∞ .



As usual, fixing isotropic splittings of any ECA of the sequence, we can rewrite this definition as follows.

The sequence ((TM_i ⊕ T^*M_i)_H_i, G(g_i,b_i),p_i)_i≥ 1 converges to ((T M ⊕ T^*  M)_ H, G( g, b), p) in the generalized Cheeger–Gromov topology if and only if there exists an exhaustion {U_i}_i≥ 1 of M consisting in neighbourhoods of p, diffeomorphisms f_i U_i → f_i(U_i)⊂ M_i, and 2-forms a_i ∈Ω^2(U_i) such that

	
  * i_U_i^* H = f_i^*H_i +  a_i,  for all i ≥ 1;
	
  * f_i^-1(p_i) → p,  as i →∞;
	
  * f_i^*g_i → g in C^∞_loc( M),  as i →∞;  
	
  * f_i^*b_i - a_i → b in C^∞_loc( M),  as i →∞.


 For the first direction, we see that in these isotropic splittings, the isomorphisms in <Ref> are given by

    (F_i,f_i) = (f_i∘ e^a_i,f_i),

for some 2-forms a_i ∈Ω^2(U_i) satisfying

    i_U_i^*H = f_i^*H_i +  a_i.

In this case

    F_i^-1∘ G_i(g_i,b_i) ∘ F_i =  G(f_i^* g_i,f_i^*b_i - a_i),

so F_i ∘ G_i(g_i,b_i) ∘ F_i^-1→ G ( g, b) if and only if Items (3) and (4) hold. The converse is easily seen.

Moreover, we can specify <Ref> when the preferred isotropic splitting of 𝒢_i are chosen. 
 
The sequence ((TM_i ⊕ T^*M_i)_H_i, G(g_i,0),p_i) converges to ((T M ⊕ T^*  M)_ H, G( g,0), p) in the generalized Cheeger–Gromov topology if and only if there is an exhaustion {U_i∋ p} of M, diffeomorphisms f_i U_i → f_i(U_i)⊂ M_i, and 2-forms a_i ∈Ω^2(U_i) such that

	
  * i_U_i^* H = f_i^*H_i +  a_i,  for all i ≥ 1;
	
  * f_i^-1(p_i) → p,  as i →∞;
	
  * f_i^*g_i → g in C^∞_loc( M),  as i →∞;  
	
  * a_i → 0 in C^∞_loc( M),  as i →∞.

In particular, f_i^*H_i → H in C^∞_loc( M),  as i →∞.


We now describe how such convergence is achieved at the level of Lie brackets. Fix a vector space g and a generalized metric G =  G(g̅,0) on g ⊕ g^*. For a Dorfman bracket  on g ⊕ g^*, denote by (( g ⊕ g)_,__ g, G_) the corresponding metric ECA.
 Let   (_i)_i≥ 1 be a sequence of nilpotent Dorfman brackets converging to  on g ⊕ g^*. Then, (( g ⊕ g^*)__i,_(_i)_ g, G__i,e)_i≥ 1 converges to (( g ⊕ g^*)_,__ g, G_,e) in the generalized Cheeger–Gromov topology.

 As usual, we consider G( g,0) as a background generalized metric, hence 

    (( g ⊕ g^*)__i,_(_i)_ g, G__i,e) = (( g ⊕ g^*)_H_i,_μ_i, G(g_μ_i,0),e),

where H_i = _Λ^3 and μ_i = _ g. We also set μ = _ g and H = _Λ^3. Then by nilpotence, the exponential maps of _μ_i and _μ are diffeomorphisms. We consider

    f_i := exp_μ_i∘exp_μ^-1_μ→_μ_i,

which is a diffeomorphism with f_i(e) = e. Recall (see for instance <cit.>) that for X ∈ g, the exponential map exp_μ g →_μ of the Lie bracket μ∈Λ^2  g^* ⊗ g satisfies

    (exp_μ)_X = ( L_exp_μ(X))_e ∘𝕀 - e^-_μ X/_μ X,

where

    𝕀 - e^-_μ X/_μ X = ∑_k=0^∞(-1)^k/(k+1)!(_μ X)^k.

Hence,  f_i depends continuously on μ_i, and an argument analogous to <cit.> shows that  f_i^*g_μ_i→ g_μ and f_i^*H_i → H smoothly and uniformly on compact subsets. Then, we define

    a_i := ι_𝕀( H - f_i^*H_i),

where 𝕀∈Γ(T^n) is the identity vector field on ^n. By Cartan's formula,

    a_i = ℒ_𝕀 ( H - f_i^*H_i) =  H - f_i^*H_i.

Hence, H = f_i^*H_i +  a_i with a_i → 0 smoothly and uniformly on compact subsets. Thus, the result follows by <Ref>.


With the technical details established, we now turn to studying the long-time behaviour of the generalized Ricci flow on nilpotent Lie groups when the initial metric ECA has harmonic torsion.

Let ((t))_t ∈ [0,∞) be a solution to the generalized bracket flow (<ref>) such that (0)_ is a nilpotent Lie bracket and (0)_Λ^3 is harmonic. Then, the rescaled brackets (t)|(t)| converge as t→∞ to an algebraic soliton bracket _∞≠ 0.


Long-time existence follows immediately from <Ref>. Up to a time-reparametrization, the rescaled brackets (t) = (t)/|(t)| solve (<ref>) with ℓ = ℓ_ (defined in 7.) That is, the scalar-normalized bracket flow. Since the brackets are nilpotent, <Ref> implies that

    / t = - Θ(M_ + 6 |M_|^2𝕀) .

 On the other hand,  using the same strategy as in  <cit.>, <Ref> and similar computations as in the proof of <Ref>, one can easily see that this is, up to scaling, the negative gradient flow of the real-analytic functional 

    Λ^2 ( g ⊕ g^*)^* ⊗ ( g ⊕ g^*) ∋⟼|M_|^2/||^4∈.

By compactness and Łojasiewicz’s Theorem on real-analytic gradient flows <cit.>, the ω-limit consists of a unique fixed point _∞ to the scalar-normalized bracket flow, and hence  it is an algebraic soliton.

Let us remark some important facts.   First of all, we should observe that the group 𝖫 is not reductive. This does not allow us, in general,  to use directly  <cit.> to deduce the fact that the scalar-normalized bracket flow is a gradient flow. Moreover, it is easy to be shown that, removing the harmonicity of the initial torsion, the scalar-normalized bracket flow  is not the negative  gradient flow  the above functional.  Finally, this gives an alternative and dynamical proof of the uniqueness of generalized nilsolitons with harmonic torsion.

Let us now prove the main convergence results on nilpotent Lie groups.

Let (E→, G_t) be a left-invariant  solution of the generalized Ricci flow on a left-invariant ECA over a simply-connected, nilpotent, non-abelian Lie group . Assume that the preferred  representative of the Ševera class of E  determined by G_0 is harmonic. Then, the scaled ECA's (- S( G_t)· E, G_t) converge in the generalized Cheeger–Gromov sense to a left-invariant, nilpotent, non-abelian, generalized Ricci soliton.


After choosing an isotropic splitting, the left-invariant solution is equivalent to the bracket flow (<ref>) by <Ref>. Denoting this solution by ((t))_t∈ [0,∞), we have by (<ref>) and <Ref> that the rescaled brackets (t)/(- S ( G_t))^1/2 converge to a nilpotent, non-abelian algebraic soliton bracket. These rescaled brackets correspond to the rescaled ECA's (- S( G_t)· E, G_t) by <Ref> and <Ref>. Finally, by <Ref>, we obtain convergence of (- S( G_t)· E, G_t) to an algebraic soliton.

From a more classical perspective, we obtain the following slightly weaker statement:

Let (,H_t,g_t) be a left-invariant  solution of the generalized Ricci flow on a simply-connected, non-abelian, nilpotent Lie group  and assume that H_0 is g_0-harmonic. Then,  the rescaled family
 (,- S(H_t, g_t)H_t,- S(H_t, g_t)g_t) converges to a nilpotent, generalized soliton (, H, g) in the following sense: for every sequence of times, there is a subsequence t_k →∞, and diffeomorphisms f_k → fixing the identity, such that (f_k^*H_t_k,f_k^*g_t_k) → ( H, g),  as k →∞,  in C^∞_loc().







§ CONNECTIONS TO THE PLURICLOSED FLOW
 

 The pluriclosed flow is an important geometric flow in Hermitian Geometry firstly introduced by Streets and Tian in <cit.>.  We quickly recall its definition and its link with the generalized Ricci flow. 

Let (M, J, g_0) be a Hermitian manifold, i.e. J∈ End(TM)∩𝖮(g_0), J^2=- Id and such that 

    N_J(X, Y)=[X, Y]+J[JX, Y]+J[X, JY]-[JX, JY]=0 ,     X, Y∈Γ (TM) .
 The pluriclosed flow is the following evolution equation:

    ∂/∂ tg=-S+Q ,    g(0)=g_0
 where S_ij̅=g^kl̅R_kl̅ ij̅ is the second Chern-Ricci tensor while Q_ij̅=g^kl̅g^mn̅T_ikn̅T_j̅l̅ m .

The tensors R and T  in the definition above are, respectively, the curvature tensor and the torsion of the Chern connection ∇, namely the unique linear connection on M such that 
 
    ∇ g=0  ,   ∇ J=0      T^1,1=0 .

 
The pluriclosed flow is part of a larger class of geometric flows called Hermitian curvature flows, introduced in <cit.>,  which share many common properties such as the  short-time existence on compact complex manifolds and the preservation of  Kählerianity along the flow. However, among this class, the pluriclosed flow is the natural flow  in the realm of  strong Kähler with torsion (SKT, for short) or pluriclosed metrics. The latter are defined by requiring that their fundamental form ω(·, ·):=g(J·, ·)∈Ω^1,1(M) is ^c-closed, where ^c:=J^-1 J is the twisted exterior differential.

Let (M, J, g_0) be a  SKT Hermitian manifold. Then, the pluriclosed flow preserves the SKT condition. Moreover,  (g(t))_t∈[0, T) is a solution of the pluriclosed flow starting at g_0 if and only if the family (ω(t))_t∈[0,T) of the fundamental forms associated to g(t) is a solution of 

    ∂/∂ tω=-(ρ^B)^1,1(ω) ,   ω(0)=ω_0
 where (ρ^B)^1,1(ω) denotes the (1,1)-part of the Bismut-Ricci form of ω.

  Moreover,  we have the following Theorem. 
   Let (M, J)  be a complex manifold  and (g(t))_t∈ [0,T) be a solution of the pluriclosed flow starting from a  SKT metric g_0. Then, (g(t), H(t))_t∈ [0,T) solves the following:
  
    ∂/∂ tg=     - Ric_g, H^B-1/2ℒ_θ^♯g ,     g(0)=g_0 ,
    ∂/∂ t H=    1/2Δ_gH-1/2ℒ_θ^♯H ,    H(0)=H_0 ,
 where θ∈Ω^1(M) is the Lee form  of the time-varying metric defined by ω^n-1=θ∧ω^n-1   while H(t):=^cω(t) is the torsion of the Bismut connection.
  
  Therefore,  up to a time reparametrization, a  solution of the pluriclosed flow is a solution of  a gauged fixed generalized Ricci flow where the gauge is generated by the generalized vector field -θ^♯-θ+ι_θ^♯H_0+ι_θ^♯b∈aut(()_H_0).
  
   
   Streets and  Tian in <cit.> proved that, given a SKT Hermitian  manifold (M, J, g), the generalized Ricci flow coupled with  an evolution equation for J:
  
    ∂/∂ tJ=Δ J+ [J, g^-1 Ric_g]+  Q(DJ) ,

    where Q(DJ) is an appropriate quadratic term in the covariant derivative of J with respect to the Levi-Civita connection, see <cit.>  for the precise expression of Q(DJ), preserves the SKT condition, namely g(t) is J(t)-hermitian and SKT for all times of existence.  Then, gauging  with the family of diffeomorphisms generated by (-J(t) _g(t)^*ω(t))^♯ the generalized Ricci flow coupled with  (<ref>) and  using <cit.>, we obtain a solution for the pluriclosed flow, up to a time reparametrization. 
     
    As a direct consequence of <Ref>,  <Ref>  and <Ref>, we have the following. 
    
    Let (𝖦,J, g) be a solvable Lie group endowed with a left-invariant SKT structure. Then, the solution of the pluriclosed flow starting from g is immortal.
    
     <Ref> recovers many known results in the literature such as those in <cit.> on compact  complex surfaces, in <cit.> on SKT nilmanifolds, in <cit.> on almost-abelian SKT solvmanifolds and in <cit.> on Oeljeklaus-Toma manifolds.
     
The gauge-equivalence between pluriclosed and generalized Ricci flow allows us also  to  give an equivalent condition for a generalized Ricci soliton to be a pluriclosed soliton.
   
   Let (M, J) be a complex manifold endowed with a SKT Hermitian metric g. Then, a soliton  (g, H, X) for the generalized Ricci flow is a pluriclosed soliton if and only if H=^cω and X+θ^♯ is holomorphic. 
   
   
   We recall that (ω, X) is a pluriclosed soliton if and only if 
   
    (ρ^B)^1,1(ω)=λω+1/2 L_Xω ,    X  ,   λ∈ .
 By means of the holomorphicity of X,   we have that [ L_X, J]=0 on any form on M,  yielding that ^c L_Xω= L_X^cω. Then,  using <cit.> and defining H=^cω, we obtain 
   
    1/2Δ_gH-1/2 L_θ^♯H= -^c(ρ^B)^1,1(ω)=-λ H-1/2 L_XH ,
 which gives us 
   
    Δ_gH=-2λ H- L_X-θ^♯H .
 On the other hand, we have that
   
    Ric^B_g, H+1/2 L_θ^♯g=(ρ^B)^1,1(·, J·)=λ g+1/2 L_Xg.
 This guarantees that 
   
    Ric_g, H^B=λ g+1/2 L_X-θ^♯g ,
giving us the first claim. The converse can be proved following the same steps backwards.    
   
     
   In view of  <Ref> and <Ref>,  we readily obtain the gauge-equivalence  between  the generalized bracket flow and  the bracket flow for the pluriclosed flow introduced in <cit.> and <cit.>.  First of all, we recall the definition of the latter.
   Let (𝖦, μ_0) be  a Lie group endowed with left-invariant complex structure J and a left-invariant SKT metric ω. The bracket flow is the following evolution equation: 
  
    / tμ=-1/2θ(P)μ ,   μ(0)=μ_0
 where  P∈𝔤𝔩(, J  ) defined by ω(P·, ·)=(ρ^B)^1, 1(·, ·).
  
  

   Combining <Ref>, <Ref>, <Ref> and <cit.>, we have the following.
   Up to a time reparametrization by 2,  the bracket flow (<ref>) is gauge equivalent to the generalized bracket flow (<ref>).
  

   
   
   Finally, <Ref> can be specialized in the case of algebraic solitons,  as defined in <Ref> and in <cit.>. For the sake of clarity, we quickly  recall  the definition of algebraic solitons for the pluriclosed flow.
   A Lie group 𝖦 endowed with a left-invariant  SKT structure (J, g) is called algebraic pluriclosed soliton if there exists D ∈ Der()∩𝔤𝔩(, J) and λ∈ℝ such that 
   
    P=λ Id+1/2(D+D^t).
 
   
   
    Let 𝖦 be a Lie group endowed with a left-invariant SKT structure (J, g). Then, an algebraic soliton (g, H, D) for the generalized Ricci flow is an algebraic pluriclosed  soliton if and only if D-  ad_θ^♯∈ Der()∩𝔤𝔩(, J) and H=^cω. 
   
   
  By <Ref>,  (g, H, D) is an algebraic soliton for the generalized Ricci flow if and only if (g, H, -X_D) is a  soliton for the generalized Ricci flow where X_D is the vector field defined by 
  
    X_D=/ t|_t=0φ_t φ_t∈ Aut(𝖦) _eφ_t=e^tD .
 
  On the other hand, by <Ref>,  (g, H, -X_D) is a pluriclosed soliton if and only if H=^cω and -X_D+θ^♯ is holomorphic. But, -X_D+θ^♯ is holomorphic if and only if D- ad_θ^♯∈𝔤𝔩(, J), giving the claim.
   
   
  



§ CLASSIFICATION OF SOLITONS UP TO DIMENSION 4

 In this Section, we will provide a classification of generalized nilsolitons on nilpotent Lie algebras of dimension up to 4.  As it is clear  from the discussion in <Ref>, we can only hope to classify generalized nilsolitons up to scaling and up to generalized automorphisms of the ECA we are considering.  Classical nilsolitons were classified in <cit.>. So, we will focus on classifying the non-classical ones.
 
  
  As it is well known, besides the abelian ones, nilpotent Lie algebras of dimension up to 4 are 3: 𝔫_3, the Lie algebra associated to the Heisenberg group Heis(3, ℝ),  which is the only non-abelian nilpotent Lie algebra of dimension 3, 𝔫_3⊕ℝ and 𝔫_4, the only  indecomposable non-abelian nilpotent Lie algebra of dimension 4.  
 
 We will go through the classification case by case using (<ref>). 
 
 In dimension 3, there is only one generalized soliton  which is non-classic. This  is mainly due to the fact that H^3(𝔫_3, )=ℝ.
 
  On 𝔫_3, the unique  non-classic generalized nilsoliton  is given by:
  
    μ(e_1, e_2)= e_3 ,    H=e^123 ,   λ=-2 ,   D= diag(1,1,2) .

  
  
  Given a left-invariant  metric on  Heis(3, ℝ),  thanks to <cit.>,   we can always find an orthonormal basis {e_1, e_2, e_3} of 𝔫_3 such that 
 
    μ(e_1, e_2)=a e_3 ,    a∈ℝ\{0} .
 Moreover, thanks to <cit.>,   we know that
 
    Ric_μ=1/2a^2 diag(-1,-1,1) .
 Now, every  0 H∈Λ^3𝔫_3^*, it will be of the form H=be^123 with  b 0 and it will be trivially closed and harmonic.  On the other hand, we have that 
 
    H^2=2b^2 Id
 which implies that 
 
    Ric_μ, H^B=1/2 diag(-(a^2+b^2), -(a^2+b^2), a^2-b^2)
 Moreover,  every  D∈ Der(𝔫_3, μ) has to satisfy 
 
    D_3^3=D_1^1+D_2^2 .
  This guarantees that 
 
    Ric_μ, H^B-λ Id∈ Der(𝔫_3, μ)
 if and only if 
 
    λ=-1/2(3a^2+b^2) .

  On the other hand,  by straightforward computations we have that 
 
    0=     ρ( Ric_μ, H^B)H-Δ_μH+ℓ H
    =-(a^2-b^2)H ,
  which is satisfied if and only if a^2=b^2, giving us the claim, using <Ref>.
 
 
  Now, we will focus on the 4-dimensional case.  A first difference between the 3-dimensional one is  a  larger third cohomology group which allows a  wider  possibility for a generalized metric to be a generalized nilsoliton. 
  
 We will, first of all, analyse the case of 𝔫_3⊕ℝ. An important remark to be done is that this Lie algebra is the only one among the 4-dimensional ones which admits left-invariant complex structures, see for instance <cit.> or <cit.>. The complex structure on 𝔫_3⊕ℝ is unique  up to biholomorphisms and quotients by co-compact  lattices give rise to the so-called primary Kodaira surfaces, the only compact complex surfaces with zero  Kodaira dimension and first Betti number equal to 3.  Clearly, every left-invariant  Hermitian metric on 𝔫_3⊕ℝ is SKT. So, using <Ref>, we will be able to detected which generalized nilsoliton on 𝔫_3⊕ℝ is an algebraic pluriclosed soliton.
 
   On 𝔫_3⊕ℝ,  the only non-classic generalized nilsolitons are the following:
   
   
  *  μ(e_1, e_2)=e_3 ,    H=e^123 ,  λ =-2 ,    D= diag(1,1,2,2) .
   
  * μ(e_1, e_2)=e_3 ,    H=λ_1e^234+λ_2e^134 ,   λ_1^2+λ_2^2=1 ,   λ =-3/2 ,    D= diag(1-λ_2^2/2,1-λ_1^2/2,3/2,1) .
   
   In particular, the first one is the unique algebraic pluriclosed soliton on  𝔫_3⊕ℝ.
 
 
 Thanks to  <cit.>, every inner product on 𝔫_3⊕ℝ is isometric to an inner product such that an  orthonormal basis {e_1, …, e_4}   satisfies the following
 
    μ(e_1, e_2)=a e_3 ,    a 0 .
We know that D∈ Der(𝔫_3⊕ℝ, μ)∩ Sym(𝔫_3⊕ℝ) if and only if 
 
    D=[     a_1     a_3       0     a_5;     a_3     a_2       0     a_6;       0       0 a_1+a_2       0;     a_5     a_6       0     a_4 ] ,    a_i∈ℝ ,    i=1,…, 6 ,
 with respect to the frame {e_1, …, e_4}.
  Moreover,  using <cit.>, it is easy to see that 

    Ric_μ=1/2a^2  diag(-1, -1, 1, 0) .

 
 
 
 
It is straightforward to check  that  

 
 

  
   a generic closed H∈Λ^3(𝔫_3⊕ℝ)^* is of the form: 
 
    H=λ_4e^123+λ_3e^124+λ_2e^134+λ_1e^234 ,  λ_i∈ℝ ,    i=1,…, 4 .
Consequently, we have that 
 Δ_μH=-a^2λ_3e^124
 and 
 
    H^2=2[ λ_4^2+λ_2^2+ λ_3^2             λ_2λ_1            -λ_3λ_1             λ_4λ_1;             λ_2λ_1  λ_4^2+λ_1^2+λ_3^2             λ_2λ_3            -λ_4λ_2;            -λ_3λ_1             λ_3λ_2  λ_1^2+λ_2^2+λ_4^2             λ_4λ_3;             λ_4λ_1            -λ_4λ_2             λ_4λ_3  λ_1^2+λ_2^2+λ_3^2 ] .

  This readily implies that we need to impose that  λ_3=0 or λ_4=λ_2=λ_1=0 in order to hope for   Ric_μ, H^B-λ Id∈ Der(𝔫_3⊕ℝ, μ).


 First of all,  we suppose λ_3=0. In this case,   Ric_μ, H^B-λ Id∈ Der(𝔫_3⊕ℝ, μ)  if and only if 

 
    λ=-1/2(λ_4^2+3a^2) .
 The condition 
  
    ρ( Ric_μ, H^B)H-Δ_μH+λ H=0

  is equivalent to 
  
    λ_4(-2a^2+3λ_2^2+2λ_4^2+3λ_1^2)=0 ,
    
    λ_2(-3a^2+3λ_2^2+2λ_4^2+3λ_1^2)=0 ,
    
    λ_1(-3a^2+3λ_2^2+2λ_4^2+3λ_1^2)=0 .
 
 However, the first equation implies that a^2=1/2( 3λ_2^2+2λ_4^2+3λ_1^2) or λ_4=0 . 
 
 
 

 
 
 
On the other hand, if a^2=1/2( 3λ_2^2+2λ_4^2+3λ_1^2), we have that 

    -3a^2+3λ_2^2+2λ_4^2+3λ_1^2=-a^2 0  .
 From this, we obtain that λ_1=λ_2=0 , giving us <Ref>. 

 
 

 
 
  
  
 
  
  
  
    
  



 




  
  

  
  
  Now let λ_4=λ_3=0. In this case, the second equation in (<ref>) implies that 
 
 

 
 
  λ_2=0 or λ^2=λ_1^2+λ_2^2. If  the last condition holds, then,  the third equation is satisfied too, giving us <Ref>.  


 
 
 
 
 

  
  


  

  It remains to analyse  when λ_4=λ_3=λ_2=0. In this case,  the third equation in (<ref>) gives us that 
 
 
  a^2=λ_1^2, since λ_1 0 if we want non-classical generalized nilsoliton. Thus, we  recover a particular generalized nilsoliton belonging to <Ref>.
  
  
  
  
  
  
  It remains to analyse when λ_3 0 and λ_4=λ_2=λ_1=0. In this case, Ric_μ, H^B-λ Id∈ Der(𝔫_3⊕ℝ, μ) if and only if λ=-1/2(3a^2+2λ_3^2).
    
   
 

 
 
 


 
 
 
 
 
 
 On the other hand, the condition
 



  
    ρ( Ric_μ, H^B)H-Δ_μH+λ H =0
 is equivalent to 

 
 

  λ_3(a^2+λ_3^2)=0 which cannot happen unless λ_3=0, concluding the classification.

 To address the last part of the statement, we  

 
 
 
 define Je_1=e_2, Je_4=e_3 and  consider 
 
    ω^1=1/√(2)(e^1+√(-1)e^2) ,   ω^2=1/√(2)(e^4+√(-1)e^3)
 obtaining that 
 
    ω^1=0 ,  ω^2=a/√(2)ω^11̅ ,
 which, in particular, gives us the integrability of J.
 The fundamental form of the metric  is then ω=√(-1)(ω^11̅+ ω^22̅),  which is SKT,  while the torsion of the Bismut connection is given by 
 
 
 
 
 
    H:=^cω=√(-1)(∂̅ω-∂ω)=-1/√(2)a(ω^11̅2̅-ω^11̅2)=a e^123 .
 On the other hand, we have that 
 
    ω=√(-1)/√(2)a(ω^11̅2+ω^11̅2̅)=a e^124 .
Then, it is easy to see that Lee form associated to ω has the following form:
 
    θ=a/√(2)(ω^2+ω^2̅)=a e^4 .
 

 This gives easily that ad_μθ^♯=0.
 Moreover, given a diagonal derivation D, it will commute with J if and only if  it is of the form 
 
    D= b diag(1,1,2,2) ,   b∈ℝ .
  But, the derivation in <Ref>   is of the above form. Then, applying <Ref>,  we obtain the claim, concluding the proof. 
   
  <Ref> in <Ref> in particular provides the first example of nilpotent Lie group admitting a infinite family of generalized nilsolitons, answering negatively to <Ref>.  
 


 
 
 

 
 
 
 

 
 


 


  
  
 
 
  
 

 
  Now, we focus on the study of generalized nilsolitons in the case of 𝔫_4.
  
  On 𝔫_4, the unique non-classic generalized nilsolitons are the following:
  
  
  * μ(e_1, e_2)=e_3 ,  μ(e_1, e_3)=√(3)/2e_4 ,    H=√(3)/2e^134 ,   λ=-3/2 ,    D= diag(2/3, 1, 5/4, 3/2) ,
  
  * μ(e_1, e_2)=e_3 ,  μ(e_1, e_3)=e_4 ,    H=e^234 ,   λ=-3/2 ,    D= diag(1/2, 1/2, 1, 1/2) .
  
  
  
  Thanks to <cit.>, up to isometries, given any inner product on 𝔫_4, we can always find an orthonormal frame  {e_1, …, e_4} of 𝔫_4 such that 
  
 
 
    μ(e_1,e_2)=ae_3+be_4 ,   μ(e_1, e_3)=ce_4 ,    a, c>0 .

In this framework, a symmetric derivation D is of the form 
 
    D= diag(α, β, α+β, 2α + β) ,  β, α∈ℝ ,

 see for instance <cit.>.  Moreover, the Ricci endomorphism, using again <cit.>, takes the following form:

    Ric_μ=1/2[ -(a^2+b^2+c^2)              0              0              0;              0     -(a^2+b^2)            -bc              0;              0            -bc        a^2-c^2             ab;              0              0             ab        b^2+c^2 ] .

 Again, a generic closed H∈Λ^3𝔫_4^* will be of the following form:
 
    H=λ_4e^123+λ_3e^124+λ_2e^134+λ_1e^234 ,  λ_i∈ℝ ,    i=1,…, 4 .
 Consequently, it is easy to see that 
 
    Δ_μH
     
     =(-λ_4(c^2+b^2)+λ_3ab)e^123 +(-λ_3a^2+λ_4ab)e^124
 and the endomorphism H^2 will have the same form as in (<ref>). 




 
 

 
 
 



 
 
 
 

 
 
 








 
















Combining (<ref>) and (<ref>), we can see that,  in order for Ric_μ, H^B-λ Id to be a derivation, we need to impose that 

    λ_2λ_1=λ_3λ_1=λ_4λ_1=λ_4λ_2=0 ,    λ_4λ_3=ab ,  λ_2λ_3=-bc ,
 which gives us that b=0 and just one among λ_i's can be non-zero.






  

  















First of all, we consider  λ_1 0.  In this particular case,  we have that Ric_μ, H^B-λ Id∈ Der(𝔫_3⊕ℝ, μ) if and only if 

    λ=-3/2a^2
          c^2=a^2 .











On the other hand,





    0=ρ( Ric_μ, H^B)H-Δ_μH+λ H=3/2λ_1( λ_1^2-a^2)e^234
 which is satisfied if and only if λ_1^2=a^2, giving us <Ref>.



 

 

 
 
 
 
 
 Now, we  assume λ_2 0.  In this case, we have that Ric_μ, H^B-λ Id∈ Der(𝔫_4, μ) if and only if 




    λ=-3/2a^2      3c^2+λ_2^2=3a^2 .









 On the other hand, 
 
    ρ( Ric_μ, H^B)H-Δ_μH+λ H=0
 is equivalent to 
 
    λ_2/2(c^2+3λ_2^2-3a^2)=0
 which gives that 3a^2=c^2+3λ_2^2. Combining this last equality with the second relation in (<ref>), we obtain that c^2=λ_1^2=3/4a^2, giving us <Ref>. 










 

 
 
 




Now, we consider the case in which λ_3 0.  Then,  we have that Ric_μ, H^B-λ Id∈ Der(𝔫_4, μ) if and only if 







    λ=-1/2(2λ_3^2+3a^2)     c^2=a^2+2/3λ_3^2 .
 







Moreover,   the condition






    ρ( Ric_μ, H^B)H-Δ_μH+ℓ H=0
 is equivalent to 

    1/2λ_3(a^2+λ_3^2)=0
 which is impossible since we are assuming that   λ_30.



The left case to analyse is that in which λ_4 0. In this hypothesis, we have that Ric_μ, H^B-λ Id∈ Der(𝔫_3⊕ℝ, μ) if and only if 

    λ=-1/2(3a^2+λ_4^2)      a^2=1/3(λ_4^2+3c^2) .

Moreover, 

    ρ( Ric_μ, H^B)H-Δ_μH+λ H =0
 is equivalent to 

    λ_4(-a^2+2c^2+λ_4^2)=0

 which is satisfied if and only if 2c^2+λ_4^2=a^2. On the other hand, combining this with the second relation in (<ref>) gives that 

c^2=-2/3λ_4^2 which is not possible, concluding the proof.



























 
 
 
 


 

 















amsalpha

