# The homogeneous generalized Ricci flow

Elia Fusi  ,  Ramiro Lafuente  and  James Stanfield

###### Abstract.

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.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on generalized geometry
  3. 3 Solitons of the generalized Ricci flow
  4. 4 Homogeneous generalized geometry
  5. 5 A flow of Dorfman brackets
  6. 6 Global existence on solvmanifolds
  7. 7 Generalized nilsolitons
  8. 8 Long-time behavior on nilmanifolds
  9. 9 Connections to the pluriclosed flow
  10. 10 Classification of solitons up to dimension 4

##  1\. 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_ [Cou90, CSCW11, GF19, GFS21,
Gua11, KW07, LWX97, Šev17, ŠV17, ŠV20]. 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 $[H]\in H^{3}(M,\mathbb{R})$, whereas the former can be
thought of as a pair $(g,b)$, with $g$ a Riemannian metric on $M$, and
$b\in\Omega^{2}(M)$ a 2-form on $M$. A smooth family $(g_{t},b_{t})_{t\in I}$
of generalized Riemannian metrics on $(M,[H])$ solves the _generalized Ricci
flow_ if it satisfies the following weakly parabolic PDE:

(1) $\begin{cases}\tfrac{\partial}{\partial t}g_{t}&=\
-2\operatorname{Ric}(g_{t})+\frac{1}{2}(H+\textrm{d}b_{t})^{2}_{g_{t}},\\\
\tfrac{\partial}{\partial t}b_{t}&=\
-\textrm{d}^{*}_{g_{t}}(H+\textrm{d}b_{t}),\end{cases}$

where, for a $3$-form $H$ and Riemannian metric $g$,
$H_{g}^{2}\in\operatorname{Sym}^{2}(T^{*}M)$ is defined by
$H^{2}_{g}(X,X)=|\iota_{X}H|_{g}^{2}$ for all $X\in TM$. The classical Ricci
flow is the particular case where $H=0$ and $b_{t}\equiv 0$. By abuse of
notation, we will sometimes call the pair $(g,H+\textrm{d}b)$ 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 $(g_{t},H_{t}:=H+\textrm{d}b_{t})$ 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 $(\mathsf{G},[H])$, where
$\mathsf{G}$ is a Lie group and $[H]\in
H^{3}(\mathsf{G},\mathbb{R})^{\mathsf{G}}$ 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_ :

######  Theorem A.

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 $\mathsf{G}$ in terms of the blow-up
behaviour of the _generalized scalar curvature_ (see 6.2). Here by
_solvmanifold_ we mean any (not necessarily compact) locally homogeneous space
of the form $\mathsf{G}/\Gamma$, where $\mathsf{G}$ is a solvable Lie group,
and $\Gamma\leq\mathsf{G}$ is a discrete subgroup. It is called a
_nilmanifold_ if $\mathsf{G}$ is nilpotent. An _invariant solution_ is simply
a solution to (1) on $\mathsf{G}/\Gamma$ that lifts to a left-invariant
solution on $\mathsf{G}$. The class of solvmanifolds is known to be very rich.
Examples in four dimensions include Inoue surfaces and primary Kodaira
surfaces.

Notice that A 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,
(1) is gauge-equivalent to the pluriclosed flow on $(M,J)$ [GFS21]. As an
immediate application of A we get

######  Corollary B.

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 [Bol16, FV15, AL19, AN22, FP22, FV22]. 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 [PE20] which were recently shown to
admit SKT structures in [COS24]. Moreover, the existence of SKT metrics was
characterized by Freibert–Swann in [FS21] on a large class of 2-step
solvmanifolds, and on solvmanifolds admitting a codimension $2$ abelian ideal
by Guo–Zheng [GZ23] and Cao–Zheng [CZ23].

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
$c_{1}^{A}(M,J)\in H_{A}^{1,1}(M,J)$ is the _first Aeppli–Chern class_ of
$(M,J)$ [Str16, Str20]. It is well-known that the first Chern class of
solvmanifolds is zero, hence the first Aeppli–Chern class vanishes as well. In
particular, $-c_{1}^{A}(\mathsf{G}/\Gamma,J)$ is a nef class on compact
solvmanifolds, and thus B 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
$M=\mathsf{G}/\Gamma$ with harmonic $H_{0}$ (i.e.
$\textrm{d}\textrm{d}^{*}H_{0}=0$), we can characterise the possible limits as
follows:

######  Theorem C.

Let $(M,g_{t},H_{t}=H_{0}+\textrm{d}b_{t})_{t\in[0,\infty)}$ be an invariant
solution to the generalized Ricci flow on a nilmanifold $M=\mathsf{G}/\Gamma$
with $H_{0}$ harmonic. Then, after pulling back to the universal cover, the
rescaled generalized metrics
$(\frac{1}{1+t}g_{t},\frac{1}{1+t}H_{t})_{t\in[0,\infty)}$ on the time-
dependent exact Courant algebroids $(\mathsf{G},\frac{1}{1+t}[H_{0}])$
converge, in the _generalized Cheeger–Gromov_ topology, to a left-invariant,
expanding generalized Ricci soliton $(g_{\infty},H_{\infty})$ on a nilpotent
Lie group $\mathsf{G}_{\infty}$.

By _generalized Ricci soliton_ we mean a solution
$(g_{t},H_{t}=H+\textrm{d}b_{t})$ to (1) which, up to pull-back by time-
dependent diffeomorphisms, evolves only by scaling. That is, there exist
$c(t)\in\mathbb{R}_{>0}$, and $\varphi_{t}\in{\rm Diff}(M)$ such that, for all
$t$, it holds that

$\left(\varphi_{t}^{*}g_{t},\varphi_{t}^{*}H_{t}\right)=(c(t)\,g_{0},c(t)\,H_{0}).$

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 [GFS21,
4.4.2] (compare however with [GFS21, Remark 6.31]), but agrees with the
definition posed in [PR24b]. We adpot it here because it allows for non-
trivial, non-steady solitons, and because examples arise naturally as
asymptotic limits by Theorem C. We refer the reader to Section 2 for more
technical details regarding this notion and the scaling of exact Courant
algebroids, and to Section 8 for the definition of generalized Cheeger-Gromov
topology.

Notice that the limit group may be non-isomorphic to $\mathsf{G}$ (this occurs
already in the classical setting [Lau11]). 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 Section 10.

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 $[H_{0}]\in H^{3}(M,\mathbb{R})$ can be identified with the bundle
$(TM\oplus T^{*}M,[\cdot,\cdot]_{H_{0}})$ endowed with the _twisted Dorfmann
bracket_ $[\cdot,\cdot]_{H_{0}}$, a bracket law on $TM\oplus T^{*}M$
determined by $H_{0}$ and the usual Lie bracket of vector fields
$[\cdot,\cdot]$ on $TM$, see (6). The pair $(g,b)$ then determines a
generalized metric on $TM\oplus T^{*}M$, which is an endomorphism
$\mathcal{G}(g,b)\in\operatorname{End}(TM\oplus T^{*}M)$. The _generalized
Ricci curvature_ $\mathcal{R}c(\mathcal{G})\in\operatorname{End}(TM\oplus
T^{*}M)$ of $\mathcal{G}$ is a $\mathcal{G}$-symmetric endomorphism on
$TM\oplus T^{*}M$ (see Section 2.3 for its definition) and the generalized
Ricci flow is the following PDE:

$\frac{\partial}{\partial
t}\mathcal{G}=-2\,\mathcal{G}\,\mathcal{R}c(\mathcal{G}).$

If $M=\mathsf{G}$ is a Lie group and $[H_{0}]\in
H^{3}(\mathsf{G},\mathbb{R})^{\mathsf{G}}$ is invariant, then the exact
Courant algebroid structure $(T\mathsf{G}\oplus
T^{*}\mathsf{G},[\cdot,\cdot]_{H_{0}})$ can be identified with the pair
$(\mathfrak{g}\oplus\mathfrak{g}^{*},{\bm{\mu}}_{H_{0}})$, where
$\mathfrak{g}$ is the Lie algebra of $\mathsf{G}$, and ${\bm{\mu}}_{H_{0}}$ is
a bracket on $\mathfrak{g}\oplus\mathfrak{g}^{*}$ determined by
$H_{0}\in\Lambda^{3}\mathfrak{g}^{*}$ and $\mu$, the Lie bracket of
$\mathfrak{g}$. The invariant generalized Ricci flow is then equivalent to an
ODE on the space of endomorphisms of $\mathfrak{g}\oplus\mathfrak{g}^{*}$. We
now follow Lauret’s _bracket flow_ approach. The non-reductive group
$\mathsf{L}=\overline{\mathsf{GL}(\mathfrak{g})}\cdot\operatorname{exp}(\Lambda^{2}\mathfrak{g}^{*})\cong\mathsf{GL}(\mathfrak{g})\ltimes\Lambda^{2}\mathfrak{g}^{*}$
acts transitively on the space of invariant generalized metrics. Thus, by
fixing a background generalized metric
$\overline{\mathcal{G}}=\mathcal{G}(g,0)$, we pull-back the generalized Ricci
flow solution to an ODE on the space of brackets on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$ via the action of $\mathsf{L}$.

######  Theorem D.

The invariant generalized Ricci flow on $(\mathsf{G},[H_{0}])$ is gauge-
equivalent to the following ODE of Dorfman brackets on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$:

(2)
$\frac{\textrm{d}}{\textrm{d}t}{\bm{\mu}}(t)=-\widetilde{\mathcal{R}c}_{{\bm{\mu}}(t)}{\bm{\mu}}(t)+{\bm{\mu}}(t)(\widetilde{\mathcal{R}c}_{{\bm{\mu}}(t)}\cdot,\cdot)+{\bm{\mu}}(t)(\cdot,\widetilde{\mathcal{R}c}_{{\bm{\mu}}(t)}\cdot).$

Here, for a twisted Dorfman bracket ${\bm{\mu}}={\bm{\mu}}_{H,\mu}$,
$\widetilde{\mathcal{R}c}_{\bm{\mu}}:=\mathcal{R}c_{\bm{\mu}}-A_{{\bm{\mu}}}$
where
$\mathcal{R}c_{{\bm{\mu}}}\in\operatorname{End}(\mathfrak{g}\oplus\mathfrak{g}^{*})$
is the generalized Ricci curvature of the infinitesimal exact Courant
algebroid structure $(\mathfrak{g}\oplus\mathfrak{g}^{*},{\bm{\mu}}_{H})$, and
$A_{{\bm{\mu}}}:=\operatorname{Skew}_{\overline{\mathcal{G}}}(\textrm{d}^{*}H)$,
where we consider the two-form
$\textrm{d}^{*}H\colon\mathfrak{g}\to\mathfrak{g}^{*}$ as a map from
$\mathfrak{g}$ to $\mathfrak{g}^{*}$.

Let us remark that a bracket flow approach to the homogeneous generalized
Ricci flow was considered by Paradiso in [Par21] using the action of
$\mathsf{GL}(\mathfrak{g})\leq\mathsf{L}$. The novelty of our approach, and
what makes D work, is that we consider the bigger group $\mathsf{L}$ which
acts transitively on the space of generalized metrics. The two approaches
agree only when $H_{0}$ is harmonic (see Remark 5.4).

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:

(3)
$\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}=\operatorname{M}_{{\bm{\mu}}}-\tfrac{1}{2}\overline{B}-\overline{\operatorname{Sym}_{g}(\operatorname{ad}_{U})}.$

Here, $U\in\mathfrak{g}$ is the _mean curvature vector_ defined by
$g(U,X)=\operatorname{tr}(\operatorname{ad}_{X}),$ for $X\in\mathfrak{g}$, and
$B$ is the _Killing form_ defined by
$g(BX,Y)=\operatorname{tr}(\operatorname{ad}_{X}\circ\operatorname{ad}_{Y})$
for $X,Y\in\mathfrak{g}$. For $F\in\operatorname{End}(\mathfrak{g})$, we
define
$\overline{F}:=F\oplus-F^{*}\in\operatorname{End}(\mathfrak{g}\oplus\mathfrak{g}^{*})$.
Finally,
$\operatorname{M}_{{\bm{\mu}}}\in\operatorname{End}(\mathfrak{g}\oplus\mathfrak{g}^{*})$
is the _moment map_ for the natural action of $\mathsf{L}$ on the space of
brackets
$\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$
(see [BL20a, Definition 1.2]).

The identification of a moment map in the structure of the generalized Ricci
curvature allows us to use techniques from real geometric invariant theory
[BL20a]. In particular, for nilmanifolds with $H_{0}$ harmonic, it holds that
$\mathcal{R}c_{\bm{\mu}}=\operatorname{M}_{{\bm{\mu}}}$. This is the key
ingredient in the proof of C since after normalizing, Equation 3 becomes a
real analytic gradient flow (see 8.5). If $H_{0}$ is not harmonic, we have
$\widetilde{\mathcal{R}c}_{{\bm{\mu}}}=\operatorname{M}_{{\bm{\mu}}}$ which is
not symmetric, as $\mathsf{L}$ 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 [CS26a, CS26b, WYZ20]. 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
[Lee24], they are dynamically stable with respect to the GRF. Furthermore,
Barbaro in [Bar22, Bar23] 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 [GFJS23]. The first
examples of homogeneous Bismut–Ricci flat metrics which are non-flat were
given by Podestà–Raffero [PR22, PR24a]. Lauret–Will in [LW23] constructed
further examples in this setting. The stability of many of these examples was
studied by Gutierrez in [Gut24]. In [PR24b] Podestà–Raffero construct examples
of $\mathsf{SO}(3)$-invariant steady, gradient generalized solitons on
$\mathbb{R}^{3}$, resembling the structure of the classical Bryant soliton for
the Ricci flow, see [Bry05]. On the other hand, it was recently shown that all
compact shrinking solitons of gradient type are in fact classical solitons
[LY24]. Moreover, in [SU21], Streets–Ustinovskiy classified generalized
Kähler–Ricci solitons on complex surfaces. Beyond soliton solutions, the
generalized Ricci flow was studied on nilmanifolds by Paradiso [Par21] and on
cohomogeneity-one manifolds with nilpotent symmetry by Gindi–Streets [GS23].
The toric, Fano case of the generalized Kähler–Ricci flow was treated by
Apostolov–Streets and Ustinovskiy in [ASU22].

The rest of the article is structured as follows: In Section 2, 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 Section 3, we define and
discuss generalized Ricci solitons, and present in detail some examples on the
Heisenberg group. In Section 4 and Section 5, we treat homogeneous generalized
geometry and the generalized bracket flow, then derive Equation 3, and prove
D. In Sections 6 – 8 we prove our main theorems A and C using the generalized
bracket flow and techniques from real geometric invariant theory. We discuss
in Section 9 the relationship between the pluriclosed flow and the generalized
Ricci flow in our (not necessarily compact) setting, leading to B. Finally, in
Section 10 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.

##  2\. Preliminaries on generalized geometry

In this section we review the basic notions of generalized Geometry and setup
our notation. We mainly follow [GFS21]. 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 [Cou90, Gua11, LWX97, Šev17, ŠV17, ŠV20] and the references therein.

###  2.1. Exact Courant algebroids

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

  * •

A non-degenerate, symmetric bilinear form ${\langle\,\cdot\,,\cdot\,\rangle}$
of signature $(n,n)$, called the _neutral inner product_ ;

  * •

A bracket $[\cdot,\cdot]$ on $\Gamma(E)$, called the _Dorfman bracket_ ;

  * •

A bundle map $\pi:E\to TM$, called the _anchor map_ ;

subject to the following axioms being satisfied for all $a,b,c\in\Gamma(E)$
and $f\in C^{\infty}(M)$:

  1. (ECA0)

$[a,[b,c]]=[[a,b],c]+[b,[a,c]]$;

  2. (ECA0)

$\pi[a,b]=[\pi a,\pi b]$;

  3. (ECA0)

$[a,fb]=f[a,b]+\pi(a)fb$;

  4. (ECA0)

$\pi(a)\langle b,c\rangle=\langle[a,b],c\rangle+\langle b,[a,c]\rangle$;

  5. (ECA0)

$[a,b]+[b,a]=(\pi^{*}\circ\textrm{d})\langle a,b\rangle$;

  6. (ECA0)

there is an exact sequence of vector bundles

(4) $0\longrightarrow
T^{*}M\overset{\pi^{*}}{\longrightarrow}E\overset{\pi}{\longrightarrow}TM\longrightarrow
0\,.$

In the last two axioms we have abused notation and considered
$\pi^{*}:T^{*}M\to E^{*}$ as a map $\pi^{*}:T^{*}M\to E$, after composing with
the isomorphism $E^{*}\simeq E$ given by ${\langle\,\cdot\,,\cdot\,\rangle}$.
Notice also that d 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 [GFS21, Lemma 2.5] 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:

$[a,b]_{c}:=\frac{1}{2}([a,b]-[b,a])\,,\quad a,b\in\Gamma(E)\,.$

Despite being skew-symmetric, the Courant bracket does not satisfy the Jacobi
identity. Indeed, one can show that

$[a,[b,c]_{c}]_{c}+[c,[a,b]_{c}]_{c}+[b,[c,a]_{c}]_{c}=\frac{1}{3}\textrm{d}(\langle[a,b]_{c},c\rangle+\langle[b,c]_{c},a\rangle+\langle[c,a]_{c},b\rangle)$

holds for all $a,b,c\in\Gamma(E)$.

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 $M$.

######  Example 2.1.

The generalized tangent bundle $TM\oplus T^{*}M$ has a natural structure of
exact Courant algebroid, with data given by:

(5) $\langle
X+\xi,Y+\eta\rangle:=\tfrac{1}{2}(\xi(Y)+\eta(X)),\qquad[X+\xi,Y+\eta]:=[X,Y]+\mathcal{L}_{X}\eta-\iota_{Y}\textrm{d}\xi,\qquad\pi(X+\xi):=X,$

for $X+\xi,Y+\eta\in\Gamma(TM\oplus T^{*}M)$.

Moreover, for any closed $3$-form $H\in\Omega^{3}(M)$, we can consider another
ECA structure on ${T\oplus T^{*}}$, denoted by

${(T\oplus T^{*})_{H}}:=(TM\oplus
T^{*}M,{\langle\,\cdot\,,\cdot\,\rangle},[\cdot,\cdot]_{H},\pi).$

Here, ${\langle\,\cdot\,,\cdot\,\rangle}$ and $\pi$ are as in Equation 5, and
$[\cdot,\cdot]_{H}$ denotes the _$H$ -twisted Dorfman bracket_ defined by:

(6)
$[X+\xi,Y+\eta]_{H}:=[X,Y]+\mathcal{L}_{X}\eta-\iota_{Y}\textrm{d}\xi+\iota_{Y}\iota_{X}H,\qquad
X+\xi,Y+\eta\in\Gamma(TM\oplus T^{*}M).$

Let $(E_{i}\to
M_{i},{\langle\,\cdot\,,\cdot\,\rangle}_{i},[\cdot,\cdot]_{i},\pi_{i})$,
$i=1,2$, be two exact Courant algebroids. Recall that a vector bundle
isomorphism is a pair $(F,f)$ where $f:M_{1}\to M_{2}$, $F:E_{1}\to E_{2}$ are
diffeomorphisms, and, for each $p\in M_{1}$, $F$ 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
$a,b\in\Gamma(E_{1})$, we have

  * •

$\langle Fa,Fb\rangle_{2}=\langle a,b\rangle_{1}$;

  * •

$[Fa,Fb]_{2}=F[a,b]_{1}$.

It follows from the second condition above that
$\textrm{d}f\circ\pi_{1}=\pi_{2}\circ F$, see for instance [KW07, Lemma 2.5]
for the detailed proof.
The next Proposition guarantees that any exact Courant algebroid is isomorphic
to one described in Example 2.1.

######  Proposition 2.2.

[GFS21, Proposition 2.10] Let $E$ be an exact Courant algebroid. Any isotropic
splitting $\sigma$ of (4) induces an isomorphism of exact Courant algebroids

$E\simeq_{\sigma}({T\oplus T^{*}})_{H}\,.$

Here $H\in\Omega^{3}(M)$ is the closed $3$-form given by

(7) $H(X,Y,Z)=2\,\langle[\sigma X,\sigma Y],\sigma Z\rangle,\qquad
X,Y,Z\in\Gamma(TM).$

Moreover, given another isotropic splitting $\tilde{\sigma}$, the
corresponding isomorphism satisfies

$E\simeq_{\tilde{\sigma}}({T\oplus T^{*}})_{H+\textrm{d}b}\,,$

for some $b\in\Omega^{2}(M)$.

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

######  Theorem 2.3.

[Šev17] The isomorphism classes of exact Courant algebroids over $M$ are in
one-to-one correspondence with
$H^{3}(M,\mathbb{R})/\operatorname{Diff}(M)=H^{3}(M,\mathbb{R})/\Gamma_{M}$,
where $\Gamma_{M}=\operatorname{Diff}(M)/\operatorname{Diff}_{0}(M)$ is the
mapping class group of $M$.

Given the correspondence in 2.2, 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
$b\in\Omega^{2}(M)$ may be viewed as a map $b:TM\to T^{*}M$ and thus it gives
rise to an endomorphism of the generalized tangent bundle which in matrix form
is represented by

$\begin{pmatrix}0&0\\\ b&0\end{pmatrix}:T\oplus T^{*}\longrightarrow T\oplus
T^{*}.$

We then set

$e^{b}:=\begin{pmatrix}\operatorname{Id}&0\\\
b&\operatorname{Id}\end{pmatrix}\in\operatorname{End}(T\oplus T^{*}),\qquad
b\in\Omega^{2}(M).$

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 $f\in{\rm Diff}(M)$, we can produce a bundle map of the
generalized tangent bundle covering $f$, as follows:

$\bar{f}:=\begin{pmatrix}\textrm{d}f&0\\\
0&(f^{-1})^{*}\end{pmatrix}\colon{T\oplus T^{*}}\to{T\oplus T^{*}}\,.$

Then, the group of automorphism of $({T\oplus T^{*}})_{H}$, and then of any
ECA, consists of bundle maps which are composition of maps introduced above.

######  Theorem 2.4.

[GFS21, Proposition 2.21] For any closed $3$-form $H\in\Omega^{3}(M)$, the
group of automorphisms of the exact Courant algebroid ${(T\oplus T^{*})_{H}}$
is given by:

$\operatorname{Aut}({(T\oplus
T^{*})_{H}})=\left\\{\left(\bar{f}e^{b},f\right):f\in\operatorname{Diff}(M),\quad
b\in\Omega^{2}(M),\quad f^{*}H=H-\textrm{d}b\right\\}.$

Moreover, the Lie algebra of $\operatorname{Aut}({(T\oplus T^{*})_{H}})$ is

$\mathfrak{aut}({(T\oplus
T^{*})_{H}})=\\{X+b\in\Gamma(TM)\oplus\Omega^{2}(M):\mathcal{L}_{X}H=-\textrm{d}b\\}.$

We refer the reader to [RT20] for a detailed study of the ILH Lie group
structure on $\operatorname{Aut}({(T\oplus T^{*})_{H}})$.
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.

######  Lemma 2.5.

Let $(E,{\langle\,\cdot\,,\cdot\,\rangle},[\cdot,\cdot],\pi)$ be an exact
Courant algebroid and let $c\in\mathbb{R}\backslash\\{0\\}$. Then, the data
$(c\,{\langle\,\cdot\,,\cdot\,\rangle},[\cdot,\cdot],\pi)$ defines a new exact
Courant algebroid structure on $E$.

###### Proof.

Trivially, $c{\langle\,\cdot\,,\cdot\,\rangle}$ 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 ${\langle\,\cdot\,,\cdot\,\rangle}$. As regards (ECA5), we can
notice that the $\pi^{*}$ in the right-hand-side involves a composition with
the isomorphism $\varphi_{{\langle\,\cdot\,,\cdot\,\rangle}}:E^{*}\simeq E$
induced by the neutral inner product. For
$c{\langle\,\cdot\,,\cdot\,\rangle}$, this isomorphism is simply

(8)
$\varphi_{c{\langle\,\cdot\,,\cdot\,\rangle}}=c^{-1}\varphi_{{\langle\,\cdot\,,\cdot\,\rangle}},$

which gives the claim. ∎

######  Definition 2.6.

Let $(E,{\langle\,\cdot\,,\cdot\,\rangle},[\cdot,\cdot],\pi)$ be an exact
Courant algebroid and $c\in\mathbb{R}\backslash\\{0\\}$. We define the _scaled
exact Courant algebroid_ $c\cdot E$ to be the one associated with the data
$(E,c\,{\langle\,\cdot\,,\cdot\,\rangle},[\cdot,\cdot],\pi)$.

We now study the effect of scaling on the $3$-form $H$ after choosing an
isotropic splitting:

######  Lemma 2.7.

Let $E$ be an ECA with isotropic splitting $\sigma$. Then, the isomorphism
$E\simeq_{\sigma}{(T\oplus T^{*})_{H}}$, $H\in\Omega^{3}(M)$ closed, induces
an isomorphism

$c\cdot E\simeq_{\sigma}({T\oplus T^{*}})_{cH}.$

###### Proof.

This follows immediately from 2.2 and (7). ∎

###  2.2. Generalized metrics

A _generalized metric_ on an ECA $E$ is an orthogonal endomorphism
$\mathcal{G}\in\mathsf{O}(E,{\langle\,\cdot\,,\cdot\,\rangle})$ such that the
bilinear form

$(a,b)\mapsto\langle\mathcal{G}a,b\rangle,\qquad a,b\in\Gamma(E),$

is symmetric and positive definite. We call the pair $(E,\mathcal{G})$ a
_metric Courant algebroid_. A Courant algebroid isomorphism
$(F,f)\colon(E_{1},\mathcal{G}_{1})\to(E_{2},\mathcal{G}_{2})$ is a _metric
Courant algebroid isometry_ (or simply _isometry_) if
$F\circ\mathcal{G}_{1}=\mathcal{G}_{2}\circ F$.

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 [GFS21, Section 2.3] for the
detailed description. Since $\mathcal{G}^{2}={\rm Id}$, we can consider the
eigenbundles $E_{\pm}$ of $E$ associated to $\pm 1$. It is not hard to see
that $\pi\colon E_{\pm}\to TM$ is an isomorphism of vector bundles. Then, we
can define

$\sigma_{\pm}:=(\pi_{|E_{\pm}})^{-1}\colon TM\to
E_{\pm}\,,\quad\tau_{\pm}(X,Y)=\langle\sigma_{\pm}X,\sigma_{\pm}Y\rangle\,,\quad
X,Y\in\Gamma(TM)\,.$

With these ingredients, we can consider the following splitting:

$\sigma=\sigma_{\pm}-\frac{1}{2}\pi^{*}\tau_{\pm}\,.$

It is easy to see that $\sigma$ is isotropic and it does not depend on the
choices of $\sigma_{\pm}$, see [GFS21, Lemma 2.29] for the proof. Moreover, we
have that

(9) $\tau_{+}(X,X)=\langle\mathcal{G}\sigma_{+}X,\sigma_{+}X\rangle>0\,,\quad
X\in\Gamma(TM)\,,X\neq 0\,,$

thus determining a Riemannian metric $g$ on $M$.

######  Proposition 2.8.

[GFS21, Proposition 2.38] Let $E$ be an ECA with isotropic splitting $\sigma$.
Under the isomorphism $E\simeq_{\sigma}{(T\oplus T^{*})_{H}}$, any generalized
metric on $E$ corresponds to a generalized metric on ${(T\oplus T^{*})_{H}}$
of the form

$\mathcal{G}(g,b):=e^{b}\begin{pmatrix}0&g^{-1}\\\ g&0\end{pmatrix}e^{-b},$

where $b\in\Omega^{2}(M)$ and $g$ is a Riemannian metric on $M$. Moreover, the
bundle map

(10) $\left(e^{-b},\operatorname{Id}_{M}\right)\colon\left({(T\oplus
T^{*})_{H}},\mathcal{G}(g,b)\right)\to\left(({T\oplus
T^{*}})_{H+\textrm{d}b},\mathcal{G}(g,0)\right),$

is an isometry.

######  Remark 2.9.

The isometry in (10) indicates that a generalized metric $\mathcal{G}$ on an
ECA $E$ with Ševera class $\alpha\in H^{3}(M,\mathbb{R})$ is equivalent to a
choice of Riemannian metric $g$ on $M$ and a _preferred representative_
$H\in\alpha$. This can be also be observed by considering the _preferred_
isotropic splitting induced by $\mathcal{G}$ and the corresponding isomorphism
$E\simeq{(T\oplus T^{*})_{H}}$ under which $\mathcal{G}$ corresponds to
$\mathcal{G}(g,0)$.

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

######  Proposition 2.10 ([GFS21, Proposition 2.41]).

The group of generalized isometries of the metric Courant algebroid
$({(T\oplus T^{*})_{H}},\mathcal{G}(g,0))$ is given by

$\mathsf{Iso}({(T\oplus
T^{*})_{H}},\mathcal{G}(g,0))=\left\\{\left(\bar{f},f\right):f\in\mathsf{Iso}(M,g),\quad
f^{*}H=H\right\\}\subset\operatorname{Aut}({(T\oplus T^{*})_{H}}).$

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

######  Lemma 2.11.

Let $(E,\mathcal{G})\simeq_{\sigma}({(T\oplus T^{*})_{H}},\mathcal{G}(g,b))$
be a metric ECA and $c>0$. Then, $\pm\mathcal{G}$ is a generalized metric on
$\pm c\cdot E$ and the isomorphism $\pm c\cdot E\simeq_{\sigma}({T\oplus
T^{*}})_{\pm cH}$ gives an isometry

$(\pm c\cdot E,\pm\mathcal{G})\simeq_{\sigma}(({T\oplus T^{*}})_{\pm
cH},\mathcal{G}(cg,\pm cb)).$

###### Proof.

The first claim is clear, since the bilinear form $\pm
c\langle\pm\mathcal{G}\cdot,\cdot\rangle=c\,\langle\mathcal{G}\cdot,\cdot\rangle$
is symmetric and positive definite. As regards the second one, we note that
the scaling on the Riemannian metric follows directly from (9). On the other
hand, considering $\sigma_{1}$ to be the preferred isotropic splitting induced
by $\mathcal{G}$, then $(\sigma_{1}-\sigma)(X)\in\ker\pi={\rm Im}(\pi^{*})$,
for any $X\in\Gamma(TM)$. Hence, we can write

(11) $b=(\pi^{*})^{-1}(\sigma_{1}-\sigma)\colon TM\to T^{*}M\,,$

where the fact that $b$ is a $2$-form is straightforward to check, using that
both $\sigma_{1}$ and $\sigma$ are isotropic. Then, the scaling on $b$ follows
from (8). ∎

######  Remark 2.12.

The isometry in the second claim in 2.11 can be also explicitly constructed.
Assuming $c>0$, we can use the isotropic splitting $\sigma$ to obtain the
isometry $(\pm c\cdot E,\mathcal{G})\simeq_{\sigma}(\pm c\cdot({T\oplus
T^{*}})_{H},\mathcal{G}(g,b))$. Then, one can easily see that

$\begin{pmatrix}{\rm Id}&0\\\ 0&\pm c{\rm Id}\end{pmatrix}\colon(\pm
c\cdot({T\oplus T^{*}})_{H},\mathcal{G}(g,b))\to(({T\oplus T^{*}})_{\pm
cH},\mathcal{G}(cg,\pm cb))$

is an isometry. Hence, composing the two isometries above, we obtain the
claim.

###  2.3. The generalized Ricci curvature

In this short Section, we will briefly discuss the explicit expression of the
generalized Ricci curvature in the case of $(({T\oplus
T^{*}})_{H},\mathcal{G}(g,0))$ which will be central in what follows. For a
detailed investigation of the general definition, we refer to [CSCW11],
[GF19], [GFS21], [ŠV17] and [ŠV20]. Before doing that, we recall that if
$(M,g)$ is a Riemannian manifold and $H\in\Omega^{3}(M)$, there exist unique
connections $\nabla^{\pm}$, called _Bismut connections_ , such that

$\nabla g=0\,,\quad gT^{\pm}=\pm H\,,$

where the tensor $T^{\pm}$ is the torsion of $\nabla^{\pm}$. In what follows,
we will abuse the notation denoting with

${\rm Ric}^{B}_{g,H}:={\rm Ric}_{g}-\frac{1}{4}H^{2}$

the symmetric part of the Ricci tensor of the Bismut connections
$\nabla^{\pm}$ associated to $g$ and $H$. The symmetric $(0,2)$-tensor $H^{2}$
is defined by $H^{2}(X,Y)=g(\iota_{X}H,\iota_{Y}H)$, for all
$X,Y\in\Gamma(TM)$.
Then, following the notation in [GFS21], the generalized metric
$\mathcal{G}(g,0)$ determines two eigenbundles as in Section 2.2, which, in
this particular case, takes the following form:

$E_{\pm}=\\{X\pm g(X):X\in\Gamma(TM)\\}\,.$

Now, using [GFS21, Definition 3.31] and [GFS21, Proposition 3.30], it is easy
to see that, for any $X\in\Gamma(TM)$,

(12) $\displaystyle\mathcal{R}c^{+}(X-g(X))=$ $\displaystyle\,g^{-1}{\rm
Ric}^{B}_{g,H}X-\frac{1}{2}g^{-1}\textrm{d}_{g}^{*}HX+{\rm
Ric}^{B}_{g,H}X-\frac{1}{2}\textrm{d}_{g}^{*}HX\,,$
$\displaystyle\mathcal{R}c^{-}(X+g(X))=$ $\displaystyle\,-g^{-1}{\rm
Ric}^{B}_{g,H}X-\frac{1}{2}g^{-1}\textrm{d}_{g}^{*}HX+{\rm
Ric}^{B}_{g,H}X+\frac{1}{2}\textrm{d}_{g}^{*}HX\,.$

$\mathcal{R}c(\mathcal{G})=\mathcal{R}c^{+}-\mathcal{R}c^{-}=\begin{pmatrix}g^{-1}{\rm
Ric}^{B}_{g,H}&\frac{1}{2}\,g^{\,-1}\,\textrm{d}^{*}_{g}Hg^{-1}\\\
-\frac{1}{2}\textrm{d}^{*}_{g}H&-{\rm Ric}^{B}_{g,H}g^{-1}\end{pmatrix}.$

In [ŠV17, Theorem 3], the authors prove the ${\rm Aut}(E)$-equivariance of the
generalized Ricci curvature which will be extensively used in the next
Sections.

###  2.4. The generalized Ricci flow

A one-parameter family $(\mathcal{G}(t))_{t\in[0,T)}$ of generalized metrics
on an ECA $E$ is a solution to the generalized Ricci flow if it satisfies

(13) $\mathcal{G}^{-1}\frac{\partial}{\partial
t}\mathcal{G}=-2{\mathcal{R}c}(\mathcal{G}).$

If we use the isotropic splitting $\sigma_{0}$ associated to the initial
metric $\mathcal{G}(0)$ to identify $E\simeq_{\sigma_{0}}({T\oplus
T^{*}})_{H_{0}}$, then $\mathcal{G}(t)$ corresponds to
$\mathcal{G}(g(t),b(t))$ for some one-parameter families of Riemannian metrics
$g(t)$ and two-forms $b(t)$ on $M$. Then, Equation (13) is equivalent to the
following coupled system:

(14) $\displaystyle\frac{\partial}{\partial t}g$ $\displaystyle=-2\,{\rm
Ric}_{g,H}^{B},$ $\displaystyle\frac{\partial}{\partial t}b$
$\displaystyle=-\textrm{d}^{*}_{g}H,$

where $H(t)=H_{0}+\textrm{d}b(t)$. Since $H_{0}$ is closed, it follows that
$H(t)$ evolves accordingly to the classical heat equation:

$\frac{\partial}{\partial t}H=\Delta_{g}H.$

We should emphasize that Equation 14 firstly appeared in [CFMP85] as the
_renormalization group flow_. We refer to [Str17] for a detailed explanation
of the relation between the renormalization group flow and generalized
Geometry.

Since the generalized Ricci curvature is ${\rm Aut}(E)$-equivariant, the PDE
system (13) 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 [GFS21, Theorem 5.6]
for details.

More generally, a one-parameter family $(\mathcal{G}(t))_{t\in[0,T)}$ of
generalized metrics on an ECA $E$ solves the _gauge-fixed generalized Ricci
flow_ if there exists a one-parameter family $(F_{t})_{t\in[0,T)}\subseteq{\rm
Aut}(E)$ so that $F_{t}\mathcal{G}(t)F_{t}^{-1}$ is a solution to the
generalized Ricci flow (13). Using again the isotropic splitting $\sigma_{0}$
induced by $\mathcal{G}(0)$ to obtain
$(E,\mathcal{G}(t))\simeq_{\sigma_{0}}(({T\oplus
T^{*}})_{H_{0}},\mathcal{G}(g(t),b(t)))$ as above, it can be seen that
$\mathcal{G}(t)$ is a solution of the gauge-fixed generalized Ricci flow if
only if there exists a one-parameter family of generalized vector fields
$X_{t}+B_{t}\in\mathfrak{aut}(({T\oplus T^{*}})_{H_{0}})$ such that
$(g(t),b(t))_{t\in[0,T)}$ is a solution of the following coupled system:

(15) $\displaystyle\frac{\partial}{\partial t}g=$ $\displaystyle\,-2\,{\rm
Ric}^{B}_{g,H}+\mathcal{L}_{X}g\,,$ $\displaystyle\frac{\partial}{\partial
t}b=$ $\displaystyle\,-\textrm{d}^{*}_{g}H-B+\mathcal{L}_{X}b\,.$

It follows that $H(t)$ will evolve by

(16) $\frac{\partial}{\partial t}H=\Delta_{g}H+\mathcal{L}_{X}H\,.$

##  3\. Solitons of the generalized Ricci flow

Motivated by 2.6 and by the effect of scaling on the $3$-form $H$ described in
2.11, we give the following new definition of solitons for the generalized
Ricci flow.

######  Definition 3.1.

A solution $(E,\mathcal{G}(t))_{t\in[0,T)}$ to the generalized Ricci flow (13)
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

$F_{t}:(c(t)\cdot E,\mathcal{G}(0))\longrightarrow(E,\mathcal{G}(t)).$

Differentiating the family $F_{t}$ at $t=0$ gives rise to a pair
$(X,\bm{D})\in\Gamma(TM)\times{\rm End}(E)$ satisfying the following
properties:

  1. (1)

$\bm{D}\in{\rm Der}([\cdot,\cdot])$, i.e.
$\bm{D}[a,b]=[\bm{D}a,b]+[a,\bm{D}b]$, for any $a,b\in\Gamma(E)$;

  2. (2)

$X{\langle\,\cdot\,,\cdot\,\rangle}=\langle({\bm{D}}+\lambda{\rm
Id})\cdot,\cdot\rangle+\langle\cdot,({\bm{D}}+\lambda{\rm Id})\cdot\rangle,$
where $c^{\prime}(0)=-2\lambda$;

  3. (3)

$[\mathcal{R}c(\mathcal{G}(0))+\bm{D},\mathcal{G}(0)]=0.$

As one can easily see, the dependence of (2) on the scalings is only through
$c^{\prime}(0)$. This motivates the following definition.

######  Definition 3.2.

Let $E$ be a ECA and $\lambda\in\mathbb{R}$. We say that
$(X,\bm{D})\in\Gamma(TM)\times{\rm End}(E)$ is a $\lambda$-derivation if Item
1 and Item 2 are satisfied. We denote by ${\rm Der}_{\lambda}([\cdot,\cdot])$
the set of all $\lambda$-derivations.

It can easily be observed that $0$-derivations are precisely the derivations
of the Dorfman bracket introduced in [Gua11]. Then, $\lambda$-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
$c\cdot({T\oplus T^{*}})_{H}$ and $({T\oplus T^{*}})_{H}$ and the set of
$\lambda$-derivations which will be useful in the next Sections.

######  Lemma 3.3.

Let $H\in\Omega^{3}(M)$ be closed and $c\in\mathbb{R}\backslash\\{0\\}$. Then,

${\rm Aut}(c\cdot({T\oplus T^{*}})_{H},({T\oplus
T^{*}})_{H})=\\{\bar{f}_{c}e^{b}:b\in\Omega^{2}(M),\quad f\in{\rm
Diff}(M),\quad f^{*}H=c(H-\textrm{d}b)\\}\,,$

where

$\bar{f}_{c}=\begin{pmatrix}\textrm{d}f&0\\\ 0&c(f^{-1})^{*}\end{pmatrix}\,.$

Moreover, if $\lambda\in\mathbb{R}$,

${\rm
Der}_{\lambda}([\cdot,\cdot])=\\{X+b\in\Gamma(TM)\oplus\Omega^{2}(M):\mathcal{L}_{X}H=-\textrm{d}b-2\lambda
H\\}\,.$

###### Proof.

Let $(F,f)\in{\rm Aut}(c\cdot({T\oplus T^{*}})_{H},({T\oplus T^{*}})_{H})$.
Since $F$ preserves the Dorfman bracket, we have that $\pi\circ
F=\textrm{d}f\circ\pi$. This naturally gives us that if

$F=e^{B}\begin{pmatrix}\textrm{d}f&0\\\ 0&h^{*}\end{pmatrix}e^{b}$

then $B=0$. On the other hand, the condition $\langle
F\cdot,F\cdot\rangle=c\langle\cdot,\cdot\rangle$ forces $b\in\Omega^{2}(M)$
and $\textrm{d}h=c\textrm{d}f^{-1}\,.$ Moreover, we have that, for any
$X+\xi,Y+\eta\in\Gamma({T\oplus T^{*}})$,

$\displaystyle\bar{f}_{c}\left[(\bar{f}_{c})^{-1}(X+\xi),(\bar{f}_{c})^{-1}(Y+\eta)\right]=$
$\displaystyle\,[X,Y]+\mathcal{L}_{X}\eta-\iota_{Y}\textrm{d}\xi+c\iota_{Y}\iota_{X}(f^{-1})^{*}H.$

Composing this with the usual action of $e^{b}$, we have that if $F\in{\rm
Aut}(c\cdot({T\oplus T^{*}})_{H},({T\oplus T^{*}})_{H})$ then

$F=\bar{f}_{c}e^{b}\,,\qquad f^{*}H=c(H-\textrm{d}b)\,.$

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
$c^{\prime}(0)=-2\lambda$ and a one parameter family $(F_{t},f_{t})\in{\rm
Aut}(c(t)\cdot({T\oplus T^{*}})_{H},({T\oplus T^{*}})_{H}).$ Differentiating
$(F_{t},f_{t})$ at $t=0$ gives $X+b\in\Gamma(TM)\oplus\Omega^{2}(M)$ such that

$\mathcal{L}_{X}H=-\textrm{d}b-2\lambda H\,.$

Viceversa, given $X+b\in\Gamma(TM)\oplus\Omega^{2}(M)$ such that
$\mathcal{L}_{X}H=-\textrm{d}b-2\lambda H$, 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 $c^{\prime}(0)=-2\lambda$. Hence, we can
define

$\bar{b}_{t}=\int_{0}^{t}c(s)f_{s}^{*}b_{s}ds\,,$

where $b_{s}\in\Omega^{2}(M)$ such that
$\mathcal{L}_{X_{s}}H=-\textrm{d}b_{s}+\frac{c^{\prime}(s)}{c(s)}H$. Then, we
have that

$\displaystyle\textrm{d}\bar{b}_{t}=$
$\displaystyle\,-\int_{0}^{t}\frac{1}{c(s)}f_{s}^{*}\left(-\frac{c^{\prime}(s)}{c(s)}H+\mathcal{L}_{X_{s}}H\right)\textrm{d}s=-\int_{0}^{t}\frac{\textrm{d}}{\textrm{d}s}\left(\frac{1}{c(s)}f_{s}^{*}H\right)\textrm{d}s=-\frac{1}{c(t)}f_{t}^{*}H+H\,.$

Then, $F=\bar{f}_{t,c(t)}e^{\bar{b}_{t}}\in{\rm Aut}(c(t)\cdot({T\oplus
T^{*}})_{H},({T\oplus T^{*}})_{H})$, as we wanted. ∎

3.1 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.

######  Proposition 3.4.

Let $(E,\mathcal{G}(t))$ be a generalized Ricci soliton, and consider the
time-independent isometry
$(E,\mathcal{G}(t))\simeq_{\sigma_{0}}\left(({T\oplus
T^{*}})_{H_{0}},\,\mathcal{G}(g(t),b(t))\right)$ induced by the isotropic
splitting $\sigma_{0}$ associated to $\mathcal{G}(0)$. Then, $g_{0}:=g(0)$ and
$H_{0}$ satisfy

(17) $\displaystyle\begin{split}{\rm
Ric}_{g_{0},H_{0}}^{B}&=\lambda\,g_{0}+\tfrac{1}{2}\,\mathcal{L}_{X}g_{0},\\\
\Delta_{g_{0}}H_{0}&=-2\,\lambda\,H_{0}-{\mathcal{L}}_{X}H_{0},\end{split}$

where $\lambda=-\tfrac{1}{2}c^{\prime}(0)$ and $c(t)>0$ are the scalings from
3.1. Conversely, if $\left(({T\oplus
T^{*}})_{H_{0}},\,\mathcal{G}(g_{0},0)\right)$ satisfies (17), then there
exists a soliton solution $\mathcal{G}(g(t),b(t))$ to (13) on $({T\oplus
T^{*}})_{H_{0}}$ with $\mathcal{G}(g(0),b(0))=\mathcal{G}(g_{0},0).$

###### Proof.

By 2.11, the isomorphism associated to $\sigma_{0}$ also induces time-
independent isometries

$\displaystyle(c(t)\cdot E,\mathcal{G}(0))$
$\displaystyle\simeq_{\sigma_{0}}\left(({T\oplus
T^{*}})_{c(t)H_{0}},\,\mathcal{G}(c(t)g_{0},0)\right).$

It follows that $c(t)g_{0}$ and $c(t)H_{0}$ must solve the gauged-fixed
equations (15) and (16).

Thus,

$\displaystyle c^{\prime}g_{0}$ $\displaystyle=-2\,{\rm
Ric}^{B}_{g_{0},H_{0}}+c\,\mathcal{L}_{X}g_{0},$ $\displaystyle
c^{\prime}H_{0}$
$\displaystyle=\Delta_{g_{0}}H_{0}+c\,{\mathcal{L}}_{X}H_{0}.$

Evaluating at $t=0$ and using $c(0)=1$, $c^{\prime}(0)=-2\lambda$, (17)
follows. ∎

######  Remark 3.5.

When $H_{0}=0$, 3.4 implies that 3.1 is equivalent to the classical definition
of Ricci soliton for the Ricci flow.

On the other hand, if $c(t)\equiv 1$ is constant, we recover the definition of
soliton solution given in [GFS21, $\S$4.4], which only allows for steady
solitons (i.e. $\lambda=0$). Recently, in [PR24b, Definition 2.1], a
definition of generalized Ricci soliton, equivalent to 3.1, was given.

Moreover, in the non-steady case, the second equation in (17), together with
$\textrm{d}H_{0}=0$ 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.

######  Example 3.6.

Let us consider $M_{0}=H_{3}:={\rm Heis}(3,\mathbb{R})$ the $3$-dimensional
Heisenberg group. It is well known that on $M_{0}$ we can choose coordinates
$\\{x,y,z\\}$ so that

$e_{1}=\frac{\partial}{\partial x}\,,\quad e_{2}=\frac{\partial}{\partial
y}+x\frac{\partial}{\partial z}\,,\quad e_{3}=\frac{\partial}{\partial z}$

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:

$e^{1}=\textrm{d}x\,,\quad e^{2}=\textrm{d}y\,,\quad
e^{3}=\textrm{d}z-x\textrm{d}y\,.$

We next consider the following one parameter family of diffeomorphisms of
$M_{0}$:

$\varphi_{t}(x,y,z)=((1+4t)^{\frac{1}{4}}x,(1+4t)^{\frac{1}{4}}y,(1+4t)^{\frac{1}{2}}z)\,,\quad(x,y,z)\in
M_{0}\,,\quad t\geq 0\,.$

It is easy to see that $\varphi_{t}$ is actually an automorphism of $M_{0}$
viewed as a Lie group. Another easy computation allows us to state that

$\textrm{d}\varphi_{t}=(1+4t)^{\frac{1}{4}}e^{1}+(1+4t)^{\frac{1}{4}}e^{2}+(1+4t)^{\frac{1}{2}}e^{3}\,,\quad
t\geq 0\,.$

Now we consider the closed left invariant $3$-form
$H=\textrm{d}x\wedge\textrm{d}y\wedge\textrm{d}z$ and observe that $H$ is
harmonic with respect to any metric on $M_{0}$. Then, one can easily see that

$\varphi_{t}^{*}H=(1+4t)H\,,\quad t\geq 0\,.$

Then, denoting $c(t)=1+4t>0$ and comparing with 3.3, we are in the position to
promote $\varphi_{t}$ to

(18) $F_{t}=\bar{\varphi}_{t,c(t)}\in{\rm Aut}(c(t)\cdot({T\oplus
T^{*}})_{H},({T\oplus T^{*}})_{H})\,,\quad t\geq 0\,.$

Moreover, we easily compute the vector field generated by $\varphi_{t}$:

$X=\frac{\textrm{d}}{\textrm{d}t}\Big{|}_{t=0}\varphi_{t}=x\frac{\partial}{\partial
x}+y\frac{\partial}{\partial y}+2z\frac{\partial}{\partial
z}=xe_{1}+ye_{2}+(2z-xy)e_{3}\,,$

which, in particular, satisfies the following bracket relations:

$[X,e_{1}]=-e_{1}\,,\quad[X,e_{2}]=-e_{2}\,,\quad[X,e_{3}]=-2e_{3}\,.$

Finally, $X\in{\rm Der}_{-2}([\cdot,\cdot])$ satisfies

$\mathcal{L}_{X}H=4H$

which is precisely the second equation in (17) with
$-2\lambda=c^{\prime}(0)=4$, since $H$ is harmonic with respect to any
Riemannian metric on $M_{0}$. Now, we consider the following Riemannian
metric:

$g=e^{1}\odot e^{1}+e^{2}\odot e^{2}+e^{3}\odot
e^{3}=\textrm{d}x\odot\textrm{d}x+(1+x^{2})\textrm{d}y\odot\textrm{d}y+\textrm{d}z\odot\textrm{d}z-2x\textrm{d}y\odot\textrm{d}z\,,$

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

${\rm Ric}^{B}_{g,H}=-(e^{1}\odot e^{1}+e^{2}\odot e^{2})\mbox{ and
}\mathcal{L}_{X}g=2(e^{1}\odot e^{1}+e^{2}\odot e^{2}+2e^{3}\odot e^{3})\,.$

Then,

${\rm Ric}^{B}_{g,H}=-2g+\frac{1}{2}\mathcal{L}_{X}g$

giving precisely the first equation in (17). Now, we can use 3.4 to infer that
$(({T\oplus T^{*}})_{H},\mathcal{G}(g,0))$ 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 3.1. Indeed, considering $F_{t}$
as in (18), we have

$F_{t}\mathcal{G}(g,0)F_{t}^{-1}=\mathcal{G}((1+4t)(\varphi_{t}^{-1})^{*}g,0)$

but

$g(t)=(1+4t)(\varphi_{t}^{-1})^{*}g=(1+4t)^{\frac{1}{2}}e^{1}\odot
e^{1}+(1+4t)^{\frac{1}{2}}e^{2}\odot e^{2}+e^{3}\odot e^{3}$

is precisely the solution of the generalized Ricci flow starting from $g$, see
[Par21, Section 6].

In the same fashion and with a bit more effort, we can construct examples of
expanding solitons for the generalized Ricci flow on
$M_{k}:=H_{3}\times\mathbb{R}^{k}$, for any $k\geq 1$. On $M_{k}$, we consider
the following left-invariant frame
$\\{e_{1},e_{2},e_{3},e_{4},\ldots,e_{k+3}\\}$ of the Lie algebra of
$H_{3}\times\mathbb{R}^{k}$ where $e_{1},e_{2},e_{3}$ are the same defined
above while $e_{3+j}=\frac{\partial}{\partial x_{3+j}}$ where $x_{3+j}$ is the
standard coordinate on the $j$-th copy of $\mathbb{R}$ in the abelian factor
$\mathbb{R}^{k}$.

Then, it is sufficient to consider $c(t)=1+4t$ as scaling,

$\varphi_{t}(x,y,z,x_{4},\ldots,x_{3+k})=((1+4t)^{\frac{1}{4}}x,(1+4t)^{\frac{1}{4}}y,(1+4t)^{\frac{1}{2}}z,(1+4t)^{\frac{1}{2}}x_{4},\ldots,(1+4t)^{\frac{1}{2}}x_{3+k})\,,$

for all $(x,y,z,x_{4},\ldots,x_{3+k})\in M_{k}$, as the one parameter family
of diffeomorphisms of $M_{k}$ , $H=e^{1}\wedge e^{2}\wedge e^{3}$ and finally
the standard left-invariant metric

$g=\sum_{i=1}^{k+3}e^{i}\odot e^{i}\,,$

with respect to the coframe $\\{e^{1},\ldots,e^{3+k}\\}$ dual with respect to
$\\{e_{1},\ldots,e_{k+3}\\}$ and obtain an expanding soliton of the
generalized Ricci flow on $M_{k}$.

##  4\. Homogeneous generalized geometry

In this Section we specialize our study to the case where the underlying
manifold $M=\mathsf{G}$ is a simply-connected Lie group. We will denote by $e$
the identity of $\mathsf{G}$ and $\mathfrak{g}=T_{e}\mathsf{G}$ its Lie
algebra. We also ask the ECA to be compatible with the algebraic structure in
the following sense:

######  Definition 4.1.

We say that a metric ECA $(E\to\mathsf{G},\mathcal{G})$ is _left-invariant_ if
the action of $\mathsf{G}$ on itself by left-translations lifts to an action
on $(E,\mathcal{G})$ by isometries (as defined in $\S$2.2).

The action of $\mathsf{G}$ on $E$ in 4.1 is a particular instance of a wider
class of Lie group actions on ECAs called _lifted_ or _extended_ actions, see
[BH15, Definition 2.4] and [BCG07, Definition 2.6].
The action of $\mathsf{G}$ on a left-invariant metric ECA allows to consider
$\mathsf{G}$-equivariant isotropic splittings. The choice of such isotropic
splittings, using 2.8, gives rise to left-invariant classical data as the
following Proposition highlights.

######  Proposition 4.2.

Let $(E\to\mathsf{G},\mathcal{G})$ be a left-invariant metric ECA. Then, any
$\mathsf{G}$-equivariant isotropic splitting $\sigma\colon T\mathsf{G}\to E$
of (4) induces a $\mathsf{G}$-equivariant isometry

$(E,\mathcal{G})\simeq_{\sigma}({(T\oplus T^{*})_{H}},\mathcal{G}(g,b)),$

where the tensors $g$, $H$, $b$ on $M$ are left-invariant and $H$ depends only
on $\sigma$ and not on $\mathcal{G}$.

###### Proof.

If $\sigma$ is the isotropic splitting given by $\mathcal{G}$, firstly, we
note that, by 2.8, $(E,\mathcal{G})$ is isometric to $({(T\oplus
T^{*})_{H}},\mathcal{G}(g,0))$, for some closed $3$-form $H$ and Riemannian
metric $g$. Using this isomorphism, we can define an action of $\mathsf{G}$ on
${(T\oplus T^{*})_{H}}$, and by definition this action will be by isometries
of $\mathcal{G}(g,0)$. By 2.10, $H$ and $g$ must be left-invariant. It remains
to prove the invariance of $b$. As in (11), the $2$-form $b$ can be expressed
as

$b=(\pi^{*})^{-1}(\sigma_{1}-\sigma)\,,$

where $\sigma_{1}$ is the isotropic splitting induced by $\mathcal{G}$, which
is $\mathsf{G}$-equivariant. Then, the claim follows directly from the fact
that, for any $(F,f)\in{\rm Aut}(E)$, we have that
$F\circ\pi^{*}=\pi^{*}\circ(f^{-1})^{*}$ and from the expression of $b$ above.
∎

######  Remark 4.3.

As said, the preferred isotropic splitting induced by $\mathcal{G}$ is
$\mathsf{G}$-equivariant. Thus, any homogeneous metric ECA is isometric to
$({(T\oplus T^{*})_{H}},\mathcal{G}(g,0))$ for a left-invariant Riemannian
metric $g$, and a closed left-invariant $3$-form
$H\in\Omega^{3}(M)^{\mathsf{G}}$.

Notice that as a vector bundle any left-invariant ECA is trivial, since after
choosing an isotropic splitting we get an isomorphism with ${(T\oplus
T^{*})_{H}}$, where

(19) $T\mathsf{G}\oplus
T^{*}\mathsf{G}\simeq(\mathfrak{g}\oplus\mathfrak{g}^{*})\times\mathsf{G}$

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

$(\mathfrak{g}\oplus\mathfrak{g}^{*})_{H}:={(T\oplus T^{*})_{H}},$

where
$H\in\Omega^{3}(\mathsf{G})^{\mathsf{G}}\simeq\Lambda^{3}\mathfrak{g}^{*}$ is
a closed, left-invariant $3$-form on $\mathsf{G}$.

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

Let $E\to\mathsf{G}$ be a left-invariant ECA, and let us fix a background
left-invariant generalized metric $\overline{\mathcal{G}}$ on $E$. The
preferred isotropic splitting $\bar{\sigma}:T\mathsf{G}\to E$ induced by
$\overline{\mathcal{G}}$ yields an isometry

$(E\to\mathsf{G},\overline{\mathcal{G}})\simeq_{\bar{\sigma}}((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bar{H}},\mathcal{G}(\bar{g},0)),$

for some left-invariant metric $\bar{g}$ and closed $3$-form $\bar{H}$ on
$\mathsf{G}$. By abuse of notation we write
$\overline{\mathcal{G}}=\mathcal{G}(\bar{g},0)$. Let us set

$\displaystyle\mathcal{M}^{\mathsf{G}}$
$\displaystyle:=\\{\mathcal{G}:\mathcal{G}\hbox{ is a left-invariant
generalized metric on }(\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bar{H}}\\}$
$\displaystyle\simeq\\{\mathcal{G}(g,b):g\hbox{ left-invariant metric on
}\mathsf{G},b\in\Omega^{2}(\mathsf{G})^{\mathsf{G}}\\}$

where the last identification is due to 4.2. Next, we consider the following
Lie subgroup of
$\mathsf{O}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$:

(20) $\mathsf{L}:=\left\\{e^{b}\cdot\begin{pmatrix}h&0\\\
0&(h^{-1})^{*}\end{pmatrix}:h\in\mathsf{GL}(\mathfrak{g}),\,\,b\in\Lambda^{2}\mathfrak{g}^{*}\right\\},\qquad
e^{b}:=\begin{pmatrix}\operatorname{Id}&0\\\
b&\operatorname{Id}\end{pmatrix}.$

As an abstract Lie group, $\mathsf{L}$ is isomorphic to the semi-direct
product $\Lambda^{2}\mathfrak{g}^{*}\rtimes\mathsf{GL}(\mathfrak{g})$.
Moreover, its Lie algebra is

$\mathfrak{l}:=\left\\{\begin{pmatrix}A&0\\\
\alpha&-A^{*}\end{pmatrix}:A\in\mathfrak{gl(g)},\,\,\alpha\in\Lambda^{2}\mathfrak{g}^{*}\right\\}.$

The standard representation of
$\mathsf{O}(\mathfrak{g}\oplus\mathfrak{g}^{*},\left\langle\cdot,\cdot\right\rangle)$
on $\mathfrak{g}\oplus\mathfrak{g}^{*}$ gives rise to a representation of
$\mathsf{L}$ on $\mathfrak{g}\oplus\mathfrak{g}^{*}$, and this extends in a
standard way to an action of $\mathsf{L}$ on all tensor powers of
$\mathfrak{g}\oplus\mathfrak{g}^{*}$. In particular, $\mathsf{L}$ acts on
generalized metrics and Dorfman brackets via the following formulas:

$\ell\cdot\mathcal{G}:=\ell\circ\mathcal{G}\circ\ell^{-1},\qquad\ell\cdot[\cdot,\cdot]:=\ell[\ell^{-1}\cdot,\ell^{-1}\cdot].$

Recall that we also have the ‘change of basis’ actions of
$\mathsf{GL}(\mathfrak{g})$ on the spaces of brackets and $3$-forms:

$h\cdot\mu(\cdot,\cdot)=h\mu(h^{-1}\cdot,h^{-1}\cdot),\qquad
h\cdot\omega(\cdot,\cdot,\cdot)=\omega(h^{-1}\cdot,h^{-1}\cdot,h^{-1}\cdot),\qquad
h\in\mathsf{GL}(\mathfrak{g}),$

for $\mu\in\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g}$ and
$\omega\in\Lambda^{3}\mathfrak{g}^{*}$.
The next Proposition allows to present the space of left-invariant metric on
$E$ as a homogeneous space.

######  Proposition 4.4.

The action of $\mathsf{L}$ on the space of left-invariant generalized metrics
$\mathcal{M}^{\mathsf{G}}$ is transitive.

###### Proof.

For any $\ell\simeq(b,h)\in\mathsf{L}$, its action on the background metric is
given by

$\displaystyle\ell\cdot\overline{\mathcal{G}}$
$\displaystyle=e^{b}\begin{pmatrix}h&0\\\
0&(h^{-1})^{*}\end{pmatrix}\begin{pmatrix}0&\bar{g}^{-1}\\\
\bar{g}&0\end{pmatrix}\begin{pmatrix}h^{-1}&0\\\
0&h^{*}\end{pmatrix}e^{-b}=e^{b}\begin{pmatrix}0&g^{-1}\\\
g&0\end{pmatrix}e^{-b},$

where

$g=(h^{-1})^{*}\bar{g}h^{-1}=h\cdot\bar{g}(\cdot,\cdot).$

Since $\mathsf{GL}(\mathfrak{g})$ acts transitively on left-invariant metrics,
the claim now follows from 4.2. ∎

###  4.2. 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 [Par21]. Then, we will focus on describing the effect of the
$\mathsf{L}$-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
$\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$
of all skew-symmetric bilinear maps

${\bm{\mu}}:(\mathfrak{g}\oplus\mathfrak{g}^{*})\times(\mathfrak{g}\oplus\mathfrak{g}^{*})\to(\mathfrak{g}\oplus\mathfrak{g}^{*}).$

Using 2.2, we know that choosing an isotropic splitting $\sigma$ gives rise to
an isomorphism $E\simeq_{\sigma}(\mathfrak{g}\oplus\mathfrak{g}^{*})_{H}$.
Then, from left-invariance and (6), it follows that the Dorfman bracket of $E$
gets identified with
$[\cdot,\cdot]_{H}\in\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$
given by

(21)
$[X+\xi,Y+\eta]_{H}=[X,Y]_{\mathfrak{g}}-\eta\circ\operatorname{ad}_{\mathfrak{g}}(X)+\xi\circ\operatorname{ad}_{\mathfrak{g}}(Y)+\iota_{Y}\iota_{X}H,$

where
$\operatorname{ad}_{\mathfrak{g}}(X)=[X,\cdot]_{\mathfrak{g}}:\mathfrak{g}\to\mathfrak{g}$
and $H\in\Lambda^{3}\mathfrak{g}^{*}$ is closed. It is not hard to see that
$\langle[\cdot,\cdot]_{H},\cdot\rangle$ is totally skew-symmetric. This
motivates the following.

######  Definition 4.5.

The _space of Dorfman brackets_ is the algebraic subset of
$\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$
given by

$\mathcal{D}:=\\{{\bm{\mu}}\in\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})\,\,:\,\,\langle{\bm{\mu}}(\cdot,\cdot),\cdot\rangle\in\Lambda^{3}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\,,\,\,{\bm{\mu}}(\mathfrak{g}^{*},\mathfrak{g}^{*})=0,\,\,\mathcal{J}({\bm{\mu}})=0\\}.$

Here,
$\mathcal{J}({\bm{\mu}})(a,b,c):={\bm{\mu}}({\bm{\mu}}(a,b),c)+{\bm{\mu}}({\bm{\mu}}(b,c),a)+{\bm{\mu}}({\bm{\mu}}(c,a),b)=0$
is the Jacobi identity.

We can show that all elements in $\mathcal{D}$ can be written as in (21) for
suitable choices of the $3$-form $H$ and the Lie bracket on $\mathfrak{g}$.

######  Lemma 4.6.

We have that ${\bm{\mu}}\in\mathcal{D}$ if and only if there exists a Lie
bracket $\mu$ on $\mathfrak{g}$ and a closed $3$-form
$H\in\Lambda^{3}\mathfrak{g}^{*}$ such that, for any
$X+\xi,Y+\eta\in\mathfrak{g}\oplus\mathfrak{g}^{*}$,

(22)
${\bm{\mu}}(X+\xi,Y+\eta)=\mu(X,Y)-\eta\circ\mu_{X}+\xi\circ\mu_{Y}+\iota_{Y}\iota_{X}H,\qquad\mu_{X}:=\mu(X,\cdot).$

###### Proof.

Sufficiency was observed above. Regarding necessity, for each
$X,Y,Z\in\mathfrak{g}$, we set

$\mu(X,Y):=\pi\circ{\bm{\mu}}(X,Y),\qquad
H(X,Y,Z):=2\,\langle{\bm{\mu}}(X,Y),Z\rangle.$

The linear conditions on ${\bm{\mu}}$ imply that, for every
$\xi\in\mathfrak{g}^{*}$, we have ${\bm{\mu}}(\xi,Y)\in\mathfrak{g}^{*}$.
Thus,

${\bm{\mu}}(\xi,Y)(X)=2\,\langle{\bm{\mu}}(\xi,Y),X\rangle=2\,\langle{\bm{\mu}}(Y,X),\xi\rangle=\xi\mu(Y,X)\,=\xi\circ\mu_{Y}(X).$

By skew-symmetry, ${\bm{\mu}}(X,\eta)=-\eta\circ\mu_{X}$, and from this (22)
follows. A simple computation shows that the Jacobi identity for ${\bm{\mu}}$
implies the Jacobi identity for $\mu$ and that $\textrm{d}_{\mu}H=0$. ∎

Moreover, one can notice that the following conditions hold for brackets in
$\mathcal{D}$:

(23)
${\bm{\mu}}(\mathfrak{g},\mathfrak{g})\subset\mathfrak{g}\oplus\mathfrak{g}^{*},\qquad{\bm{\mu}}(\mathfrak{g},\mathfrak{g}^{*})\subset\mathfrak{g}^{*},\qquad{\bm{\mu}}(\mathfrak{g}^{*},\mathfrak{g}^{*})=0.$

That is, $(\mathfrak{g}\oplus\mathfrak{g}^{*},{\bm{\mu}})$ is a Lie algebra
which is a central extension of $(\mathfrak{g},\mu)$. We thus have that the
space of Dorfman brackets is contained in the linear subspace:

(24)
$V_{\mathcal{D}}:=\left\\{{\bm{\mu}}\in\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})\,:\,{\bm{\mu}}\hbox{
satisfies (\ref{eqn_centralext}) and
}\langle{\bm{\mu}}(\cdot,\cdot),\cdot\rangle\in\Lambda^{3}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\right\\}.$

We then have two projections
$(\cdot)_{\mathfrak{g}}:\mathcal{D}\to\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g}$
and $(\cdot)_{\Lambda^{3}}:\mathcal{D}\to\Lambda^{3}\mathfrak{g}^{*}$,

(25)
${\bm{\mu}}\mapsto{\bm{\mu}}_{\mathfrak{g}}=\mu\in\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g},\qquad{\bm{\mu}}\mapsto{\bm{\mu}}_{\Lambda^{3}}=H\in\Lambda^{3}\mathfrak{g}^{*},$

where here $\mu$ is simply the Lie bracket of $\mathfrak{g}$. Furthermore, we
notice that, by (22), these projections completely determine a Dorfman bracket
${\bm{\mu}}\in\mathcal{D}$.

Let us first study how the action of $\mathsf{L}$ relates to the projections
in (25):

######  Lemma 4.7.

Let ${\bm{\mu}}\in\mathcal{D}$ with $\mu:={\bm{\mu}}_{\mathfrak{g}}$,
$H:={\bm{\mu}}_{\Lambda^{3}}$. Then, for any
$\ell=\bar{h}\,e^{\alpha}\in\mathsf{L}$, we have

(26)
$(\ell\cdot{\bm{\mu}})_{\mathfrak{g}}=h\cdot\mu,\qquad(\ell\cdot{\bm{\mu}})_{\Lambda^{3}}=h\cdot(H-\textrm{d}_{\mu}\alpha).$

###### Proof.

First, we compute for $X+\xi,Y+\eta\in\mathfrak{g}\oplus\mathfrak{g}^{*}$:

$\displaystyle(e^{\alpha}\cdot{\bm{\mu}})(X+\xi,Y+\eta)$
$\displaystyle=e^{\alpha}{\bm{\mu}}(X+\xi-i_{X}\alpha,\,Y+\eta-i_{Y}\alpha)$
$\displaystyle=e^{\alpha}\left({\bm{\mu}}(X+\xi,Y+\eta)+\alpha(Y,\mu(X,\cdot))-\alpha(X,\mu(Y,\cdot))\right)$
$\displaystyle={\bm{\mu}}(X+\xi,Y+\eta)-i_{Y}i_{X}(\textrm{d}_{\mu}\alpha).$

Similarly,

$\displaystyle\left(\begin{pmatrix}h&0\\\
0&(h^{-1})^{*}\end{pmatrix}\cdot{\bm{\mu}}\right)(X+\xi,Y+\eta)$
$\displaystyle=\begin{pmatrix}h&0\\\
0&(h^{-1})^{*}\end{pmatrix}{\bm{\mu}}(h^{-1}X+\xi\circ h,h^{-1}Y+\eta\circ h)$
$\displaystyle=(h\cdot\mu)(X,Y)-\eta\circ(h\cdot\mu)_{X}+\xi\circ(h\cdot\mu)+i_{Y}i_{X}(h\cdot
H).$

The result follows from composing the two actions. ∎

Given the actions of $\mathsf{L}$ on Dorfman brackets on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$, and of $\mathsf{GL}(\mathfrak{g})$ on
brackets and $3$-forms on $\mathfrak{g}$, we denote the corresponding Lie
algebra representations (or ‘infinitesimal actions’) by

$\Theta:\mathfrak{l}\to\operatorname{End}(\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})),\qquad\theta:\mathfrak{gl(g)}\to\operatorname{End}(\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g}),\qquad\rho:\mathfrak{gl(g)}\to\operatorname{End}(\Lambda^{3}\mathfrak{g}^{*}).$

More precisely, they are given by

$\displaystyle\Theta(A){\bm{\mu}}(\cdot,\cdot)$
$\displaystyle=A{\bm{\mu}}(\cdot,\cdot)-{\bm{\mu}}(A\cdot,\cdot)-{\bm{\mu}}(\cdot,A\cdot),\qquad
A\in\mathfrak{gl}(\mathfrak{g}\oplus\mathfrak{g}^{*}),$
$\displaystyle\theta(A)\mu(\cdot,\cdot)$
$\displaystyle=A\mu(\cdot,\cdot)-\mu(A\cdot,\cdot)-\mu(\cdot,A\cdot),\qquad
A\in\mathfrak{gl}(\mathfrak{g}),$
$\displaystyle\rho(A)\omega(\cdot,\cdot,\cdot)$
$\displaystyle=-\omega(A\cdot,\cdot,\cdot)-\omega(\cdot,A\cdot,\cdot)-\omega(\cdot,\cdot,A\cdot),\qquad
A\in\mathfrak{gl}(\mathfrak{g}).$

From 4.7 we immediately deduce:

######  Corollary 4.8.

For any
$L\simeq(\alpha,A)\in\mathfrak{l}\simeq\Lambda^{2}\mathfrak{g}^{*}\rtimes\mathfrak{gl(g)}$,
we have

$(\Theta(L){\bm{\mu}})_{\mathfrak{g}}=\theta(A)\mu,\qquad(\Theta(L){\bm{\mu}})_{\Lambda^{3}}=\rho(A)H-\textrm{d}_{\mu}\alpha.$

Since we will be moving the Dorfman and Lie brackets on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$ and $\mathfrak{g}$ respectively, it is
convenient to introduce the following notation: given ${\bm{\mu}}$ a Dorfman
bracket as in (22), we denote by

$(\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}}:=(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle},{\bm{\mu}},\pi)$

the left-invariant ECA structure given by the data
$({\langle\,\cdot\,,\cdot\,\rangle},{\bm{\mu}},\pi)$, where
${\langle\,\cdot\,,\cdot\,\rangle}$ and $\pi$ are defined in (5). Notice that
in order for (ECA2) to be satisfied, we must necessarily have that the Lie
bracket on $\mathfrak{g}$ is $\mu:={\bm{\mu}}_{\mathfrak{g}}$.

######  Proposition 4.9.

Let $\ell=\bar{h}\,e^{\alpha}\in\mathsf{L}$, for
$h\in\mathsf{GL}(\mathfrak{g})$ and $\alpha\in\Lambda^{2}\mathfrak{g}^{*}$.
Then, the bundle map

$(\ell,h):\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},(\ell^{-1})\cdot\mathcal{G}\right)\to\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\ell\cdot{\bm{\mu}}},\mathcal{G}\right)$

is an isometry. Here ${\bm{\mu}}_{\mathfrak{g}}=[\cdot,\cdot]_{\mathfrak{g}}$
is the original Lie bracket of $\mathfrak{g}$ and
${\bm{\mu}}_{\Lambda^{3}}=H$.

###### Proof.

We first prove the claim for $\ell=\bar{h}$ covering
$h\in\mathsf{GL}(\mathfrak{g})$. By definition of the action, the map
$h:(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})\to(\mathfrak{g},h\cdot[\cdot,\cdot]_{\mathfrak{g}})$
is a Lie algebra isomorphism. Set $\mu:=h\cdot[\cdot,\cdot]_{\mathfrak{g}}$,
let $\mathsf{G},\mathsf{G}_{\mu}$ be respectively the simply-connected Lie
groups with Lie algebras $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ and
$(\mathfrak{g},\mu)$. Then, there exists a Lie group isomorphism
$\varphi:\mathsf{G}\to\mathsf{G}_{\mu}$ for which $\textrm{d}_{e}\varphi=h$,
where $e\in\mathsf{G}$ is the identity. The corresponding bundle map
$\bar{\varphi}:T\mathsf{G}\oplus T^{*}\mathsf{G}\to T\mathsf{G}_{\mu}\oplus
T^{*}\mathsf{G}_{\mu}$ covering $\varphi$ clearly maps left-invariant sections
to left-invariant sections, and since the trivialization (19) is done via
left-invariant sections, it follows that $\bar{\varphi}$ may be represented by
an element of $\mathsf{GL}(\mathfrak{g}\oplus\mathfrak{g}^{*})$. The latter is
precisely $\bar{h}$, and the pair $(\bar{h},h)$ is the infinitesimal data
corresponding to the ECA isomorphism

$\bar{\varphi}:(\mathfrak{g}\oplus\mathfrak{g}^{*})_{H}\to(\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bar{h}\cdot{\bm{\mu}}}.$

Indeed, $\bar{h}$ preserves the Dorfman bracket by definition of the action,
and it preserves the neutral inner product because
$\bar{h}\in\mathsf{O}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$.
Finally,

$\bar{h}\cdot(\bar{h}^{-1}\cdot\mathcal{G})=\mathcal{G},$

thus we have an isometry.

The proof for $\ell=e^{\alpha}$ covering
$\operatorname{Id}\in\mathsf{GL}(\mathfrak{g})$ is similar (in this case,
$\mathsf{G}=\mathsf{G}_{\mu}$). ∎

Let us now consider the effect of scaling when Dorfman brackets are moving.
Recall that, by 2.11, a choice of isotropic splitting $\sigma$ induces an
isometry $(c\cdot
E,\mathcal{G})\simeq_{\sigma}((\mathfrak{g}\oplus\mathfrak{g}^{*})_{cH},\mathcal{G}(cg,cb))$,
for any $c>0$.

######  Proposition 4.10.

Let $c>0$ and let ${\bm{\mu}}$ be the Dorfman bracket determined by
${\bm{\mu}}_{\mathfrak{g}}=[\cdot,\cdot]_{\mathfrak{g}}$ and
${\bm{\mu}}_{\Lambda^{3}}=H$. Then, the following bundle map is an isometry:

$\left(\pm\overline{c^{1/2}\operatorname{Id}}\,,\,\pm
c^{1/2}\operatorname{Id}\right):((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\pm
cH},\mathcal{G}(cg,\pm
cb))\longrightarrow((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\pm
c^{-1/2}{\bm{\mu}}},\mathcal{G}(g,b)).$

###### Proof.

First, we see that it is an ECA isomorphism. Indeed, we have that
$\pm\overline{c^{1/2}\operatorname{Id}}\in\mathsf{O}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$,
and

$\pm\overline{{c^{1/2}}\operatorname{Id}}\left[(\pm\overline{c^{1/2}\operatorname{Id}})^{-1}\,\cdot\,,(\pm\overline{c^{1/2}\operatorname{Id}})^{-1}\,\cdot\,\right]_{cH}=\pm
c^{-1/2}{\bm{\mu}}(\cdot,\cdot)$

can be easily verified using (21) and (22) (with
$\mu=[\cdot,\cdot]_{\mathfrak{g}}$). Regarding the isometry claim, we directly
compute:

$\displaystyle\left(\overline{c^{1/2}\operatorname{Id}}\right)\,\mathcal{G}(cg,cb)\,\left(\overline{c^{1/2}\operatorname{Id}}\right)^{-1}$
$\displaystyle=\begin{pmatrix}c^{1/2}\operatorname{Id}&0\\\
0&c^{-1/2}\operatorname{Id}\end{pmatrix}\,e^{cb}\,\begin{pmatrix}0&c^{-1}g^{-1}\\\
cg&0\end{pmatrix}\,e^{-cb}\,\begin{pmatrix}c^{-1/2}\operatorname{Id}&0\\\
0&c^{1/2}\operatorname{Id}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-g^{-1}b&g^{-1}\\\
g-bg^{-1}b&bg^{-1}\end{pmatrix}=\mathcal{G}(g,b).$

∎

###  4.3. The moment map

In this section we define a moment map for the action of $\mathsf{L}$ on
Dorfman brackets, in the sense of real GIT (see [RS90, HS07, BL20a]). In 4.13
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 $\overline{\mathcal{G}}$ on
a left-invariant ECA, yielding an isomorphism with
$((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bar{H}},\mathcal{G}(\bar{g},0))$. The
metric $\bar{g}$ gives rise to an inner product on
$\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$,
also denoted by $\bar{g}$. To define it, we fix a $\bar{g}$-orthonormal basis
$\\{e_{i}\\}$ of $\mathfrak{g}$ with dual basis $\\{e^{i}\\}$ and set

$\bar{g}({\bm{\mu}},{\bm{\nu}}):=2\,\sum_{i,j}\left\langle\overline{\mathcal{G}}{\bm{\mu}}(e_{i},e_{j}),{\bm{\nu}}(e_{i},e_{j})\right\rangle+4\,\sum_{i,j}\left\langle\overline{\mathcal{G}}{\bm{\mu}}(e_{i},e^{j}),{\bm{\nu}}(e_{i},e^{j})\right\rangle+2\,\sum_{i,j}\left\langle\overline{\mathcal{G}}{\bm{\mu}}(e^{i},e^{j}),{\bm{\nu}}(e^{i},e^{j})\right\rangle.$

The overall factor of $2$ is due to the $\tfrac{1}{2}$ factor in the
definition of ${\langle\,\cdot\,,\cdot\,\rangle}$. Since we are only
interested in Dorfman brackets and their infinitesimal variations, we will
mostly work with brackets in $V_{\mathcal{D}}$ (see (24)). Then, the only non-
vanishing structure coefficients are:

${\bm{\mu}}(e_{i},e_{j})={\bm{\mu}}_{ij}^{k}e_{k}+{\bm{\mu}}_{ijk}e^{k},\qquad{\bm{\mu}}(e_{i},e^{j})=-{\bm{\mu}}_{ik}^{j}e^{k}=-{\bm{\mu}}(e^{j},e_{i}),$

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 ${\bm{\mu}}(e_{i},e^{j})$
is related to that of $e_{j}$ in ${\bm{\mu}}(e_{i},e_{k})$ follows from
$\langle{\bm{\mu}}(\cdot,\cdot),\cdot\rangle\in\Lambda^{3}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}$.
We then have

(27)
$\bar{g}({\bm{\mu}},{\bm{\nu}}):=\sum_{i,j,k}\left(3\,{\bm{\mu}}_{ij}^{k}{\bm{\nu}}_{ij}^{k}+{\bm{\mu}}_{ijk}{\bm{\nu}}_{ijk}\right)=3\,\bar{g}(\mu,\nu)+\bar{g}(H,\tilde{H}),\qquad{\bm{\mu}},{\bm{\nu}}\in
V_{\mathcal{D}},$

where
$\mu={\bm{\mu}}_{\mathfrak{g}},\nu={\bm{\nu}}_{\mathfrak{g}},H={\bm{\mu}}_{\Lambda^{3}},\tilde{H}={\bm{\nu}}_{\Lambda^{3}}$.
Here we have used the extension of $\bar{g}$ to a positive-definite inner
product on $\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g}$, and on
$\Lambda^{k}\mathfrak{g}^{*}$, given respectively by

$\bar{g}(\mu,\nu):=\sum_{i,j,k}\mu_{ij}^{k}\nu_{ij}^{k},\qquad\mu,\nu\in\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g},\qquad\mu_{ij}^{k}:=\bar{g}(\mu(e_{i},e_{j}),e_{k}).$

and

$\bar{g}(\alpha,\beta):=\sum_{i_{1},\ldots,i_{k}}\alpha_{i_{1}\cdots
i_{k}}\beta_{i_{1}\cdots
i_{k}},\qquad\alpha,\beta\in\Lambda^{k}\mathfrak{g}^{*},\qquad\alpha_{i_{1}\cdots
i_{k}}:=\alpha(e_{i_{1}},\ldots,e_{i_{k}}).$

The metric $\bar{g}$ also determines a maximal compact subgroup
$\mathsf{O}(\mathfrak{g},\bar{g})\leq\mathsf{L}$. We fix on $\mathfrak{l}$ the
$\operatorname{Ad}(\mathsf{O}(\mathfrak{g},\bar{g}))$-invariant, positive-
definite inner product $\bar{g}_{\mathfrak{l}}$ given by

(28) $\bar{g}_{\mathfrak{l}}\left(\begin{pmatrix}A&0\\\
\alpha&-A^{*}\end{pmatrix},\begin{pmatrix}B&0\\\
\beta&-B^{*}\end{pmatrix}\right):=2\operatorname{tr}AB^{T}+\frac{1}{6}\operatorname{tr}\beta^{*}\alpha=2\,\sum_{i,j}A_{ij}B_{ij}+\frac{1}{6}\sum_{i,j}\alpha_{ij}\beta_{ij},$

where $A_{ij}:=g(Ae_{i},e_{j})$, $\alpha_{ij}=\alpha(e_{i},e_{j})$, 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.

######  Definition 4.11.

The _moment map_ for the action of $\mathsf{L}$ on
$(\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*}),\bar{g})$
is the map

$\operatorname{M}:\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})\to\mathfrak{l},\qquad{\bm{\mu}}\mapsto\operatorname{M}_{\bm{\mu}},$

implicitly defined by

$\bar{g}_{\mathfrak{l}}(\operatorname{M}_{\bm{\mu}},L)=\frac{1}{6}\,\bar{g}(\Theta(L){\bm{\mu}},{\bm{\mu}}),\qquad
L\in\mathfrak{l}.$

######  Proposition 4.12.

The moment map is $\mathsf{O}(\mathfrak{g},\bar{g})$-equivariant. Moreover,
for each
${\bm{\mu}}\in\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$
and $L\in\mathfrak{l}$, it satisfies

$\bar{g}_{\mathfrak{l}}(\operatorname{M}_{\bm{\mu}},L)=\frac{1}{12}\,\frac{\textrm{d}}{\textrm{d}t}\Big{|}_{t=0}\lvert\exp(tL)\cdot{\bm{\mu}}\rvert^{2}_{g}\,.$

###### Proof.

Let $K\in\mathsf{O}(\mathfrak{g},\bar{g})$, for every $L\in\mathfrak{l}$, we
have that

$\bar{g}_{\mathfrak{l}}(\operatorname{M}_{\overline{K}\cdot{\bm{\mu}}},L)=\frac{1}{6}\bar{g}(\overline{K}^{-1}\cdot\Theta(L)(\overline{K}\cdot{\bm{\mu}}),{\bm{\mu}})=\frac{1}{6}\bar{g}(\Theta(\overline{K}^{-1}L\overline{K}){\bm{\mu}},{\bm{\mu}})=\bar{g}_{\mathfrak{l}}(\operatorname{M}_{{\bm{\mu}}},\overline{K}^{-1}L\overline{K})=\bar{g}_{\mathfrak{l}}(\overline{K}\operatorname{M}_{{\bm{\mu}}}\overline{K}^{-1},L)$

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

We now define
${\mathcal{R}c}_{\bm{\mu}}\in\operatorname{End}(\mathfrak{g}\oplus\mathfrak{g}^{*})$
to be the generalized Ricci curvature of
$((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bm{\mu}},\overline{\mathcal{G}})$ (at
the identity). We also denote by ${\rm
Ric}_{\mu}\in\operatorname{End}(\mathfrak{g})$ the $(1,1)$-Ricci tensor of
$(\mathfrak{g},\mu,\bar{g})$, and by
$\operatorname{M}_{\mu}\in\operatorname{End}(\mathfrak{g})$ the moment map for
the action of $\mathsf{GL}(\mathfrak{g})$ on
$\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g}$:

$\bar{g}(\operatorname{M}_{\mu},A)=\frac{1}{4}\bar{g}(\theta(A)\mu,\mu),\qquad
A\in\mathfrak{gl(g)},\qquad\mu\in\Lambda^{2}\mathfrak{g}^{*}\otimes\mathfrak{g}.$

The factor $\tfrac{1}{4}$ makes this consistent with the standard notation in
the homogeneous Riemannian setting, see for instance [LL14]. In this notation,
for ${\bm{\mu}}\in{\mathcal{D}}$ with ${\bm{\mu}}_{\mathfrak{g}}=\mu$,
${\bm{\mu}}_{\Lambda^{3}}=H$, we have

(29)
${\mathcal{R}c}_{\bm{\mu}}=\begin{pmatrix}\operatorname{Ric}^{B}_{\mu,H}&\frac{1}{2}\bar{g}^{-1}(\textrm{d}_{\mu}^{*}H)\,\bar{g}^{-1}\\\
-\frac{1}{2}\textrm{d}^{*}_{\mu}H&-(\operatorname{Ric}^{B}_{\mu,H})^{*}\end{pmatrix},\qquad\operatorname{Ric}^{B}_{\mu,H}:={\rm
Ric}_{\mu}-\frac{1}{4}\,\bar{g}^{-1}H^{2}.$

######  Proposition 4.13.

The generalized Ricci curvature and the moment map for the action of
$\mathsf{L}$ on Dorfman brackets are related by

${\mathcal{R}c}_{\bm{\mu}}=\operatorname{M}_{\bm{\mu}}+\overline{{\rm
Ric}_{\mu}}-\overline{\operatorname{M}_{\mu}}+A_{\bm{\mu}}\,,$

for some
$A_{\bm{\mu}}\in\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},\overline{\mathcal{G}})\cap\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$.

###### Proof.

Let us set

(30)
$A_{\bm{\mu}}:=\begin{pmatrix}0&\frac{1}{2}\bar{g}^{-1}(\textrm{d}_{\mu}^{*}H)\,\bar{g}^{-1}\\\
\frac{1}{2}\textrm{d}^{*}_{\mu}H&0\end{pmatrix}.$

It is not hard to see that indeed
$A_{\bm{\mu}}\in\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},\overline{\mathcal{G}})\cap\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$.
We are thus left with

${\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}=\begin{pmatrix}\operatorname{Ric}^{B}_{\mu,H}&0\\\
-\textrm{d}^{*}_{\mu}H&-(\operatorname{Ric}^{B}_{\mu,H})^{*}\end{pmatrix}\in\mathfrak{l}.$

Furthermore, one can easily see that (30) defines the unique element in
$\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},\bar{\mathcal{G}})\cap\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$
such that ${\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}\in\mathfrak{l}$.

Moreover, for any $L\simeq\left(B,\beta\right)\in\mathfrak{l}$, we compute
using the definition of $\bar{g}_{\mathfrak{l}}$:

$\displaystyle\bar{g}_{\mathfrak{l}}\left({\mathcal{R}c}_{\bm{\mu}}-A_{\mu},L\right)$
$\displaystyle=2\operatorname{tr}(\operatorname{Ric}_{\mu}B)-\frac{1}{2}\operatorname{tr}(H^{2}B)-\frac{1}{6}\bar{g}(\textrm{d}^{*}_{\mu}H,\beta).$

Furthermore, we notice that

$\displaystyle\bar{g}(\rho(B)H,H)$
$\displaystyle=\sum_{i,j,k}(\rho(B)H)(e_{i},e_{j},e_{k})H(e_{i},e_{j},e_{k})=-3\sum_{i,j,k}H(Be_{i},e_{j},e_{k})H(e_{i},e_{j},e_{k})$
$\displaystyle=-3\sum_{i}\bar{g}(\iota_{e_{i}}H,\iota_{Be_{i}}H)=-3\bar{g}(H^{2}e_{i},Be_{i})=-3\operatorname{tr}(H^{2}B),$

from which we get

$\bar{g}_{\mathfrak{l}}\left({\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}},L\right)=\bar{g}_{\mathfrak{l}}\left(\overline{{\rm
Ric}_{\mu}},L\right)+\frac{1}{6}\,\bar{g}(\rho(B)H-\textrm{d}_{\mu}\beta,H).$

On the other hand, by 4.8 and (27) the moment map satisfies:

(31)
$\displaystyle\bar{g}_{\mathfrak{l}}\left(\operatorname{M}_{\bm{\mu}},L\right)=\frac{1}{6}\,\bar{g}(\Theta(L){\bm{\mu}},{\bm{\mu}})$
$\displaystyle=\frac{1}{2}\,\bar{g}(\theta(B)\mu,\mu)+\frac{1}{6}\,\bar{g}(\rho(B)H-\textrm{d}_{\mu}\beta,H)$
$\displaystyle=2\,\bar{g}(\operatorname{M}_{\mu},B)+\frac{1}{6}\,\bar{g}(\rho(B)H-\textrm{d}_{\mu}\beta,H)$
$\displaystyle=\bar{g}_{\mathfrak{l}}\left(\overline{\operatorname{M}_{\mu}},L\right)+\frac{1}{6}\,\bar{g}(\rho(B)H-\textrm{d}_{\mu}\beta,H),$

and the Proposition follows. ∎

######  Corollary 4.14.

If the Lie algebra $(\mathfrak{g},\mu)$ is nilpotent, then

${\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}=\operatorname{M}_{\bm{\mu}},\qquad
A_{\bm{\mu}}\in\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},\overline{\mathcal{G}})\cap\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle}).$

###### Proof.

Lauret’s formula for the Ricci curvature of left-invariant metrics in terms of
the moment map on the variety of Lie algebras [Lau06] yields

${\rm
Ric}_{\mu}=\operatorname{M}_{\mu}-\frac{1}{2}\,B_{\mu}-\frac{1}{2}(\operatorname{ad}_{\mu}U+(\operatorname{ad}_{\mu}U)^{t}),$

where $\bar{g}(B_{\mu}\cdot,\cdot)$ is the Killing form of
$(\mathfrak{g},\mu)$, and $U\in\mathfrak{g}$ denotes the _mean curvature
vector_ of the metric Lie algebra $(\mathfrak{g},\mu,\bar{g})$, which vanishes
if and only if $(\mathfrak{g},\mu)$ is unimodular. In particular, for
nilpotent Lie algebras –for which it is known that the Killing form vanishes–
this gives ${{\rm Ric}_{\mu}}=\operatorname{M}_{\mu}$, and the stated formula
follows from 4.13. ∎

##  5\. A flow of Dorfman brackets

The fact that the Lie group $\mathsf{L}$ (see (20)) acts transitively on the
space of generalized left-invariant metrics $\mathcal{M}^{\mathsf{G}}$ on
$(\mathfrak{g}\oplus\mathfrak{g}^{*})_{H}$ (4.4) and the equivalence between
acting on generalized metrics and acting on Dorfman brackets (4.9) 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
$\mathcal{D}\subset\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$.

As in previous Sections, we fix a left-invariant ECA
$(\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bar{H}}$ and a background left-
invariant generalized metric $\overline{\mathcal{G}}=\mathcal{G}(\bar{g},0)$
on it.

######  Definition 5.1.

The _generalized bracket flow_ with background
$\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\overline{\mathcal{G}}\right)$
is the initial value problem for a curve of brackets ${\bm{\mu}}(t)$ in
$\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})^{*}\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$:

(32)
$\frac{\textrm{d}}{\textrm{d}t}{\bm{\mu}}=-\Theta({\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}){\bm{\mu}},\qquad{\bm{\mu}}(0)={\bm{\mu}}_{0},$

where ${\mathcal{R}c}_{\bm{\mu}}$ and $A_{\bm{\mu}}$ are respectively defined
in (29) and (30).

Since a Dorfman bracket ${\bm{\mu}}\in\mathcal{D}$ (see 4.5) is determined by
its two projections ${\bm{\mu}}_{\mathfrak{g}},{\bm{\mu}}_{\Lambda^{3}}$, it
is natural to expect the generalized bracket flow to be equivalent to a
coupled flow of Lie brackets on $\mathfrak{g}$ and of closed $3$-forms.
Indeed:

######  Proposition 5.2.

The generalized bracket flow (32) with initial condition a Dorfman bracket
${\bm{\mu}}_{0}\in\mathcal{D}$ is equivalent to the following coupled flow of
brackets $\mu(t)$ and $3$-forms $H(t)$:

(33)
$\begin{cases}\frac{\textrm{d}}{\textrm{d}t}\mu=-\theta(\operatorname{Ric}^{B}_{\mu,H})\mu,\qquad&\mu(0)=\mu_{0}:=({\bm{\mu}}_{0})_{\mathfrak{g}},\\\
\frac{\textrm{d}}{\textrm{d}t}H=\Delta_{\mu}H-\rho(\operatorname{Ric}^{B}_{\mu,H})H,\qquad&H(0)=H_{0}:=({\bm{\mu}}_{0})_{\Lambda^{3}}.\end{cases}$

###### Proof.

Let $(\mu_{t},H_{t})_{t\in[0,T)}$ be a solution of (33). Then, we define

${\bm{\mu}}_{t}(X+\xi,Y+\eta)=\mu_{t}(X,Y)-\eta\circ(\mu_{t})_{X}+\xi\circ(\mu_{t})_{Y}+\iota_{Y}\iota_{X}H_{t}\,,\quad
X,Y\in\mathfrak{g}\,,\,\,\xi,\eta\in\mathfrak{g}^{*},\,\,t\in[0,T).$

Using 4.6, we have that ${\bm{\mu}}_{t}\in\mathcal{D}$ , for all $t\in[0,T)$.
Now, using (33) and 4.8, we have that

$\frac{\textrm{d}}{\textrm{d}t}{\bm{\mu}}_{t}=-\Theta(\mathcal{R}c_{{\bm{\mu}}_{t}}-A_{{\bm{\mu}}_{t}}){\bm{\mu}}_{t}\,,\quad{\bm{\mu}}(0)={\bm{\mu}}_{0}\,.$

Then, since they solve the same ODE with the same initial datum,
$({\bm{\mu}}_{t})_{t\in[0,T)}$ coincides with the solution of the generalized
bracket flow starting from ${\bm{\mu}}_{0}$.

Viceversa, let $({\bm{\mu}}_{t})_{t\in[0,T^{\prime})}$ be a solution of the
generalized bracket flow starting from ${\bm{\mu}}_{0}$. For the sake of
simplicity, we will denote $\tilde{\mu}_{t}:=({\bm{\mu}}_{t})_{\mathfrak{g}}$
and $\tilde{H}_{t}:=({\bm{\mu}}_{t})_{\Lambda^{3}}$. By definition,
${\bm{\mu}}_{t}$ belongs to the $\mathsf{L}$-orbit of ${\bm{\mu}}_{0}$. Using
the fact that the $\mathsf{L}$-action does not change $\pi$ and 4.8, we obtain
that

(34)
$\frac{\textrm{d}}{\textrm{d}t}\tilde{\mu}=\pi\frac{\textrm{d}}{\textrm{d}t}{\bm{\mu}}=-\pi\Theta(\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}){\bm{\mu}}=-\theta({\rm
Ric}_{\tilde{\mu},\tilde{H}}^{B})\tilde{\mu}\,.$

Moreover, using again the fact that the $\mathsf{L}$-action acts trivially on
${\langle\,\cdot\,,\cdot\,\rangle}$, we have, for all $X,Y,Z\in\mathfrak{g}$,

(35) $\displaystyle\frac{\textrm{d}}{\textrm{d}t}\tilde{H}(X,Y,Z)=$
$\displaystyle\,2\frac{\textrm{d}}{\textrm{d}t}\langle{\bm{\mu}}(X,Y),Z\rangle=-2\langle\Theta(\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}){\bm{\mu}}(X,Y),Z\rangle$
$\displaystyle=$
$\displaystyle\,-(\Theta(\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}){\bm{\mu}})_{\Lambda^{3}}(X,Y,Z)=(\Delta_{\tilde{\mu}}\tilde{H}-\rho({\rm
Ric}_{\tilde{\mu},\tilde{H}}^{B})\tilde{H})(X,Y,Z)\,.$

where the last equality is due again to 4.8. Equation 35 and Equation 34
readily guarantees that $(\tilde{\mu_{t}},\tilde{H}_{t})_{t\in[0,T^{\prime})}$
coincides with the solution of (33), 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.

######  Lemma 5.3.

Let
$((\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}(t)},\bar{\mathcal{G}})_{t\in[0,T)}$
be a solution of the generalized bracket flow starting from ${\bm{\mu}}$.
Then,

$\frac{\textrm{d}}{\textrm{d}t}\lvert\textrm{d}_{\mu}^{*}H\rvert^{2}=2(\bar{g}(\rho({\rm
Ric}_{\mu,H}^{B})\textrm{d}_{\mu}^{*}H,\textrm{d}_{\mu}^{*}H)-2\bar{g}(\rho({\rm
Ric}^{B}_{\mu,H})H,\textrm{d}_{\mu}\textrm{d}^{*}_{\mu}H)-\lvert\Delta_{\mu}H\rvert^{2})\,.$

In particular, if $\textrm{d}_{\mu(0)}^{*}H(0)=0$, then
$\textrm{d}_{\mu}^{*}H=0$, for any $t\in[0,T)$.

###### Proof.

First of all, we easily see that, for all $A\in\mathfrak{gl}(\mathfrak{g})$
and brackets $\mu$, one has

(36) $\textrm{d}_{\theta(A)\mu}^{*}=[\textrm{d}_{\mu}^{*},\rho(A^{t})].$

Then, we can use (36) to infer that

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\textrm{d}^{*}_{\mu}H=$
$\displaystyle\,[\rho({\rm
Ric}_{\mu,H}^{B}),\textrm{d}^{*}_{\mu}]H-\textrm{d}^{*}_{\mu}\rho({\rm
Ric}_{\mu,H}^{B})H+\Delta_{\mu}\textrm{d}^{*}_{\mu}H=\rho({\rm
Ric}_{\mu,H}^{B})\textrm{d}^{*}_{\mu}H-2\textrm{d}^{*}_{\mu}\rho({\rm
Ric}_{\mu,H}^{B})H+\Delta_{\mu}\textrm{d}^{*}_{\mu}H\,.$

Then, we easily obtain that

$\displaystyle\frac{\textrm{d}}{\textrm{d}t}\lvert\textrm{d}_{\mu}^{*}H\rvert^{2}=$
$\displaystyle\,2(\bar{g}(\rho({\rm
Ric}_{\mu,H}^{B})\textrm{d}_{\mu}^{*}H,\textrm{d}_{\mu}^{*}H)-2\bar{g}(\rho({\rm
Ric}^{B}_{\mu,H})H,\textrm{d}_{\mu}\textrm{d}^{*}_{\mu}H)-\lvert\Delta_{\mu}H\rvert^{2})\,,$

as claimed. The second part follows trivially from the first one. ∎

######  Remark 5.4.

A similar approach was firstly introduced by Paradiso in [Par21]. As a main
difference from our setting, the author considered the action of
$\mathsf{GL}(\mathfrak{g})\subset\mathsf{L}$ on $\mathcal{D}$. This choice
yields a coupled flow as follows:

$\begin{cases}\frac{\textrm{d}}{\textrm{d}t}\mu=-\theta(\operatorname{Ric}^{B}_{\mu,H})\mu,\qquad&\mu(0)=\mu_{0}:=({\bm{\mu}}_{0})_{\mathfrak{g}},\\\
\frac{\textrm{d}}{\textrm{d}t}H=-\rho(\operatorname{Ric}^{B}_{\mu,H})H,\qquad&H(0)=H_{0}:=({\bm{\mu}}_{0})_{\Lambda^{3}}\end{cases}$

which, in view of 5.3, is equivalent to (33) 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.

######  Theorem 5.5.

Let
$\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H_{0}},\mathcal{G}_{t}\right)_{t\in[0,T)}$
be a left-invariant solution to the generalized Ricci flow equation (13) and
let $({\bm{\mu}}(t))_{t\in[0,T^{\prime})}$ be the solution to the generalized
bracket flow (32) with background
$\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H_{0}},\mathcal{G}_{0}\right)$
and initial condition ${\bm{\mu}}(0)$ defined by
${\bm{\mu}}(0)_{\mathfrak{g}}=[\cdot,\cdot]_{\mathfrak{g}}$,
${\bm{\mu}}(0)_{\Lambda^{3}}=H_{0}$, with both solutions defined on a maximal
interval of time. Then, $T=T^{\prime}$ and there exists a one-parameter family
of generalized isometries

$F_{t}:\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H_{0}},\mathcal{G}_{t}\right)\longrightarrow\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}(t)},\mathcal{G}_{0}\right),\qquad
t\in[0,T).$

###### Proof.

First, given any metric ECA
$((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bm{\mu}},\mathcal{G})$ , we define
the endomorphism

$A_{{\bm{\mu}},\mathcal{G}}:=\operatorname{Skew}_{\mathcal{G}}(\textrm{d}^{*_{g}}_{\mu}(H+\textrm{d}_{\mu}b))\in\mathfrak{o}(\mathfrak{g}\oplus\mathfrak{g}^{*},\left\langle\mathcal{G}\cdot,\cdot\right\rangle),$

where $\mu={\bm{\mu}}_{\mathfrak{g}},H:={\bm{\mu}}_{\Lambda^{3}}$, and
$g\in\operatorname{Sym}^{2}_{+}(\mathfrak{g}),b\in\Lambda^{2}\mathfrak{g}^{*}$
satisfy

$\mathcal{G}=e^{b}\begin{pmatrix}0&g^{-1}\\\ g&0\end{pmatrix}e^{-b}.$

One readily checks that the map $({\bm{\mu}},\mathcal{G})\mapsto
A_{{\bm{\mu}},\mathcal{G}}$ is $\mathsf{L}$-equivariant:
$A_{F\cdot{\bm{\mu}},F\cdot\mathcal{G}}=F\cdot A_{{\bm{\mu}},\mathcal{G}}$.
Then, since the generalized Ricci curvature is similarly
$\mathsf{L}$-equivariant, it follows from Equation 30 that

$\mathcal{R}c({\bm{\mu}},\mathcal{G})-A_{{\bm{\mu}},\mathcal{G}}\in\mathfrak{l},$

where $\mathcal{R}c({\bm{\mu}},\mathcal{G})$ is the generalized Ricci
curvature of $((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bm{\mu}},\mathcal{G})$.

Now, given a solution to the generalized bracket flow
$({\bm{\mu}}(t))_{[0,T^{\prime})}$ (32), we define the family of left-
invariant endomorphisms $(\mathcal{G}_{{\bm{\mu}}(t)})_{t\in[0,T^{\prime})}$
as follows:

(37) $\mathcal{G}_{{\bm{\mu}}(t)}(x)=\overline{{\rm
L}_{{\bm{\mu}}(t)}(x)}\cdot\mathcal{G}_{0}(e)\,,\quad x\in\mathsf{G},$

where $\overline{{\rm L}_{{\bm{\mu}}(t)}(x)}$ is the lift on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$ of the left-translation in
$\mathsf{G}_{\mu(t)}$ by $x$ while $\mathcal{G}_{0}(e)$ and
$\mathcal{G}_{{\bm{\mu}}(t)}(x)$ are, respectively, the endomorphisms on the
fibers in $e$ and $x$. We easily see that $\mathcal{G}_{{\bm{\mu}}(t)}$ are
left-invariant generalized metric on $\mathfrak{g}\oplus\mathfrak{g}^{*}$. Let
us then consider

$\frac{\textrm{d}}{\textrm{d}t}F_{t}=F_{t}(\mathcal{R}c_{{\bm{\mu}}(t)}-A_{{\bm{\mu}}(t)})\,,\quad
F_{0}={\rm Id}\,,$

where
$A_{{\bm{\mu}}(t)}\in\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},\mathcal{G}_{0})\cap\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$
is defined as in (30). By a standard result in ODE theory, the family $F_{t}$
is defined on $[0,T^{\prime})$ and
$(F_{t})_{t\in[0,T^{\prime})}\subseteq\mathsf{L}$. We then define
$\bm{\lambda}(t):=F_{t}^{-1}\cdot{\bm{\mu}}(0)$ yielding that

(38)
$\frac{\textrm{d}}{\textrm{d}t}\bm{\lambda}=-\Theta(F_{t}^{-1}F_{t}^{\prime})\bm{\lambda}=-\Theta(\mathcal{R}c_{{\bm{\mu}}(t)}-A_{{\bm{\mu}}(t)})\bm{\lambda}\,.$

Thus, $\bm{\lambda}(t)$ and ${\bm{\mu}}(t)$ satisfy the same ODE with equal
initial datum concluding that ${\bm{\mu}}(t)=F_{t}^{-1}\cdot{\bm{\mu}}(0).$
From this, we can deduce that

$F_{t}\colon((\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}(t)},\mathcal{G}_{{\bm{\mu}}(t)})\to((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H_{0}},F_{t}\cdot\mathcal{G}_{{\bm{\mu}}(t)})\,,$

is an isometry. So, denoting
$\tilde{\mathcal{G}}_{t}:=F_{t}\cdot\mathcal{G}_{{\bm{\mu}}(t)}$ we obtain
that, in $e$,

$\displaystyle\tilde{\mathcal{G}}_{t}^{-1}\frac{\textrm{d}}{\textrm{d}t}\tilde{\mathcal{G}}_{t}=$
$\displaystyle\,F_{t}\mathcal{G}^{-1}_{{\bm{\mu}}(t)}(\mathcal{R}c_{{\bm{\mu}}(t)}-A_{{\bm{\mu}}(t)})\mathcal{G}_{{\bm{\mu}}(t)}F_{t}^{-1}-F_{t}(\mathcal{R}c_{{\bm{\mu}}(t)}-A_{{\bm{\mu}}(t)})F_{t}^{-1}=-2\mathcal{R}c(\tilde{\mathcal{G}}_{t})\,,$

where the last equality follows from
$\mathcal{R}c(\mathcal{G})\mathcal{G}=-\mathcal{G}\mathcal{R}c(\mathcal{G})$
and from $[A_{{\bm{\mu}}(t)},\mathcal{G}_{{\bm{\mu}}(t)}]=0$. This allows us
to conclude that $\mathcal{G}_{t}=\tilde{\mathcal{G}_{t}}$, since they solve
the same ODE with the same initial condition.

Conversely, we define

$\frac{\textrm{d}}{\textrm{d}t}F_{t}=(\mathcal{R}c(\mathcal{G}_{t})-A(\mathcal{G}_{t}))F_{t}\,,\quad
F(0)={\rm Id},$

where $A(\mathcal{G}_{t}):=A_{{\bm{\mu}},\mathcal{G}_{t}}$. As before, we
consider $\tilde{\mathcal{G}_{t}}:=F_{t}\cdot\mathcal{G}_{0}\ $ and easily
note that

$\tilde{\mathcal{G}_{t}}^{-1}\frac{\textrm{d}}{\textrm{d}t}\tilde{\mathcal{G}_{t}}=\tilde{\mathcal{G}_{t}}^{-1}(\mathcal{R}c(\mathcal{G}_{t})-A(\mathcal{G}_{t}))\tilde{\mathcal{G}_{t}}-(\mathcal{R}c(\mathcal{G}_{t})-A(\mathcal{G}_{t}))\,.$

On the other hand,

$\mathcal{G}_{t}^{-1}\frac{\textrm{d}}{\textrm{d}t}\mathcal{G}_{t}=-2\mathcal{R}c(\mathcal{G}_{t})=\mathcal{G}_{t}^{-1}(\mathcal{R}c(\mathcal{G}_{t})-A(\mathcal{G}_{t}))\mathcal{G}_{t}-(\mathcal{R}c(\mathcal{G}_{t})-A(\mathcal{G}_{t})),$

since, again,
$\mathcal{R}c(\mathcal{G}_{t})\mathcal{G}_{t}=-\mathcal{G}_{t}\mathcal{R}c(\mathcal{G}_{t})$
and $[A(\mathcal{G}_{t}),\mathcal{G}_{t}]=0$. So, in particular, we have that

$\frac{\textrm{d}}{\textrm{d}t}\tilde{\mathcal{G}_{t}}=[\mathcal{R}c(\mathcal{G}_{t})-A(\mathcal{G}_{t}),\tilde{\mathcal{G}_{t}}]\,,\quad\frac{\textrm{d}}{\textrm{d}t}\mathcal{G}_{t}=[\mathcal{R}c(\mathcal{G}_{t})-A(\mathcal{G}_{t}),\mathcal{G}_{t}]\,,$

Then, $\tilde{\mathcal{G}_{t}}$ and $\mathcal{G}_{t}$ satisfy the same ODE
with the same initial condition yielding that
$\mathcal{G}_{t}=F_{t}\cdot\mathcal{G}_{0}$. Therefore, if we let
$\bm{\lambda}(t)=F_{t}^{-1}\cdot{\bm{\mu}}(0)$,

$F_{t}\colon((\mathfrak{g}\oplus\mathfrak{g}^{*})_{\bm{\lambda}(t)},\mathcal{G}_{\bm{\lambda}(t)})\to((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H_{0}},\mathcal{G}_{t})$

is an isometry. In particular, this implies that

$F_{t}\mathcal{R}c_{\bm{\lambda}(t)}F_{t}^{-1}=\mathcal{R}c(\mathcal{G}_{t})\,,\quad
F_{t}A_{\bm{\lambda}(t)}F_{t}^{-1}=A(\mathcal{G}_{t}),$

using the ${\rm Aut}(E)$-equivariance of the generalized Ricci curvature and
the uniqueness of
$A(\mathcal{G})\in\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},\mathcal{G})\cap\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$
such that $\mathcal{R}c(\mathcal{G})-A(\mathcal{G})\in\mathfrak{l}$. Then we
have that

$\frac{\textrm{d}}{\textrm{d}t}F_{t}=F_{t}(\mathcal{R}c_{\bm{\lambda}(t)}-A_{\bm{\lambda}(t)})\,,\quad
F_{0}={\rm Id}$

and repeating the computations in (38), we infer that
${\bm{\mu}}(t)=\bm{\lambda}(t)$, concluding the proof. ∎

######  Remark 5.6.

We do not necessarily need to use $\mathcal{G}_{0}$ as background metric, but
this simplifies the formulas.

##  6\. Global existence on solvmanifolds

As a first application of 5.5, we prove in this Section the long-time
existence of invariant solutions on any solvmanifold. Recall from [GFS21] that
the _generalized scalar curvature_ of a metric Courant algebroid $({(T\oplus
T^{*})_{H}},\mathcal{G}(g,0))$ is given by

(39) $\mathcal{S}_{g,H}:=\operatorname{R}_{g}-\tfrac{1}{12}|H|^{2},$

where $\operatorname{R}_{g}$ 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 [GFS21, §3.9]. Within the
moving Dorfman bracket framework, a direct consequence of [Lau11, Lemma 4.2]
allows us to infer that on a nilpotent Lie group

(40) $\mathcal{S}_{{\bm{\mu}}}=-\frac{1}{12}(3\lvert\mu\rvert^{2}+\lvert
H\lvert^{2})=-\frac{1}{12}\lvert{\bm{\mu}}\rvert^{2}\,.$

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

######  Lemma 6.1.

Let $({\bm{\mu}}(t))_{t\in(\varepsilon_{-},\varepsilon_{+})}$ be a solution of
the generalized bracket flow defined on its maximal time interval. Then, there
exists a uniform constant $C>0$ such that:

  1. (1)

if $\varepsilon_{+}<\infty$, then

$\lvert{\bm{\mu}}(t)\rvert\geq\frac{C}{(\varepsilon_{+}-t)^{\frac{1}{2}}}\,,\quad
t\in[0,\varepsilon_{+})\,;$

  2. (2)

if $\varepsilon_{-}>-\infty$, then

$\lvert{\bm{\mu}}(t)\rvert\geq\frac{C}{(t-\varepsilon_{-})^{\frac{1}{2}}}\,,\quad
t\in(\varepsilon_{-},0]\,.$

###### Proof.

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 [Laf15, Lemma 3.1], we have $\lvert{\rm Ric}_{\mu}\rvert^{2}\leq
C\lvert\mu\rvert^{4}$. Moreover, one can easily show that $\lvert
H^{2}\rvert^{2}\leq C\lvert H\rvert^{4}$. Finally, we obtain that
$\lvert\textrm{d}_{\mu}^{*}H\rvert^{2}\leq C\lvert\mu\rvert^{2}\lvert
H\rvert^{2}\leq C\lvert{\bm{\mu}}\rvert^{4}$, using that $*_{\mu}$ is an
isometry and that $\lvert\textrm{d}_{\mu}\alpha\rvert^{2}\leq
C\lvert\mu\rvert^{2}\lvert\alpha\rvert^{2}$, for any form $\alpha$ and any Lie
bracket $\mu$. Then, one can deduce

(41) $\lvert\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}\rvert^{2}=2\lvert{\rm
Ric}^{B}_{\mu,H}\rvert^{2}+\frac{1}{6}\lvert\textrm{d}_{\mu}^{*}H\rvert^{2}\leq
C\rvert{\bm{\mu}}\lvert^{4}\,.$

Now, we fix $t_{0}\in[0,\varepsilon_{+})$ and we can use (41) and the
linearity of the infinitesimal $\mathsf{L}$-action $\Theta$ to infer that

$\frac{\textrm{d}}{\textrm{d}t}\lvert{\bm{\mu}}\rvert^{2}=2\bar{g}\left({\bm{\mu}},\frac{\textrm{d}}{\textrm{d}t}{\bm{\mu}}\right)\leq
C\lvert{\bm{\mu}}\rvert^{4}\,,\quad t\in[t_{0},\varepsilon_{+})\,.$

This implies, by comparison, that

(42)
$\lvert{\bm{\mu}}\rvert^{2}\leq\frac{1}{-C(t-t_{0})+\lvert{\bm{\mu}}(t_{0})\rvert^{-2}}\,,\quad
t\in[t_{0},\varepsilon_{+})\,.$

On the other hand, (42) readily guarantees that $\varepsilon_{+}\geq
t_{0}+\frac{1}{C}\lvert{\bm{\mu}}(t_{0})\rvert^{-2}$, 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:

######  Theorem 6.2.

Let $(E,\mathcal{G}_{t})_{t\in(\varepsilon_{-},\varepsilon_{+})}$ be a left-
invariant solution of the generalized Ricci flow over a simply-connected Lie
group $\mathsf{G}$ defined in its maximal time interval. Then,

  1. (1)

if $\varepsilon_{+}<\infty$, then $\mathcal{S}_{t}\to\infty$, as
$t\to\varepsilon_{+}$ ;

  2. (2)

if $\varepsilon_{-}>-\infty$, then $\mathcal{S}_{t}\to-\infty$, as
$t\to\varepsilon_{-}$ .

###### Proof.

Fixing the preferred isotropic splitting $\sigma_{0}$ associate with
$\mathcal{G}_{0}$, we have the time-varying isometry
$(E,\mathcal{G}_{t})_{t\in(\varepsilon_{-},\varepsilon_{+})}\simeq_{\sigma}((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H_{t}},\mathcal{G}(g_{t},0))_{t\in(\varepsilon_{-},\varepsilon_{+})}$.
By [Str23, Proposition 1.1], if
$\phi\colon\mathsf{G}\times(\varepsilon_{-},\varepsilon_{+})\to\mathbb{R}$
satisfies

(43) $\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\phi_{t}=\tfrac{1}{6}|H_{t}|^{2}_{g_{t}};\qquad
t\in(\varepsilon_{-},\varepsilon_{+}),$

then,

$\left(\frac{\partial}{\partial
t}-\Delta_{g_{t}}\right)\left(\mathcal{S}_{g_{t},H_{t}}+2\Delta_{g_{t}}\phi_{t}-|\nabla_{g_{t}}\phi_{t}|^{2}\right)=2|\operatorname{Ric}^{B}_{g_{t},H_{t}}+\nabla^{2}_{g_{t}}\phi_{t}-\tfrac{1}{2}(\textrm{d}^{*}_{g_{t}}H_{t}+\iota_{\nabla_{g_{t}}\phi_{t}}H_{t})|^{2};\qquad
t\in(\varepsilon_{-},\varepsilon_{+}).$

In this case, we set
$\phi_{t}:=\int_{0}^{t}\tfrac{1}{6}|H_{s}|^{2}\textrm{d}s$, which is constant
in space and hence satisfies Equation 43. Thus, by left-invariance, we have

(44)
$\frac{\textrm{d}}{\textrm{d}t}\mathcal{S}_{g_{t},H_{t}}=2|\operatorname{Ric}^{B}_{g_{t},H_{t}}-\tfrac{1}{2}\textrm{d}^{*}_{g_{t}}H_{t}|^{2}=|\mathcal{R}c(\mathcal{G}_{t})|^{2}.$

The remainder of the proof follows [Laf15] for the classical Ricci flow. By
5.5,
$((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H_{0}},\mathcal{G}_{t})_{t\in(\varepsilon_{-},\varepsilon_{+})}$
is isometric to
$\left((\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}(t)},\mathcal{G}(g_{0},0)\right)_{t\in(\varepsilon_{-},\varepsilon_{+})}$,
where ${\bm{\mu}}$ satisfies Equation 32. Let
$\mathcal{S}_{{\bm{\mu}}(t)}=\mathcal{S}_{g_{t}}$ denote the generalized
scalar curvature of this latter metric ECA and note that by Equation 44 it
satisfies
$\tfrac{\textrm{d}}{\textrm{d}t}\mathcal{S}_{{\bm{\mu}}(t)}=|\mathcal{R}c_{{\bm{\mu}}(t)}|^{2}$.
Also observe that

$\tfrac{\textrm{d}}{\textrm{d}t}|{\bm{\mu}}|^{2}=-2\left\langle\Theta(\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}){\bm{\mu}},{\bm{\mu}}\right\rangle\leq
C|\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}||{\bm{\mu}}|^{2},$

for some constant $C>0$. Hence, observing that
$|\mathcal{R}c_{\bm{\mu}}-A_{\bm{\mu}}|^{2}\leq
2|\mathcal{R}c_{\bm{\mu}}|^{2}$, we have, for
$t\in(\varepsilon_{-},\varepsilon_{+})$,

$\displaystyle\log|{\bm{\mu}}(t)|^{2}-\log|{\bm{\mu}}(0)|^{2}$
$\displaystyle=\int_{0}^{t}\tfrac{\textrm{d}}{\textrm{d}s}\log|{\bm{\mu}}(s)|^{2}\textrm{d}s\leq
C\int_{0}^{t}|\mathcal{R}c_{{\bm{\mu}}(s)}-A_{{\bm{\mu}}(s)}|\textrm{d}s\leq
C\int_{0}^{t}|\mathcal{R}c_{{\bm{\mu}}(s)}|\textrm{d}s$ $\displaystyle\leq
C\int_{0}^{t}1+|\mathcal{R}c_{{\bm{\mu}}(s)}-A_{\bm{\mu}}(s)|^{2}\textrm{d}s=C\left(t+\mathcal{S}_{{\bm{\mu}}(t)}-\mathcal{S}_{{\bm{\mu}}(0)}\right).$

On the other hand, by 6.1, we have that $|{\bm{\mu}}(t)|^{2}\to\infty$, as
$t\to\varepsilon_{+}$, forcing $\mathcal{S}_{{\bm{\mu}}(t)}\to\infty$, as
$t\to\varepsilon_{+}$. A similar argument shows the statement for
$t\to\varepsilon_{-}$. ∎

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
[BB78]. Hence, we yield the immediate Corollary:

######  Corollary 6.3.

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 4.14. In particular, we can describe the precise
asymptotic behaviour of the generalized scalar curvature:

######  Theorem 6.4.

For any left-invariant solution $(g_{t},H_{t})$ of the generalized Ricci flow
on a nilpotent Lie group $\mathsf{G}$, the generalized scalar curvature
satisfies $\mathcal{S}_{g_{t},H_{t}}\sim-\frac{1}{t}$, as $t\to\infty$.

###### Proof.

Let $(g_{t},H_{t})$ be a solution of the generalized Ricci flow on
$\mathsf{G}$ and ${\bm{\mu}}(t)$ the corresponding generalized bracket flow.
Then by Equation 32, 4.14 and 4.11, we obtain

$\frac{\textrm{d}}{\textrm{d}t}|{\bm{\mu}}(t)|^{2}=-2\left\langle\Theta(\operatorname{M}_{{\bm{\mu}}(t)}){\bm{\mu}}(t),{\bm{\mu}}(t)\right\rangle=-6\overline{g}_{\mathfrak{l}}(\operatorname{M}_{{\bm{\mu}}(t)},\operatorname{M}_{{\bm{\mu}}(t)})=-6|M_{{\bm{\mu}}(t)}|^{2}.$

We now claim that $|\operatorname{M}_{{\bm{\mu}}(t)}|^{2}\geq
c|{\bm{\mu}}|^{4}$, for some small constant $c>0$. To see this, first note
that the map

$\mathbb{S}(\Lambda^{2}(\mathfrak{g}\oplus\mathfrak{g}^{*})\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*}))\ni{\bm{\mu}}\mapsto|\operatorname{M}_{\bm{\mu}}|^{2},$

is never zero on the unit sphere. Indeed $\operatorname{M}_{{\bm{\mu}}}=0$
implies that
$0=\operatorname{tr}(\operatorname{M}_{{\bm{\mu}}})=-\frac{1}{6}|{\bm{\mu}}|^{2}=-\frac{1}{6}$,
which is a contradiction. The claim therefore follows from compactness of the
unit sphere. Thus,

$-C|{\bm{\mu}}(t)|^{4}\leq\frac{\textrm{d}}{\textrm{d}t}|{\bm{\mu}}(t)|^{2}\leq-c|{\bm{\mu}}(t)|^{4},$

so $|{\bm{\mu}}(t)|^{2}\sim\frac{1}{t}$ by ODE comparison. The result now
follows from Equation 40.

∎

##  7\. Generalized nilsolitons

In this Section, following the approach by Lauret in [Lau13] and [Lau01], 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.

######  Definition 7.1.

The $\ell$-_normalized generalized bracket flow_ with background data
$((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\overline{\mathcal{G}})$ is the
initial value problem for a curve of brackets ${\bm{\mu}}(t)$ and
$\ell=\ell(t)$:

(45)
$\frac{\textrm{d}}{\textrm{d}t}{\bm{\mu}}=-\Theta({\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}){\bm{\mu}}+\ell{\bm{\mu}}\,,\qquad{\bm{\mu}}(0)={\bm{\mu}}_{0}\,.$

The multiplication of ${\bm{\mu}}$ in (45) by the factor $\ell$ has to be
interpreted using 2.11 and 4.10. On the other hand, given a function $\ell$,
one can obtain a solution of the $\ell$-normalized generalized bracket flow
from a solution of the generalized bracket flow (32) via a time
reparametrization and a scaling.

######  Lemma 7.2.

Let $((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\overline{\mathcal{G}})$ be a
left-invariant metric ECA over a Lie group $\mathsf{G}$. Let
${\bm{\mu}}^{\ell}(t)$ and ${\bm{\mu}}(t)$, respectively, be the solution of
the $\ell$-normalized generalized bracket flow and the solution of the
generalized bracket flow starting from ${\bm{\mu}}_{0}$. Then, there exist a
family of scaling $c(t)>0$ such that $c(0)=1$ and a time reparametrization
$\tau=\tau(t)$ so that

${\bm{\mu}}^{\ell}(t)=c(t){\bm{\mu}}(\tau)\,.$

###### Proof.

Following [Lau13], we choose

$c^{\prime}=\ell c\,,\quad
c(0)=1\quad\mbox{and}\quad\tau^{\prime}=c^{2}\,,\quad\tau(0)=0.$

Then, we have

$\frac{\textrm{d}}{\textrm{d}t}(c(t){\bm{\mu}}(\tau))=-c^{3}\Theta({\mathcal{R}c}_{{\bm{\mu}}(\tau)}-A_{{\bm{\mu}}(\tau)}){\bm{\mu}}(\tau)+\ell
c{\bm{\mu}}(\tau).$

Using that
${\mathcal{R}c}_{c{\bm{\mu}}}-A_{c{\bm{\mu}}}=c^{2}({\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}})$,
we obtain the claim. ∎

The case in which we will be interested the most is when we choose
$c=\frac{1}{\lvert{\bm{\mu}}\rvert}.$ Applying 7.2, we have that
$\bar{\bm{\mu}}:=\frac{{\bm{\mu}}}{\lvert{\bm{\mu}}\rvert}$ is a solution of
the $\ell$-normalized generalized bracket flow with

$\ell_{\bar{\bm{\mu}}}=\frac{\bar{g}(\Theta({\mathcal{R}c}_{{\bm{\mu}}}-A_{{\bm{\mu}}}){\bm{\mu}},{\bm{\mu}})}{\lvert{\bm{\mu}}\rvert^{4}}=6\bar{g}(\mathcal{R}c_{\bar{\bm{\mu}}}-A_{\bar{\bm{\mu}}},\operatorname{M}_{\bar{\bm{\mu}}})\,.$

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,

$\frac{\textrm{d}}{\textrm{d}t}\lvert{\bm{\mu}}\rvert^{2}=2\bar{g}(\Theta({\mathcal{R}c}_{{\bm{\mu}}}-A_{{\bm{\mu}}}){\bm{\mu}},{\bm{\mu}})(-1+\lvert{\bm{\mu}}\rvert^{2})\,,$

then $\lvert{\bm{\mu}}\rvert^{2}=1$, since $\lvert{\bm{\mu}}_{0}\rvert^{2}=1$.
Recalling (40), this also implies that
$\mathcal{S}_{{\bm{\mu}}}=\mathcal{S}_{{\bm{\mu}}_{0}}=-\frac{1}{12}$ along
the scalar-normalized generalized bracket flow.

With this in mind, we can give the definition of generalized algebraic
soliton.

######  Definition 7.3.

Let ${\bm{\mu}}\in\mathcal{D}$. We say that ${\bm{\mu}}$ 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

(46)
$\Theta({\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}){\bm{\mu}}=\ell_{{\bm{\mu}}}{\bm{\mu}}\,.$

We say that ${\bm{\mu}}$ is expanding, steady or shrinking if, respectively,
$\ell_{{\bm{\mu}}}>0$, $\ell_{{\bm{\mu}}}=0$ or $\ell_{{\bm{\mu}}}<0$.

Building from this definition, we can derive equivalent conditions for a
Dorfman bracket to be an algebraic soliton for the generalized Ricci flow.

######  Proposition 7.4.

Let ${\bm{\mu}}\in\mathcal{D}$. Then, the following are equivalent:

  1. (1)

${\bm{\mu}}$ is an algebraic soliton;

  2. (2)

the generalized bracket flow starting at ${\bm{\mu}}$ evolves only by scaling;

  3. (3)

${\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}=\lambda{\rm Id}+\bm{D}$ with
$\bm{D}\in{\rm Der}_{\lambda}({\bm{\mu}})$ and $\lambda\in\mathbb{R}$;

  4. (4)

there exist $D\in{\rm Der}(\mu)$ and $\lambda\in\mathbb{R}$ such that

(47) $\begin{cases}{\rm Ric}_{\mu,H}^{B}=\lambda{\rm Id}+D\,,\\\
\Delta_{\mu}H=\lambda H+\rho({\rm Ric}_{\mu,H}^{B})H\,.\end{cases}$

###### Proof.

The equivalence between Item 1 and Item 2 is trivial using (46) and 7.2. On
the other hand, if ${\bm{\mu}}$ is an algebraic soliton, the fact that
$\Theta({\rm Id}){\bm{\mu}}=-{\bm{\mu}}$, for all ${\bm{\mu}}\in\mathcal{D}$,
implies ${\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}=\lambda{\rm Id}+\bm{D}$ with
$\bm{D}\in{\rm Der}({\bm{\mu}})$. Moreover,
${\mathcal{R}c}_{\bm{\mu}}-A_{\bm{\mu}}\in\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},{\langle\,\cdot\,,\cdot\,\rangle})$,
then, $\bm{D}$ has to satisfy

$2\lambda{\langle\,\cdot\,,\cdot\,\rangle}+\langle{\bm{D}}\cdot,\cdot\rangle+\langle\cdot,{\bm{D}}\cdot\rangle=0\,,$

which gives Item 3. Viceversa, it is easy to show that if Item 3 holds, then
(46) is satisfied with $\ell_{{\bm{\mu}}}=-\lambda$. Furthermore, 4.8
guarantees the equivalence between (46) and

$\begin{cases}\theta({\rm Ric}^{B}_{\mu,H}+\ell_{{\bm{\mu}}}{\rm
Id})\mu=0\,,\\\ \rho({\rm
Ric}_{\mu,H}^{B})H-\Delta_{\mu}H=\ell_{{\bm{\mu}}}H\,\end{cases}$

which is equivalent to (47) with $\lambda=-\ell_{{\bm{\mu}}}$, concluding the
proof. ∎

Item 3 and Item 4 in 7.4 can be considered as the static definition, in terms
of both generalized and classical objects, of algebraic solitons and so 7.4
has to be regarded as the analogue of 3.4 in the varying Dorfman bracket
framework.

Moreover, 7.4 allows us to construct from algebraic solitons for the
generalized Ricci flow generalized metrics which are generalized Ricci
solitons as in 3.1.

######  Lemma 7.5.

Let ${\bm{\mu}}$ be an algebraic soliton for the generalized Ricci flow. Then,
$\mathcal{G}_{{\bm{\mu}}}$, defined as in (37), is a generalized Ricci
soliton.

###### Proof.

The first equation in (47) can be equivalently written, in terms of symmetric
$(0,2)$-tensors, as

${\rm Ric}_{g_{\mu},H}^{B}=\lambda
g_{\mu}+\frac{1}{2}(g_{\mu}(D\cdot,\cdot)+g_{\mu}(\cdot,D\cdot)=\lambda
g_{\mu}-\frac{1}{2}\mathcal{L}_{X_{D}}g_{\mu}\,,$

where $X_{D}=\frac{\textrm{d}}{\textrm{d}t}\big{|}_{t=0}\varphi_{t}$ with
$\varphi_{t}\in{\rm Aut}(\mathsf{G}_{\mu})$ such that
$\textrm{d}_{e}\varphi_{t}=e^{tD}$. Moreover, plugging the first equation of
(47) into the second one, we have

$\Delta_{\mu}H=\lambda H+\rho(\lambda{\rm Id}+D)H=-2\lambda
H+\rho(D)H=-2\lambda H+\mathcal{L}_{X_{D}}H\,.$

The claim follows using 3.4. ∎

Before going into the discussion of algebraic solitons in the nilpotent case,
we will need the following preliminary Lemma about $\lambda$-derivations,
which is nothing but an adaptation of 3.3 to the invariant case.

######  Lemma 7.6.

Let $(\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}}$ be a left-invariant
ECA over a Lie group $\mathsf{G}$. Then, we have that, for any
$c\in\mathbb{R}\backslash\\{0\\}$ and $\lambda\in\mathbb{R}$,

${\rm
Aut}(c\cdot(\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}},(\mathfrak{g}\oplus\mathfrak{g}^{*})_{{\bm{\mu}}})=\left\\{\bar{A}_{c}e^{b}:b\in\Lambda^{2}\mathfrak{g}^{*}\,,\quad
A\in{\rm Aut}(\mu)\,,\quad A^{-1}\cdot H=c(H-\textrm{d}b)\right\\}$

and

${\rm Der}_{\lambda}({\bm{\mu}})=\left\\{\begin{pmatrix}D&0\\\
\alpha&-2\lambda{\rm Id}-D^{*}\end{pmatrix}:D\in{\rm
Der}(\mu),\quad\rho(D)H-\textrm{d}_{\mu}\alpha-2\lambda H=0\right\\}\,.$

###### Proof.

The first assertion follows directly from the discussion in the non-invariant
case. 3.2 guarantees that a $\lambda$-derivation has to be of the form

$\bm{D}=\begin{pmatrix}D&0\\\ \alpha&-2\lambda{\rm Id}-D^{*}\end{pmatrix}\,,$

where $D\in{\rm Der}(\mu)$ and $\rho(D)H-\textrm{d}\alpha-2\lambda H=0$ . ∎

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 [Lau01].

######  Proposition 7.7.

Let ${\bm{\mu}}\in\mathcal{D}$ be a generalized nilsoliton. Then, it is
expanding. Moreover, the derivation $D$ satisfies the following property:

${\rm tr}\left(\left(D+\frac{1}{4}H^{2}\right)D^{\prime}\right)=-\lambda{\rm
tr}(D^{\prime})\,,$

for all $D^{\prime}\in{\rm Der}(\mu)$ defining
$\bm{D}^{\prime}=(D^{\prime},\alpha)\in{\rm
Der}_{\lambda^{\prime}}({\bm{\mu}})$, for some
$\lambda^{\prime}\in\mathbb{R}$. Finally,
$\bar{g}(\iota_{X}H,\textrm{d}_{\mu}^{*}H)=0$, for all $X\in\mathfrak{g}$.

###### Proof.

We have that

$\ell_{{\bm{\mu}}}=6\bar{g}(\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}},\operatorname{M}_{{\bm{\mu}}})=6\lvert\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}\rvert^{2}\geq
0\,,$

where the last equality follows from 4.14. On the other hand,
$\lvert\mathcal{R}c_{{\bm{\mu}}}-A_{{\bm{\mu}}}\rvert^{2}=0$ would, in
particular, imply that

${\rm Ric}_{\mu}=\frac{1}{4}H^{2}\geq 0\,,$

which cannot happen in the nilpotent case, see [Lau11, Lemma 4.2, (iii)],
giving us the first claim.

As regards the second, fixed $\bm{D}^{\prime}=(D^{\prime},\alpha)\in{\rm
Der}_{\lambda^{\prime}}({\bm{\mu}})$, we have that

(48) $\bar{g}_{\mathfrak{l}}(\lambda{\rm Id}+\bm{D},\lambda^{\prime}{\rm
Id}+\bm{D}^{\prime})=\frac{1}{6}\bar{g}(\Theta(\lambda^{\prime}{\rm
Id}){\bm{\mu}},{\bm{\mu}})=-\frac{\lambda^{\prime}}{6}\lvert{\bm{\mu}}\rvert^{2}=\lambda^{\prime}\left(2{\rm
R}_{\mu}-\frac{1}{6}\lvert H\rvert^{2}\right)\,.$

On the other hand, we can use 7.4 to deduce that
$\bm{D}=(D,-\textrm{d}_{\mu}^{*}H)$ and then obtain

$\bar{g}_{\mathfrak{l}}(\lambda{\rm Id}+\bm{D},\lambda^{\prime}{\rm
Id}+\bm{D}^{\prime})=2{\rm tr}((\lambda{\rm Id}+D)(\lambda^{\prime}{\rm
Id}+D^{\prime}))-\frac{1}{6}\bar{g}(\textrm{d}_{\mu}^{*}H,\alpha)\,.$

The fact that $\bm{D}^{\prime}\in{\rm Der}_{\lambda^{\prime}}({\bm{\mu}})$
implies, thanks to 7.6, that
$\textrm{d}_{\mu}\alpha=\rho(D^{\prime})H-2\lambda^{\prime}H$. This can be
used to infer that

(49) $\displaystyle\bar{g}_{\mathfrak{l}}(\lambda{\rm
Id}+\bm{D},\lambda^{\prime}{\rm Id}+\bm{D}^{\prime})=$
$\displaystyle\,2\lambda^{\prime}{\rm tr}({\rm Ric}_{\mu,H}^{B})+2\lambda{\rm
tr}(D^{\prime})+2{\rm
tr}(DD^{\prime})-\frac{1}{6}\bar{g}(\alpha,\textrm{d}_{\mu}^{*}H)$
$\displaystyle=$ $\displaystyle\,2\lambda^{\prime}\left({\rm
R_{\mu}}-\frac{1}{4}\lvert H\rvert^{2}\right)+2\lambda{\rm
tr}(D^{\prime})+2{\rm tr}(DD^{\prime})+\frac{1}{2}{\rm
tr}(H^{2}D^{\prime})+\frac{1}{3}\lambda^{\prime}\lvert H\rvert^{2}$
$\displaystyle=$ $\displaystyle\,\lambda^{\prime}\left(2{\rm
R}_{\mu}-\frac{1}{6}\lvert H\rvert^{2}\right)+2\lambda{\rm
tr}(D^{\prime})+2{\rm tr}(DD^{\prime})+\frac{1}{2}{\rm
tr}(H^{2}D^{\prime})\,.$

Now, it is sufficient to use (49) in (48) to obtain the claim. Finally, recall
that for metric nilpotent Lie algebras, any symmetric derivation $D$ satisfies
$\operatorname{tr}(\operatorname{ad}_{X}D)=0$, for all $X\in\mathfrak{g}$.
Then, the soliton equation in $\mu$ implies

$0=\operatorname{tr}(\operatorname{ad}_{X}H^{2})=-\frac{1}{3}\bar{g}(\rho(\operatorname{ad}_{X})H,H)=-\frac{1}{3}\bar{g}(\mathcal{L}_{X}H,H)=-\frac{1}{3}\bar{g}(\textrm{d}_{\mu}\iota_{X}H,H)=-\frac{1}{3}\bar{g}(\iota_{X}H,\textrm{d}_{\mu}^{*}H),$

as claimed. ∎

Motivated by the wide range of results concerning classical nilsolitons proved
throughout the years, see for instance [Lau01], many questions about
generalized nilsolitons can be raised. First of all, as usual, let
$(E,\overline{\mathcal{G}})$ be a left-invariant metric ECA over a nilpotent
Lie group $\mathsf{G}$ and assume that $\overline{\mathcal{G}}$ is a
generalized nilsoliton. Using the preferred isotropic splitting induced by
$\overline{\mathcal{G}}$, we can identify
$(E,\overline{\mathcal{G}})\simeq((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\mathcal{G}(\bar{g},0))$,
for some closed $H\in\Lambda^{3}\mathfrak{g}^{*}$. So, in this setting, one
can wonder if $\overline{\mathcal{G}}$ is unique, up to scaling and up to the
action of ${\rm Aut}([\cdot,\cdot])$, among left-invariant generalized metrics
defining a preferred representative $H^{\prime}\in[H]\in
H^{3}(\mathfrak{g},\mathbb{R})$.

######  Remark 7.8.

Considering $[H]\in H^{3}(\mathsf{G},\mathbb{R})$, the uniqueness is not true
in general. Indeed, on $\mathfrak{n}_{3}={\rm Lie}({\rm Heis}(3,\mathbb{R}))$,
the following

$\mu(e_{1},e_{2})=e_{3}\,,\quad H=e^{123}\,,\quad\lambda=-2\,,\quad D={\rm
diag}(1,1,2)\,$

defines a generalized nilsoliton, see 10.1 for the proof, with $[H]=0\in
H^{3}({\rm Heis}(3,\mathbb{R}),\mathbb{R})$. One can easily observe that the
above generalized nilsoliton defines exactly the expanding soliton for the
generalized Ricci flow described in Example 3.6. On the other hand,
$\mu(e_{1},e_{2})=e_{3}$ defines also the classical nilsoliton, see [Lau02,
Theorem 4.2], 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 4.4, fixed $E$ a left-invariant ECA over a Lie group
$\mathsf{G}$, not necessarily nilpotent, the space of left-invariant
generalized metric $\mathcal{M}^{\mathsf{G}}$ can be presented as a
homogeneous space where $\mathsf{L}$ is acting transitively by conjugation.

Of course, 4.4 gives also that the action of
$\mathsf{SO}(\langle\cdot,\cdot\rangle)$ is transitive on
$\mathcal{M}^{\mathsf{G}}$. This allows us to present
$\mathcal{M}^{\mathsf{G}}$ as a homogeneous space in a different manner.
Clearly, the isotropy of $\mathsf{SO}(\langle\cdot,\cdot\rangle)$ in
$\overline{\mathcal{G}}\in\mathcal{M}^{\mathsf{G}}$ is

$\overline{\mathcal{G}}_{\mathsf{SO}(\langle\cdot,\cdot\rangle)}=\left\\{F\in\mathsf{SO}(\langle\cdot,\cdot\rangle):F\overline{\mathcal{G}}F^{-1}=\overline{\mathcal{G}}\right\\}=:{\rm
Isom}(\overline{\mathcal{G}},\mathsf{SO})\,.$

On the other hand, we observe that

$\theta\colon\mathsf{SO}(\langle\cdot,\cdot\rangle)\to\mathsf{SO}(\langle\cdot,\cdot\rangle)\,,\quad\sigma(F)=\overline{\mathcal{G}}F\overline{\mathcal{G}}\,,$

is an automorphism of $\mathsf{SO}(\langle\cdot,\cdot\rangle)$ which is
clearly involutive, and the set of fixed points of $\theta$ precisely
coincides with ${\rm Isom}(\overline{\mathcal{G}},\mathsf{SO}).$ Thus, we
obtain that $(\mathsf{SO}(\langle\cdot,\cdot\rangle),{\rm
Isom}(\overline{\mathcal{G}},\mathsf{SO}))$ is a Riemannian symmetric pair, as
defined in [Hel01, IV 3.4]. As a consequence, we obtain a reductive
decomposition of
$\mathfrak{so}(\langle\cdot,\cdot\rangle)=\mathfrak{h}\oplus\mathfrak{m}$ such
that $[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}$, where, using
$(E,\overline{\mathcal{G}})\simeq((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\mathcal{G}(\bar{g},0))$,

$\mathfrak{h}:={\rm Lie}({\rm
Isom}(\overline{\mathcal{G}},\mathsf{SO}))=\mathfrak{so}(\mathfrak{g}\oplus\mathfrak{g}^{*},\overline{\mathcal{G}})=\left\\{\begin{pmatrix}A&\bar{g}^{-1}\alpha\bar{g}^{-1}\\\
\alpha&-A^{*}\end{pmatrix}:A\in\mathfrak{so}(n)\,,\quad\alpha\in\Lambda^{2}\mathfrak{g}^{*}\right\\}$

and

$\mathfrak{m}:=\ker(\textrm{d}\theta+{\rm
Id})=\left\\{\begin{pmatrix}A&-\bar{g}^{-1}\alpha\bar{g}^{-1}\\\
\alpha&-A^{*}\end{pmatrix}:A\in{\rm
Sym}(n)\,,\quad\alpha\in\Lambda^{2}\mathfrak{g}^{*}\right\\}\,.$

As usual, we will be identifying $\mathfrak{m}$ with
$T_{\overline{\mathcal{G}}}\mathcal{M}^{\mathsf{G}}$. Now, using again the
identification
$(E,\overline{\mathcal{G}})\simeq((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\mathcal{G}(\bar{g},0))$,
we can define a non degenerate bilinear form $\bar{g}$ on
$\mathfrak{so}({\langle\,\cdot\,,\cdot\,\rangle})$ such that, for all
$(A,\alpha),(A^{\prime},\alpha^{\prime})\in\mathfrak{m}$ and
$(B,\beta),(B^{\prime},\beta^{\prime})\in\mathfrak{h}$,

$\bar{g}((A,\alpha),(A^{\prime},\alpha^{\prime}))=2{\rm
tr}(AA^{\prime})+2\bar{g}(\alpha,\alpha^{\prime})\,,\quad\bar{g}((B,\beta),(B^{\prime},\beta^{\prime}))=-\frac{3}{4}({\rm
tr}(BB^{\prime})+\bar{g}(\beta,\beta^{\prime}))\,.$

It is not hard to see that $\bar{g}$ is an Ad$({\rm
Isom}(\overline{\mathcal{G}},\mathsf{SO}))$-invariant bilinear form which,
when restricted to $\mathfrak{l}$, coincides with the inner product
$\bar{g}_{\mathfrak{l}}$ defined in (28). Moreover, using [KN96, Theorem 3.5],
the $\mathsf{SO}(\langle\cdot,\cdot\rangle)$-invariant Riemannian metric
induced by it on $\mathcal{M}^{\mathsf{G}}$ will be naturally reductive. This
readily guarantees that all the geodesics emanating from
$\overline{\mathcal{G}}$ are of the form
$\exp(tX)\cdot\overline{\mathcal{G}}$, for some $X\in\mathfrak{m}$, see [KN96,
Theorem 3.3]. Finally, [KN96, Theorem 3.5] guarantees that
$(\mathcal{M}^{\mathsf{G}},\bar{g})$ has all non-positive sectional
curvatures.

Keeping in mind this and building from [Heb98], we can define the generalized
scalar functional

$\mathcal{S}\colon\mathcal{M}^{\mathsf{G}}\to\mathbb{R}\,,$

which associates to a given left-invariant generalized metric its generalized
scalar curvature as in (39). We will study some analytic properties of this
functional.

######  Proposition 7.9.

Let $E$ be a left-invariant ECA over a nilpotent Lie group $\mathsf{G}$. Then,
we have:

  1. (1)

the gradient of $\mathcal{S}$ at any
$\overline{\mathcal{G}}\in\mathcal{M}^{\mathsf{G}}$ is given by

$\nabla_{\overline{\mathcal{G}}}\mathcal{S}=\tilde{\operatorname{M}}(\overline{\mathcal{G}}):=\begin{pmatrix}{\rm
Ric}^{B}_{\bar{g},H}&\frac{1}{6}\bar{g}^{-1}\textrm{d}_{\bar{g}}^{*}H\bar{g}^{-1}\\\
-\frac{1}{6}\textrm{d}_{\bar{g}}^{*}H&-({\rm
Ric}_{\bar{g},H}^{B})^{*}\end{pmatrix}$

after using
$(E,\overline{\mathcal{G}})\simeq_{\sigma}((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\mathcal{G}(\bar{g},0))$;

  2. (2)

$\mathcal{S}$ is concave along geodesics, i.e. if $\gamma$ is a geodesic
emanating from $\overline{\mathcal{G}}\in\mathcal{M}^{\mathsf{G}}$, then
$(\mathcal{S}\circ\gamma)^{{}^{\prime\prime}}(0)\leq 0$;

  3. (3)

$(\mathcal{S}\circ\gamma)^{{}^{\prime\prime}}(0)=0$ if and only if
$\gamma(t)=\exp(tX)\cdot\overline{\mathcal{G}}$ where $X\in{\rm
Der}_{0}([\cdot,\cdot])\cap\mathfrak{m}$. In this case,
$\mathcal{S}\circ\gamma$ is constant.

###### Proof.

We fix $\overline{\mathcal{G}}\in\mathcal{M}^{\mathsf{G}}$ and, as usual, we
identify
$(E,\overline{\mathcal{G}})\simeq_{\sigma}((\mathfrak{g}\oplus\mathfrak{g}^{*})_{H},\mathcal{G}(\bar{g},0))$.
We will make use of the moving Dorfman brackets approach. First of all, we
observe that, if $\alpha\in\Lambda^{2}\mathfrak{g}^{*}$ and
${\bm{\mu}}\in\mathcal{D}$, we have that

$2\left\langle\overline{\mathcal{G}}\Theta\begin{pmatrix}0&-\bar{g}^{-1}\alpha\bar{g}^{-1}\\\
0&0\end{pmatrix}{\bm{\mu}}(e_{i},e_{j}),{\bm{\mu}}(e_{i},e_{j})\right\rangle=-{\bm{\mu}}_{ijk}{\bm{\mu}}_{ij}^{s}\alpha_{ks}\,,$

while

$4\left\langle\overline{\mathcal{G}}\Theta\begin{pmatrix}0&-\bar{g}^{-1}\alpha\bar{g}^{-1}\\\
0&0\end{pmatrix}{\bm{\mu}}(e_{i},e^{j}),{\bm{\mu}}(e_{i},e^{j})\right\rangle=-2{\bm{\mu}}_{is}^{j}{\bm{\mu}}_{iks}\alpha_{jk}$

which gives us that

(50) $\bar{g}\left(\Theta\begin{pmatrix}0&-\bar{g}^{-1}\alpha\bar{g}^{-1}\\\
0&0\end{pmatrix}{\bm{\mu}},{\bm{\mu}}\right)=3{\bm{\mu}}_{ijk}{\bm{\mu}}_{ij}^{s}\alpha_{sk}=-\bar{g}(\textrm{d}_{\mu}\alpha,H)\,.$

Then, combining (50) with (31), we obtain that, for
$X=(A,\alpha)\in\mathfrak{m}$,

$\displaystyle\bar{g}(\Theta(X){\bm{\mu}},{\bm{\mu}})=$
$\displaystyle\,\bar{g}\left(\Theta\begin{pmatrix}A&0\\\
\alpha&-A^{*}\end{pmatrix}{\bm{\mu}},{\bm{\mu}}\right)+\bar{g}\left(\Theta\begin{pmatrix}0&-\bar{g}^{-1}\alpha\bar{g}^{-1}\\\
0&0\end{pmatrix}{\bm{\mu}},{\bm{\mu}}\right)=12{\rm tr}({\rm
Ric}_{\mu,H}^{B}A)-2\bar{g}(\textrm{d}_{\mu}\alpha,H)\,.$

On the other hand, considering

$\tilde{\operatorname{M}}_{{\bm{\mu}}}=\begin{pmatrix}{\rm
Ric}^{B}_{\mu,H}&\frac{1}{6}\bar{g}^{-1}\textrm{d}^{*}_{\mu}H\bar{g}^{-1}\\\
-\frac{1}{6}\textrm{d}^{*}_{\mu}H&-({\rm Ric}_{\mu,H}^{B})^{*}\end{pmatrix}$

we then obtain that

$\bar{g}(\tilde{\operatorname{M}}_{{\bm{\mu}}},X)=2{\rm tr}({\rm
Ric}_{\mu,H}^{B}A)-\frac{1}{3}\bar{g}(\alpha,\textrm{d}_{\mu}^{*}H)=\frac{1}{6}\bar{g}(\Theta(X){\bm{\mu}},{\bm{\mu}})=\frac{1}{12}\frac{\textrm{d}}{\textrm{d}t}\Big{|}_{t=0}\lvert\exp(tX)\cdot{\bm{\mu}}\rvert^{2}\,.$

We now compute the gradient of $\mathcal{S}$ at $\overline{\mathcal{G}}$. If
${\bm{\mu}}\in\mathcal{D}$ is the given Dorfman bracket on $E$, we know that,
for any $t$, the generalized metric $\exp(tX)\cdot\overline{\mathcal{G}}$ is
isometric to $\mathcal{G}_{\exp(-tX)\cdot{\bm{\mu}}}$. So, given
$X\in\mathfrak{m}$, we have that

$\displaystyle\textrm{d}_{\overline{\mathcal{G}}}\mathcal{S}(X)=$
$\displaystyle\,\frac{\textrm{d}}{\textrm{d}t}\Big{|}_{t=0}\mathcal{S}(\exp(tX)\cdot\overline{\mathcal{G}})=-\frac{1}{12}\frac{\textrm{d}}{\textrm{d}t}\Big{|}_{t=0}\lvert\exp(-tX)\cdot{\bm{\mu}}\rvert^{2}=\frac{1}{6}\bar{g}(\Theta(X){\bm{\mu}},{\bm{\mu}})=\bar{g}(\tilde{\operatorname{M}}_{{\bm{\mu}}},X)=\bar{g}(\tilde{\operatorname{M}}(\overline{\mathcal{G}}),X)\,,$

giving us the first claim.

Let now $\gamma(t)=\exp(tX)\cdot\overline{\mathcal{G}}$, for some
$X\in\mathfrak{m}$, be a geodesic emanating from $\overline{\mathcal{G}}$, we
have

(51)
$(\mathcal{S}\circ\gamma)^{{}^{\prime\prime}}(0)=-\frac{1}{12}\frac{\textrm{d}^{2}}{\textrm{d}t^{2}}\Big{|}_{t=0}\lvert\exp(-tX)\cdot{\bm{\mu}}\rvert^{2}=-\frac{1}{6}(\bar{g}(\Theta(X)\Theta(X){\bm{\mu}},{\bm{\mu}})+\lvert\Theta(X){\bm{\mu}}\rvert^{2})=-\frac{1}{3}\lvert\Theta(X){\bm{\mu}}\rvert^{2}\leq
0\,,$

giving us the second claim.

Finally, clearly, if $X\in{\rm Der}_{0}({\bm{\mu}})\cap\mathfrak{m}$,
$\exp(tX)\in{\rm Aut}({\bm{\mu}})$ which guarantees that
$\mathcal{S}\circ\gamma$ is constant for all $t$, the viceversa holds too
thanks to (51). ∎

######  Remark 7.10.

The element $\tilde{\operatorname{M}}_{{\bm{\mu}}}\in\mathfrak{m}$ can also be
viewed as the moment map of the
$\mathsf{SO}({\langle\,\cdot\,,\cdot\,\rangle})$-action on the space of
Dorfman brackets $\mathcal{D}$. Indeed, it precisely satisfies the relation:

$\bar{g}(\tilde{\operatorname{M}}_{\bm{\mu}},X)=\frac{1}{6}\,\bar{g}(\Theta(X){\bm{\mu}},{\bm{\mu}})\,,\qquad
X\in\mathfrak{m}\,.$

Unfortunately, $\tilde{\operatorname{M}}_{{\bm{\mu}}}$ coincides with the
generalized Ricci curvature only on the subspace of Dorfman bracket with
harmonic torsion.

7.9 can be considered as the analogue, in the generalized setting, of [Heb98,
Lemma 3.3] specialized in the nilpotent case. So, following the steps in
[Lau01, Theorem 3.5], we can give a partial answer to the uniqueness question,
proving that generalized nilsolitons with harmonic torsion are unique.

######  Theorem 7.11.

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

###### Proof.

Let us now consider the following closed Lie subgroup of
$\mathsf{SO}({\langle\,\cdot\,,\cdot\,\rangle})$:

$\mathsf{K}:=\pm\prod_{c\in\mathbb{R}^{+}}c^{-\frac{1}{2}}{\rm Aut}(c\cdot
E,E)\,.$

First of all, we need some clarifications about $\mathsf{K}$. The action of an
element in $\mathsf{K}$ on $\overline{\mathcal{G}}\in\mathcal{M}^{\mathsf{G}}$
is defined as follows: an element $c^{-\frac{1}{2}}F\in\mathsf{K}$ acts on
$\overline{\mathcal{G}}$ as the following composition:

(52)
$c^{-\frac{1}{2}}F\colon(E,{\langle\,\cdot\,,\cdot\,\rangle},c^{-\frac{1}{2}}{\bm{\mu}},\overline{\mathcal{G}})\to(c\cdot
E,\overline{\mathcal{G}})\to(E,F\cdot\overline{\mathcal{G}})$

where the first arrow is defined, using the isometry induced by the preferred
isotropic splitting of $\overline{\mathcal{G}}$, by the transformation

$\overline{c^{-\frac{1}{2}}{\rm
Id}}\colon((\mathfrak{g}\oplus\mathfrak{g}^{*})_{c^{-\frac{1}{2}}{\bm{\mu}}},\mathcal{G}(\bar{g},0))\to((\mathfrak{g}\oplus\mathfrak{g}^{*})_{cH},\mathcal{G}(c\bar{g},0))$

which is an isometry of ECA’s thanks to 4.10. Furthermore, we can use the
explicit isometry in Remark 2.12 to identify

$((\mathfrak{g}\oplus\mathfrak{g}^{*})_{cH},\mathcal{G}(c\bar{g},0))\simeq(c\cdot
E,\overline{\mathcal{G}})\,.$

In conclusion, the first arrow in (52) is the composition of the above
isometries and it is then induced by the transformation $c^{-\frac{1}{2}}{\rm
Id}.$ Finally, the last arrow in (52) is just the usual action of the element
$F\in{\rm Aut}(c\cdot E,E)$. Moreover, by 7.6, any element $c^{-\frac{1}{2}}F$
can be represented with

$c^{-\frac{1}{2}}{\rm
Id}\circ\bar{A}_{c}e^{b}=\begin{pmatrix}c^{-\frac{1}{2}}A&0\\\
((c^{-\frac{1}{2}}A)^{-1})^{*}b&((c^{-\frac{1}{2}}A)^{-1})^{*}\end{pmatrix}=\overline{c^{-\frac{1}{2}}A}e^{b}\in\mathsf{SO}({\langle\,\cdot\,,\cdot\,\rangle})\,,$

where $b\in\Lambda^{2}\mathfrak{g}^{*}\,,$ and $A\in{\rm Aut}(\mu)$ such that
$A^{-1}\cdot H=c(H-\textrm{d}b)$. Finally, $-c^{\frac{1}{2}}F$ is given by the
composition of

$\overline{-{\rm
Id}}\colon((\mathfrak{g}\oplus\mathfrak{g}^{*})_{-c^{-\frac{1}{2}}{\bm{\mu}}},\mathcal{G}(\bar{g},0))\to((\mathfrak{g}\oplus\mathfrak{g}^{*})_{c^{-\frac{1}{2}}{\bm{\mu}}},\mathcal{G}(\bar{g},0))$

with the previously defined action. Now, we can prove that $\mathsf{K}$ is
actually a Lie subgroup of $\mathsf{SO}({\langle\,\cdot\,,\cdot\,\rangle}).$
Clearly, it is sufficient to prove that, if $c^{-\frac{1}{2}}F\in
c^{-\frac{1}{2}}{\rm Aut}(c\cdot E,E)$ and
$\gamma^{-\frac{1}{2}}G\in\gamma^{-\frac{1}{2}}{\rm Aut}(\gamma\cdot E,E)$,
then $c^{-\frac{1}{2}}F(\gamma^{-\frac{1}{2}}G)^{-1}\in\mathsf{K}$. It is not
hard to see that