Generalized Ricci flow on nilpotent Lie groups

We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of M. Garcia-Fernandez and J. Streets. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows, such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.


Introduction
Generalized geometry, building on the work of N. Hitchin [Hit03] and M. Gualtieri [Gua04] and the structure of Courant algebroids, constitutes a rich mathematical environment. In this spirit, for example, a generalized almost complex structure on M 2m , defined by an orthogonal automorphism J of TM , J 2 = − Id TM , determines a U(m, m)-reduction of GL(TM ). The integrability of such a structure is expressed through an involutivity condition with respect to a natural bracket operation, called the Dorfman bracket: On the other hand, a generalized Riemannian metric on M n , defined by a symmetric (with respect to ·, · ) and involutive automorphism G of TM , determines an O(n) × O(n)-reduction of GL(TM ).
More generally, one can consider a Courant algebroid E over M , namely a smooth vector bundle over M endowed with a pairing ·, · and a bracket [·, ·] satisfying certain properties so that TM , endowed with (1.1) and (1.2), is a special case. On the generic Courant algebroid one can then study reductions of GL(E), such as generalized almost complex structures and generalized (pseudo-)Riemannian metrics.
In [Str17,Gar19,GS21], the authors introduced a flow of generalized (pseudo-)Riemannian metrics on a Courant algebroid E over a smooth manifold M , generalizing the classical Ricci flow of R. Hamilton [Ham82] and the B-field renormalization group flow of Type II string theory (see [Pol98]). The generalized Ricci flow, as we shall refer to this flow from now on, is actually a flow for a pair of families of generalized (pseudo-)Riemannian metrics G ∈ Aut(E) and divergence operators div : Γ(E) → C ∞ (M ), the latter of which are required in order to "gauge-fix" curvature operators associated with a generalized (pseudo-)Riemannian metric.
The paper is organized as follows: Section 2 is devoted to a review of the setting of generalized geometry -including the notions of Courant algebroid, generalized curvature tensors and the definition of generalized Ricci flow -and of the algebraic framework of nilpotent Lie groups.
In Section 3 we introduce the notion of generalized Ricci soliton, which derives from the study of self-similar (in a suitable sense) solutions to the generalized Ricci flow on exact Courant algebroids. This condition generalizes the Ricci soliton condition Rc g = λg + L X g, where Rc g denotes the Ricci tensor of g, λ ∈ R and L X g denotes the Lie derivative of g with respect to a vector field X. We show that, when working on a Lie group and considering left-invariant structures, this condition descends to an algebraic condition on the Lie algebra of the group.
Borrowing from the ideas of J. Lauret, in Section 4 we consider left-invariant Dorfman brackets on simply connected nilpotent Lie groups, describing them as elements of an algebraic subset of the vector space of skew-symmetric bilinear forms on R n ⊕ (R n ) * , for the suitable n. We then define a family of flows of such structures, showing that they generalize the constructions known in literature as bracket flows, which have been extensively used to rephrase geometric flows on (nilpotent) Lie groups (see for example [Lau11]). This justifies our definition of generalized bracket flows.
In Section 5, we perform explicit computations of generalized Ricci solitons and exhibit an example of generalized bracket flow on the three-dimensional Heisenberg group.
In Section 6, we study solutions of the generalized Ricci flow on the Heisenberg group, highlighting the differences with the classical Ricci flow. the supervision of Anna Fino. To her the author wishes to express his most sincere gratitude. The author also wishes to thank Mario Garcia-Fernandez for useful comments and Jeffrey Streets for pointing out reference [Str17]. He also thanks David Krusche for noting an imprecision in formula (2.5), and an anonymous referee for useful comments which helped improve the presentation of the paper. The author was supported by GNSAGA of INdAM.

Preliminaries
2.1. Courant algebroids. Let V be a real vector space of dimension n. We start by recalling a few facts about the algebra of the vector space V ⊕ V * ; for more details, see [Gua04].
V ⊕ V * can be endowed with a natural symmetric bilinear form of neutral signature and with a canonical orientation provided by the preimage of 1 ∈ R in the isomorphism Consider the Lie group SO(V ⊕ V * ) ∼ = SO(n, n) of automorphisms of V ⊕ V * preserving the pairing ·, · and the canonical orientation. Its Lie algebra so(V ⊕ V * ) ∼ = so(n, n) consists of endomorphisms T ∈ gl(V ⊕ V * ) which are skew-symmetric with respect to ·, · , namely (2.1) T z 1 , z 2 + z 1 , T z 2 = 0 for all z 1 , z 2 ∈ V ⊕ V * . Seeing T as a block matrix, (2.1) dictates T to be of the form for some φ ∈ gl(V ), B ∈ Λ 2 V * and β ∈ Λ 2 V , recovering the fact that where the former isomorphism is given by T → T ·, · .
Via the exponential map exp : • e φ = e φ 0 0 (e −φ ) * , which extends to an embedding of the whole GL In the case V = R n , the image of this embedding will be denoted by GL n . Let M be an oriented smooth manifold of positive dimension n.
Definition 2.1. A Courant algebroid over M is a smooth vector bundle E → M equipped with: • ·, · , a fiberwise nondegenerate bilinear form, which allows to identify E and its dual E * , viewing z ∈ E as z, · ∈ E * , • [·, ·], a bilinear operator on Γ(E), • a bundle homomorphism π : E → T M , called the anchor, which satisfy the following properties for all z, z i ∈ Γ(E), i = 1, 2, 3, f ∈ C ∞ (M ): ( is exact, namely if the anchor map is surjective and its kernel is exactly the image of π * .
By the classification of P.Ševera [Sev98], isomorphism classes of exact Courant algebroids over and (twisted) Dorfman bracket In what follows, let E be a Courant algebroid over M , with rk(E) = 2n and pairing ·, · of neutral signature. Definition 2.3. A generalized Riemannian metric on E is an O(n) × O(n) -reduction of O(E), the O(n, n)-principal subbundle of orthonormal frames of E with respect to the pairing ·, · . Explicitly, it is equivalently determined by • a subbundle E + of E, rk(E + ) = n, on which ·, · is positive-definite, • an automorphism G of E which is involutive, namely G 2 = Id E , and such that G·, · is a positive-definite metric on E. Given E + , denoting by E − its orthogonal complement with respect to ·, · , G is defined by G| E ± = ± Id E ± . E ± can then be recovered as the ±1-eigenbundles of G. Given z ∈ E, we shall denote by z ± its orthogonal projections along E ± .
Example 2.4. Every generalized Riemannian metric on the exact Courant algebroid E H is of the form for some g Riemannian metric and B 2-form on M (see [Gua04, Section 6.2]). The corresponding E ± are E ± = e B {X ± g(X), X ∈ T M }, where by g(X) we mean g(X, ·). Notice that G is of the form 0 g −1 g 0 in the splitting E H+dB .
2.2. Generalized curvature. We now recall the definition of generalized connection on a Courant algebroid E, showing how these objects can be used to associate curvature operators with a generalized Riemannian metric G. Unlike the Riemannian case, where the uniqueness of the Levi-Civita connection allows to single out canonical curvature operators for a given Riemannian metric, in the generalized setting there are plenty of torsion-free generalized connections compatible with a generalized Riemannian metric G, and these may define different curvature operators. To gauge-fix them, one needs to additionally fix a divergence operator. For further details, we refer the reader to [Gar19] and [CD19].
Definition 2.5. A generalized connection on a Courant algebroid E is a linear map which satisfies a Leibniz rule and a compatibility condition with ·, · : If T D = 0, the generalized connection D is said to be torsion-free. Given a generalized connection D on E which is compatible with a generalized Riemannian metric G, one can define curvature operators where o(E ± ) = ·, · −1 Λ 2 E * ± denotes the Lie algebra of skew-symmetric endomorphisms of E ± with respect to ·, · , by

One then has associated Ricci tensors
. Given a generalized connection D on E, one may define the associated divergence operator div D (z) = tr(Dz).
Remark 2.7. Divergence operators on E form an affine space over the vector space Γ(E) ∼ = Γ(E * ). Fixing a divergence operator div 0 , any other div is of the form div = div 0 − z, · for some z ∈ Γ(E).
Proposition 2.8. [Gar19, Proposition 4.4] Let D i , i = 1, 2, be torsion-free generalized connections on E compatible with a given generalized Riemannian metric G. Suppose div D 1 = div D 2 . Then, Rc ± D 1 = Rc ± D 2 . Moreover, for any divergence operator div and generalized Riemannian metric G on E, the set of torsion-free generalized connections D on E which are compatible with G and such that div D = div is nonempty (see [Gar19, Section 3.2]). Thus, Ricci tensors Rc ± G,div are well-defined as equal to Rc ± D for any such generalized connection D. Example 2.9. On the exact Courant algebroid E H over M , let where g is a Riemannian metric, dV g its associated Riemannian volume form and z ∈ Γ(E H ). Then, via the isomorphism π + = π| E + : E + → T M , the Ricci tensor Rc + of (G, div g,θ ) is given by • Rc g ∈ Γ(S 2 + T * M ) is the Ricci tensor associated with g, is the Hodge codifferential associated with the metric g and the fixed orientation, * g being the Hodge star operator, See [GS21, Proposition 3.30] for the proof of this fact (cf. also [Kru]).
2.3. Generalized Ricci flow. We now review the framework of the generalized Ricci flow first introduced in [Str17, Gar19] and later described and studied in [GS21] by the two authors. Consider a smooth family of generalized Riemannian metrics (G(t)) t∈I on E, I ⊂ R, with respective eigenbundles E ± | t . Its variationĠ(t) exchanges the eigenbundles Definition 2.10. [Gar19, Definition 5.1] A smooth pair of families (G(t), div(t)) t∈I of generalized Riemannian metrics and divergence operators on E is a solution to the generalized Ricci flow if it satisfiesĠ On an exact Courant algebroid, the system may be written as follows: Proposition 2.11. [Gar19, Example 5.4] Let E be an exact Courant algebroid on an oriented smooth manifold M , withŠevera class [H] ∈ H 3 (M ). Fix an isotropic splitting E H = TM for E and consider the pair of smooth families (G(t), div(t)) t∈I defined by: where (g(t)) ⊂ Γ(S 2 + T * M ), (B(t)) ⊂ Γ(Λ 2 T * M ) and (z(t)) ⊂ Γ(E). Then (G(t), div(t)) t∈I is a solution of the generalized Ricci flow on E if and only if the families (g(t), B(t), θ(t)) t∈I , with θ(t) = 2g(πz(t) + , ·) ∈ Γ(T * M ), solve the equation where Separating the symmetric and skew-symmetric part of (2.6) one gets (see [ST13]) , where one has that are respectively the symmetric and skew-symmetric parts of ∇ + g(t),H(t) θ(t).
The pair (g(t), H(t)) evolves as , where ∆ g = dd * g + d * g d denotes the Hodge Laplacian operator associated with g and the fixed orientation. Notice how, up to scaling, the pluriclosed flow introduced in [ST10] is equivalent to a particular case of the generalized Ricci flow, as is proven in Propositions 6.3 and 6.4 in [ST13]. By [ST13, Theorem 6.5] a solution to (2.7) can be pulled back to a solution of (2.8) via the one-parameter family of diffeomorphism generated by 1 4 g(t) −1 θ(t). 2.4. Simply connected nilpotent Lie groups. We briefly recall the structure of simply connected nilpotent Lie groups, in the description of J. Lauret (see for example [Lau11]).
Every simply connected nilpotent Lie group G is diffeomorphic to its Lie algebra of leftinvariant fields g via the exponential map. Identifying g with R n via the choice of a basis, denote by µ ∈ Λ 2 (R n ) * ⊗ R n the induced Lie bracket. Now, exploiting the Campbell-Baker-Hausdorff formula, exp(X) · exp(Y ) = exp(X + Y + p µ (X, Y )), X, Y ∈ g ∼ = R n , where p µ is a R n -valued polynomial in the variables X, Y , one can endow R n with the operation · µ , X · µ Y = X + Y + p µ (X, Y ), so that exp : (R n , · µ ) → G is an isomorphism of Lie groups. Therefore, the set of isomorphism classes of simply connected nilpotent Lie groups is parametrized by the set of nilpotent Lie brackets on R n : these form an algebraic subset of the vector space of skew-symmetric bilinear forms on R n , V n := Λ 2 (R n ) * ⊗ R n , which parametrizes all skew-symmetric algebra structures on R n . Coordinates for V n can be obtained by fixing a basis {e i } n i=1 for R n : this allows to determine the so-called structure constants of any fixed µ ∈ V n as the real numbers {µ k ij , i, j, k = 1 . . . n} given by µ(e i , e j ) = µ k ij e k . One can then consider L n := {µ ∈ V n , µ satisfies the Jacobi identity}, the algebraic subset of V n consisting of Lie brackets on R n , and N n := {µ ∈ L n , µ is nilpotent}, which parametrizes all nilpotent Lie algebra structures on R n . By the previous remarks, N n parametrizes all n-dimensional simply connected nilpotent Lie groups, up to isomorphism.
Let us consider the family of Riemannian metrics on R n (2.9) {g µ,q , µ ∈ N n , q positive definite bilinear form on R n }, where g µ,q coincides with q at the origin and is left-invariant with respect to the nilpotent Lie group operation · µ . The set (2.9) is actually the set of all Riemannian metrics on R n which are invariant by some transitive action of a nilpotent Lie group. By [Wil82, Theorem 3], the Riemannian manifolds (R n , g µ,q ) (varying n, µ and q) are, up to isometry, all the possible examples of simply connected homogeneous nilmanifolds, namely connected Riemannian manifolds admitting a transitive nilpotent Lie group of isometries. The Riemannian metrics in (2.9) are not all distinct, up to isometry: it was shown again in [Wil82, Theorem 3] that g µ,q is isometric to g µ ,q if and only if there exists h ∈ GL n such that µ = h * µ and q = h * q. By convention we shall denote g µ := g µ, ·,· , where ·, · denotes the standard scalar product.
Since the Riemannian metrics g µ,q are completely determined by their value at 0 and by the Lie bracket µ, so will be all curvature quantities related to g µ,q . In particular, we are interested in Riemannian metrics g µ and their Ricci tensor, which we shall encounter in two guises, which we denote by For these, explicit formulas can be computed [Lau01]. Let {e i } n i=1 be the standard basis of R n , which, in particular, is orthonormal with respect to ·, · : one has (2.10) Rc µ (X, Y ) = − 1 2 µ(X, e k ), e l µ(Y, e k ), e l + 1 4 µ(e k , e l ), X µ(e k , e l ), Y , so that, if Rc µ = (Rc µ ) ij e i ⊗ e j and Ric µ = (Ric µ ) j i e i ⊗ e j , one has Notice that one can use formulas (2.10) and (2.11) to define Rc µ ∈ S 2 (R n ) * and Ric µ ∈ gl n for any µ ∈ V n .

Generalized Ricci solitons
Just as Ricci soliton metrics arise from self-similar solutions of the Ricci flow, generalized Ricci solitons arise from self-similar solutions of the generalized Ricci flow. We focus on exact Courant algebroids, defining a family of generalized Riemannian metrics, whose initial one is determined by a Riemannian metric on the base manifold; imposing that this family (together with a family of divergence operators) is a solution of the generalized Ricci flow, we draw necessary conditions on said Riemannian metric: these conditions generalize the Ricci soliton condition, leading to the definition of generalized Ricci solitons.

By Proposition 2.11, such (G(t), div(t)) t∈I is a solution of the generalized Ricci flow if and only if
where for all t ∈ I, x ∈ M . Setting t = 0 and rearranging the terms, . Summing together the two equations of (3.1), which involve symmetric and skew-symmetric tensor fields respectively, one has which is therefore equivalent to (3.1). We can now introduce the following definition, which generalizes the notion of Ricci soliton. When working on a Lie group G, for simplicity one can assume all structures to be leftinvariant, so that the generalized Ricci soliton condition reduces to an algebraic condition on structures on the Lie algebra of G, (g, µ).
In the context of semi-algebraic Ricci solitons, it was proven in [Jab15, Theorem 1.5] that, if g 0 is a left-invariant Riemannian metric on G, the Lie derivative of g 0 with respect to a left-invariant vector field X can be written as , for some D = D X ∈ Der(g), where Der(g) denotes the algebra of derivations of g. It was then shown in [Jab14, Theorem 1] (generalizing the already known fact for the simply connected nilpotent case in [Lau01, Proposition 1.1]) that D can be chosen to be symmetric with respect to g 0 , so that one always has L X g 0 = g 0 (D) = g 0 (D·, ·), for some D = D X ∈ Der(g) ∩ Sym(g, g 0 ). (3.2) then becomes for g 0 ∈ S 2 + g * , λ ∈ R, D ∈ Der(g) ∩ Sym(g, g 0 ), H 0 ∈ Λ 3 g * (with d µ H 0 = 0, d µ : Λ 3 g * → Λ 4 g * denoting the Chevalley-Eilenberg differential of the Lie algebra (g, µ)), θ 0 ∈ g * , ω ∈ Λ 2 g * , or equivalently (3.4) is still a left-invariant form, since the Hodge star operator commutes with pull-backs via orientation-preserving isometries of g 0 , such as left translations L g , g ∈ G, by left-invariance of g 0 .

Generalized bracket flows
Bracket flows have proven to be a powerful tool in the study of geometric flows on homogeneous spaces. This technique was first fully formalized by J. Lauret to study the Ricci flow on nilpotent Lie groups [Lau11]. In particular, J. Lauret proved that the Ricci flow on an n-dimensional simply connected nilpotent Lie group G starting from a left-invariant Riemannian metric g 0 is equivalent to an ode system defined on the variety of nilpotent Lie algebras N n , (4.1) μ(t) = −π(Ric µ(t) )µ(t), where µ 0 is the nilpotent Lie bracket associated with a fixed g 0 -orthonormal left-invariant frame and π : gl n → gl(V n ), given by is the differential of the standard GL n -action on V n : More generally, in literature many other bracket flows have been considered (see for example [Arr13, EFV15, Lau15, LR15, Lau16, Lau17, AL19]): these can be written in the form (4.2) μ(t) = −π(φ(µ(t)))µ(t), for some smooth function φ : V n → gl n .

Left-invariant Dorfman brackets.
Let E be an exact Courant algebroid over a real Lie group G. We shall be interested in the case when G is simply connected and nilpotent, so that we know that G is isomorphic to (R n , · µ ), for some Lie bracket µ ∈ N n . As we have recalled, there exists a unique cohomology class [H] ∈ H 3 (R n ) such that, for any H ∈ [H], E is isomorphic to E H = T R n ⊕ T * R n , endowed with the inner product ·, · in (2.3) and Dorfman bracket [·, ·] H in (2.4).
The whole structure descends to a structure on left-invariant sections, viewed as elements of R n ⊕(R n ) * , if and only if the 3-form H is left-invariant. Explicitly, when X+ξ, Y +η ∈ R n ⊕(R n ) * , the Dorfman bracket [·, ·] H reduces to the operator We call such a bilinear operator a (nilpotent) left-invariant Dorfman bracket.
We shall denote the set of left-invariant Dorfman brackets on R n by C n . By definition, it is clear that Equivalently, a quick analysis using the axioms of Courant algebroids and the previous remarks shows that C n can be identified with the algebraic subset of V n consisting of all brackets µ ∈ V n such that • µ satisfies the Jacobi identity. Given any µ ∈ V n , one can define the structure constants with respect to the standard basis of R n as the (2n) 3 = 8n 3 real numbers µ i j k , i, j, k = 1 . . . n, given by µ(e i , e j ) = µ ijk e k + µ ijk e k , µ(e i , e j ) = µ ijk e k + µ ijk e k .
Taking µ H ∈ C n , the structure constants are skew-symmetric in all three indices and vanish when two or more indices are overlined. The remaining structure constants are determined by µ and H. More precisely, The set of nilpotent left-invariant Dorfman brackets on R n , denoted by N n , is an algebraic subset of V n contained in C n . It is easy to see that its elements are exactly those Dorfman brackets µ H for which µ ∈ N n . 4.2. Generalized bracket flows. To introduce classical bracket flows, one uses the differential of the GL n -action on V n . In the same spirit, one can consider the natural GL(R n ⊕ (R n ) * ) on V n , (F · µ)(z 1 , z 2 ) = F µ(F −1 z 1 , F −1 z 2 ), F ∈ GL(R n ⊕ (R n ) * ), µ ∈ V n , z 1 , z 2 ∈ R n ⊕ (R n ) * , which induces an action of GL n ∼ = GL n ⊂ SO(R n ⊕ (R n ) * ) on V n , preserving both C n and N n . Now, identifying µ ∈ C n with (µ, H) ∈ V n × Λ 3 (R n ) * , it is evident that this action distributes as A · (µ, H) = (A · µ, A · H), where A ∈ GL n and A · H := (A −1 ) * H.
We denote the differential of this action again by π : gl n → gl(V n ): for µ ∈ V n , φ ∈ gl n one has Since the curve s → e sφ · µ is contained in the orbit GL n · µ, in this interpretation one has (4.5) π(φ)µ ∈ T µ (GL n · µ).
Following the ideas in the work of J. Lauret (see [Lau11]), these remarks suggest the idea of defining a flow, which we shall refer to as generalized bracket flow, on the vector space V n , of the form (4.6) μ(t) = −π φ(µ(t) µ(t), µ(0) = µ 0 , for some smooth function φ : V n → gl n and some µ 0 ∈ N n . By (4.5), a solution µ(t) to (4.6) satisfiesμ(t) ∈ T µ(t) (GL n · µ(t)) ⊂ T µ(t) N n for all t, so that the curve µ(t) is entirely contained in N n . For this reason, the function φ may also be defined on N n only.
The system (4.6) may be rewritten as the ode system on N n × Λ 3 (R n ) * (4.7) where π denotes the differential of the GL n -action on V n or Λ 3 (R n ) * .
In what follows, we shall omit the time dependencies of the quantities involved. Fixing the standard basis {e i } n i=1 for R n , we shall denote by φ j i , i, j = 1 . . . n the entries of the generic φ ∈ GL n with respect to it, such that φ(e i ) = φ j i e j for all i = 1 . . . n. One can then compute the coordinate expression for the evolution equations (4.7), obtaininġ for i, j, k = 1, . . . , n.
Special generalized bracket flows are obtained when the gl n -valued smooth function φ only depends on µ, φ = φ(µ): when this happens, the first equation of (4.7) is independent from the second one and corresponds to a usual bracket flow (4.2) on N n .
Classical bracket flows have proved to be a powerful tool in the study of geometric flows on (nilpotent) Lie groups. We thus expect the generalized bracket flows we have defined to be useful in the context of geometric flows in generalized geometry.

Examples on the Heisenberg group
In this section we perform explicit computations for the constructions introduced in the previous sections. We focus in particular on the Heisenberg group.
The Heisenberg group H 3 is a three-dimensional simply connected Lie group, which can be defined as closed subgroup of GL 3 : Via the exponential map, H 3 is diffeomorphic to its Lie algebra 5.1. Generalized Ricci solitons on the Heisenberg group. Let H 3 be the Heisenberg group, and fix the basis (5.1) for its Lie algebra h 3 . In order to find generalized Ricci solitons on H 3 , we first notice that the codifferential d * g 0 is the null map for every g 0 ∈ S 2 + h * 3 , since * g 0 sends Λ 3 h * 3 to R and d : R → h * 3 is the null map. With respect to the basis {e 1 , e 2 , e 3 } in (5.1), the generic derivation D of h 3 can be written in matrix form as with a i ∈ R, i = 1 . . . 6.
Let g 0 be the standard metric g 0 = e 1 ⊗ e 1 + e 2 ⊗ e 2 + e 3 ⊗ e 3 , such that {e 1 , e 2 , e 3 } is an orthonormal basis. Now, symmetric derivations with respect to g 0 are simply represented by symmetric matrices with respect to this basis: In what follows, assume where a, θ i , ω ij ∈ R, ω ij = −ω ji , i, j = 1, 2, 3, e i 1 ...i k := e i 1 ∧ · · · ∧ e i k and ε ijk is equal to the sign of the permutation sending (1, 2, 3) into (i, j, k) whenever i, j and k are all different, and equal to 0 otherwise, by definition.
We are now ready to compute the coordinate expression for all the terms involved in (3.3): • Rc g 0 : from (2.11), since the basis {e 1 , e 2 , e 3 } is orthonormal, by a direct computation we get in the fixed basis, • g 0 (D): it is simply represented by the matrix (5.2) with respect to the orthonormal basis, • ∇ + g 0 ,H 0 θ 0 : writing ∇ + instead of ∇ + g 0 ,H 0 and by left-invariance of the quantities involved, one has ∇ + θ 0 (e i , e j ) = −θ 0 (∇ + e i e j ). Now, ∇ + = ∇ g 0 + 1 2 g −1 0 H 0 and, letting ∇ g 0 e i e j = Γ k ij e k and recalling the Koszul formula one computes The corresponding matrix with respect to the orthonormal basis is thus so that its symmetric and skew-symmetric parts are The first equation of (3.4) gives now rise to a system of six equations in the unknowns λ, a 1 , a 2 , a 3 , θ 1 , θ 2 , θ 3 : a 1 = a 3 = 1, a 2 = θ 1 = θ 2 = 0.

Generalized Ricci flow on the Heisenberg group
Let us consider the gauge-corrected generalized Ricci flow (2.8) on the three-dimensional Heisenberg group H 3 , with initial data g 0 = e 1 ⊗ e 1 + e 2 ⊗ e 2 + e 3 ⊗ e 3 , H 0 = a e 123 , a ∈ R.