Generalized Ricci flow on nilpotent Lie groups

Fabio Paradiso

doi:10.1515/forum-2020-0171

Article Open Access

Generalized Ricci flow on nilpotent Lie groups

Fabio Paradiso

Published/Copyright: June 30, 2021

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From the journal Forum Mathematicum Volume 33 Issue 4

Abstract

We define solitons for the generalized Ricci flow on an exact Courant algebroid. 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.

Keywords: Generalized geometry; generalized Ricci flow; nilpotent Lie groups

MSC 2010: 53D18; 53C44; 53C30

1 Introduction

Generalized geometry, building on the work of Hitchin [9] and Gualtieri [7] and the structure of Courant algebroids, constitutes a rich mathematical environment. The main idea behind it lies in the shift of point of view when studying structures on a differentiable manifold Mn, replacing the tangent bundle TM with the generalized tangent bundle

𝕋⁢M=T⁢M⊕T*⁢M.

More explicitly, in the language of G-structures, one studies reductions of GL⁡(𝕋⁢M), the GL2⁢n-principal bundle of frames of 𝕋⁢M.

A reduction to the orthogonal group O⁡(n,n) always exists, thanks to the nondegenerate symmetric bilinear form of neutral signature

(1.1)〈X+ξ,Y+η〉=12⁢(η⁢(X)+ξ⁢(Y)),

so that one usually only considers structures which are reductions of O⁡(𝕋⁢M), the O⁡(n,n)-reduction of GL⁡(𝕋⁢M) determined by this natural pairing.

In this spirit, for example, a generalized almost complex structure on M2⁢m, defined by an orthogonal automorphism 𝒥 of 𝕋⁢M, i.e. 𝒥2=-Id𝕋⁢M, determines a U⁡(m,m)-reduction of GL⁡(𝕋⁢M). The integrability of such a structure is expressed through an involutivity condition with respect to a natural bracket operation, called the Dorfman bracket:

(1.2)[X+ξ,Y+η]=[X,Y]+ℒX⁢η-ιY⁢d⁢ξ.

On the other hand, a generalized Riemannian metric on Mn, defined by a symmetric (with respect to 〈⋅,⋅〉) and involutive automorphism 𝒢 of 𝕋⁢M, determines an O⁡(n)×O⁡(n)-reduction of GL⁡(𝕋⁢M).

More generally, one can consider a Courant algebroidE over M, namely a smooth vector bundle over M endowed with a pairing 〈⋅,⋅〉 and a bracket [⋅,⋅] satisfying certain properties so that 𝕋⁢M, 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 [22, 5, 6], the classical Ricci flow of Hamilton [8] and the B-field renormalization group flow of Type II string theory (see [20]) were generalized to a flow of generalized (pseudo-)Riemannian metrics on a Courant algebroid E over a smooth manifold M. 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 Rcg=λ⁢g+ℒX⁢g, where Rcg denotes the Ricci tensor of g, λ∈ℝ and ℒX⁢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 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 ℝn⊕(ℝn)* 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, [15]). 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.

2 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 [7].

The vector space V⊕V* can be endowed with a natural symmetric bilinear form of neutral signature

〈X+ξ,Y+η〉=12⁢(η⁢(X)+ξ⁢(Y))

and with a canonical orientation provided by the preimage of 1∈ℝ in the isomorphism

φ:Λ2⁢n⁢(V⊕V*)=Λn⁢V⊗Λn⁢V*→ℝ,

sending (X1∧…∧Xn)⊗(ξ1∧…∧ξn) into det⁡(ξi⁢(Xj))i⁢j.

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

𝔰⁢𝔬⁢(V⊕V*)≅𝔰⁢𝔬⁢(n,n)

consists of endomorphisms T∈𝔤⁢𝔩⁢(V⊕V*) which are skew-symmetric with respect to 〈⋅,⋅〉, namely

(2.1)〈T⁢z1,z2〉+〈z1,T⁢z2〉=0

for all z1,z2∈V⊕V*. By seeing T as a block matrix, (2.1) dictates T to be of the form

T=(ϕβB-ϕ*)

for some ϕ∈𝔤⁢𝔩⁢(V), B∈Λ2⁢V* and β∈Λ2⁢V, recovering the fact that

𝔰⁢𝔬⁢(V⊕V*)≅Λ2⁢(V⊕V*)*≅Λ2⁢V*⊕(V*⊗V)⊕Λ2⁢V,

where the former isomorphism is given by T↦〈T⋅,⋅〉.

Via the exponential map

exp:𝔰⁢𝔬⁢(V⊕V*)→SO⁡(V⊕V*),

we obtain distinguished elements of SO⁡(V⊕V*):

We have
eB=(Id0BId):X+ξ↦X+ξ+ιX⁢B,
called B-field transformations.
We have
eϕ=(eϕ00(e-ϕ)*),
which extends to an embedding of the whole GL⁡(V) into SO⁡(V⊕V*), sending A∈GL⁡(V) into
𝐀=(A00(A*)-1).
In the case V=ℝn, the image of this embedding will be denoted by GLn.

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,zi∈Γ⁢(E), i=1,2,3, f∈C∞⁢(M):

[z1,[z2,z3]]=[[z1,z2],z3]+[z2,[z1,z3]] (Jacobi identity).
π⁢[z1,z2]=[π⁢(z1),π⁢(z2)].
[z1,f⁢z2]=f⁢[z1,z2]+π⁢(z1)⁢(f)⁢z2.
[z,z]=12⁢𝒟⁢〈z,z〉, where 𝒟≔π*⁢d:C∞⁢(M)→Γ⁢(E).
π⁢(z1)⁢〈z2,z3〉=〈[z1,z2],z3〉+〈z2,[z1,z3]〉.

Definition 2.2.

A Courant algebroid E over M is exact if the short sequence

(2.2)0→T*⁢M→π*E→𝜋T⁢M→0

is exact, namely if the anchor map is surjective and its kernel is exactly the image of π*.

By the classification of Ševera [21], isomorphism classes of exact Courant algebroids over M are in bijection with the elements of the third de Rham cohomology group of M, i.e. H3⁢(M): an exact Courant algebroid with Ševera class[H]∈H3⁢(M) is isomorphic to the Courant algebroid EH=𝕋⁢M≔T⁢M⊕T*⁢M over M with pairing of neutral signature

(2.3)〈X+ξ,Y+η〉=12⁢(η⁢(X)+ξ⁢(Y))

and (twisted) Dorfman bracket

(2.4)[X+ξ,Y+η]=[X,Y]+ℒX⁢η-ιY⁢d⁢ξ+ιY⁢ιX⁢H

for any H∈[H]. Such isomorphisms are obtained explicitly via the choice of an isotropic splitting to (2.2), while B-field transformations, B∈Γ⁢(Λ2⁢T*⁢M), provide explicit isomorphisms

eB:EH→EH-d⁢B.

In what follows, let E be a Courant algebroid over M, with rk⁡(E)=2⁢n 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 𝒢2=IdE, and such that 〈𝒢⋅,⋅〉 is a positive-definite metric on E.

Given E+, denoting by E- its orthogonal complement with respect to 〈⋅,⋅〉, we define 𝒢 by 𝒢|E±=±IdE±. Then E± can 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 EH is of the form

𝒢=eB⁢(0g-1g0)⁢e-B

for some Riemannian metric g and for some 2-form B on M (see [7, Section 6.2]). The corresponding E± are

E±=eB⁢{X±g⁢(X):X∈T⁢M},

where by g⁢(X) we mean g⁢(X,⋅). Notice that 𝒢 is of the form

(0g-1g0)

in the splitting EH+d⁢B.

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 [5, 3].

Definition 2.5.

A generalized connection on a Courant algebroid E is a linear map

D:Γ⁢(E)→Γ⁢(E*⊗E)

which satisfies a Leibniz rule and a compatibility condition with 〈⋅,⋅〉:

D⁢(f⁢z)=f⁢(D⁢z)+𝒟⁢f⊗z,

〈𝒟⁢〈z1,z2〉,⋅〉=〈D⋅⁢z1,z2〉+〈z1,D⋅⁢z2〉

for all z,z1,z2∈Γ⁢(E) and f∈C∞⁢(M), where Dz1⁢z2≔D⁢z2⁢(z1).

Given a generalized Riemannian metric 𝒢, a generalized connection D is compatible with 𝒢 if D⁢𝒢=0, where D denotes the induced E-connection on the tensor bundle E*⊗E≅End⁡(E). Equivalently, D is compatible with 𝒢 if D⁢(Γ⁢(E±))⊂Γ⁢(E*⊗E±).

The torsionTD∈Γ⁢(Λ2⁢E*⊗E) of a generalized connection D on E is defined by

TD⁢(z1,z2)=Dz1⁢z2-Dz2⁢z1-[z1,z2]+(D⁢z1)*⁢z2.

If TD=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

RD±∈Γ⁢(E±*⊗E∓*⊗𝔬⁢(E±)),

where 𝔬⁢(E±)=〈⋅,⋅〉-1⁢Λ2⁢E±* denotes the Lie algebra of skew-symmetric endomorphisms of E± with respect to 〈⋅,⋅〉, by

RD±⁡(z1±,z2∓)⁢z3±=Dz1±⁢Dz2∓⁢z3±-Dz2∓⁢Dz1±⁢z3±-D[z1±,z2∓]⁢z3±.

One then has associated Ricci tensors

RcD±∈Γ⁢(E∓*⊗E±*),

RicD±∈Γ⁢(E∓*⊗E±),

defined by

RcD±⁡(z1∓,z2±)=tr⁡(z±↦RD±⁡(z±,z1∓)⁢z2±),

RicD±=〈⋅,⋅〉-1⁢RcD±.

Definition 2.6.

A divergence operator on E is a first-order differential operator div:Γ⁢(E)→C∞⁢(M) satisfying the Leibniz rule

div⁡(f⁢z)=π⁢(z)⁢f+f⁢div⁡(z),

f∈C∞⁢(M), z∈Γ⁢(E). Given a generalized connection D on E, one may define the associated divergence operator

divD⁡(z)=tr⁡(D⁢z).

Remark 2.7.

Divergence operators on E form an affine space over the vector space Γ⁢(E)≅Γ⁢(E*). Fixing a divergence operator div0, any other divergence operator div is of the form

div=div0-〈z,⋅〉

for some z∈Γ⁢(E).

Proposition 2.8 ([5, Proposition 4.4]).

Let Di, i=1,2, be torsion-free generalized connections on E compatible with a given generalized Riemannian metric G. Suppose divD1=divD2. Then, RcD1±=RcD2±.

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 divD=div is nonempty (see [5, Section 3.2]). Thus, Ricci tensors Rc𝒢,div± are well-defined as equal to RcD± for any such generalized connection D.

Example 2.9.

On the exact Courant algebroid EH over M, let

𝒢=(0g-1g0)

and

divg,z⁡(X+ξ)=d⁢Vg-1⁢ℒX⁢d⁢Vg-〈z,X+ξ〉,

where g is a Riemannian metric, d⁢Vg its associated Riemannian volume form and z∈Γ⁢(EH). Then, via the isomorphism π+=π|E+:E+→T⁢M, the Ricci tensor Rc+ of (𝒢,divg,θ) is given by

(2.5)Rc𝒢,divg,z+=Rcg-∘𝑔⁡14⁢HH-12⁢dg*⁢H+12⁢∇g,H+⁡θ,

where

Rcg∈Γ⁢(S+2⁢T*⁢M) is the Ricci tensor associated with g,
∘𝑔⁡HH∈Γ⁢(S2⁢T*⁢M), with
∘𝑔⁡HH⁢(X,Y)=g⁢(ιX⁢H,ιY⁢H),
dg*=-*𝑔d*𝑔:Γ(Λ3T*M)→Γ(Λ2T*M) is the Hodge codifferential associated with the metric g and the fixed orientation, with *𝑔 being the Hodge star operator,
∇g,H+=∇g+12⁢g-1⁢H is the Bismut connection with torsion H, with ∇g denoting the Levi-Civita connection of g,
θ∈Γ⁢(T*⁢M) is given by θ=2⁢g⁢(π⁢z+,⋅)=g⁢(X,⋅)+ξ if z=X+ξ.

See [6, Proposition 3.30] for the proof of this fact (cf. also [13]).

2.3 Generalized Ricci flow

We now review the framework of the generalized Ricci flow first introduced in [22, 5] and later described and studied in [6] by Garcia-Fernandez and Streets. Consider a smooth family of generalized Riemannian metrics (𝒢⁢(t))t∈I on E, I⊂ℝ, with respective eigenbundles E±|t. Its variation 𝒢˙⁢(t) exchanges the eigenbundles E±|t, so that 𝒢˙⁢(t)=𝒢˙+⁢(t)+𝒢˙-⁢(t), with

𝒢˙±⁢(t)∈Γ⁢(E∓|t*⊗E±|t).

Definition 2.10.

[5, 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 satisfies

𝒢˙+⁢(t)=-2⁢Rict+

for all t in the interior of I, where Rict+≔Ric𝒢⁢(t),div⁡(t)+.

On an exact Courant algebroid, the system may be written as follows.

Proposition 2.11 ([5, Example 5.4]).

Let E be an exact Courant algebroid on an oriented smooth manifold M, with Ševera class [H]∈H3⁢(M). Fix an isotropic splitting EH=T⁢M for E and consider the pair of smooth families (G⁢(t),div⁡(t))t∈I defined by

𝒢⁢(t)=eB⁢(t)⁢(0g⁢(t)-1g⁢(t)0)⁢e-B⁢(t),

div⁡(t)=divg⁢(t),z⁢(t),

where (g⁢(t))⊂Γ⁢(S+2⁢T*⁢M), (B⁢(t))⊂Γ⁢(Λ2⁢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)=2⁢g⁢(π⁢z⁢(t)+,⋅)∈Γ⁢(T*⁢M),

solve the equation

(2.6)g˙⁢(t)=-2⁢(Rcg⁢(t)-∘g⁢(t)⁡14⁢H⁢(t)H⁢(t)-12⁢dg⁢(t)*⁢H⁢(t)+12⁢∇g⁢(t),H⁢(t)+⁡θ⁢(t))+B˙⁢(t),

where H⁢(t)=H+d⁢B⁢(t).

Separating the symmetric and skew-symmetric part of (2.6), one gets (see [24])

{g˙⁢(t)=-2⁢Rcg⁢(t)+∘g⁢(t)⁡12⁢H⁢(t)H⁢(t)-12⁢ℒg⁢(t)-1⁢θ⁢(t)⁢g⁢(t),B˙⁢(t)=-dg⁢(t)*⁢H⁢(t)+12⁢d⁢θ⁢(t)-12⁢ιg⁢(t)-1⁢θ⁢(t)⁢H⁢(t),

where one has that

12⁢ℒg⁢(t)-1⁢θ⁢(t)⁢g⁢(t)=S⁢(∇g⁢(t),H⁢(t)+⁡θ⁢(t)),12⁢d⁢θ⁢(t)-12⁢ιg⁢(t)-1⁢θ⁢(t)⁢H⁢(t)=A⁢(∇g⁢(t),H⁢(t)+⁡θ⁢(t))

are respectively the symmetric and skew-symmetric parts of ∇g⁢(t),H⁢(t)+⁡θ⁢(t).

The pair (g⁢(t),H⁢(t)) evolves as

(2.7){g˙⁢(t)=-2⁢Rcg⁢(t)+∘g⁢(t)⁡12⁢H⁢(t)H⁢(t)-12⁢ℒg⁢(t)-1⁢θ⁢(t)⁢g⁢(t),H˙⁢(t)=-Δg⁢(t)⁢H⁢(t)-12⁢ℒg⁢(t)-1⁢θ⁢(t)⁢H⁢(t),

where Δg=d⁢dg*+dg*⁢d denotes the Hodge–Laplacian operator associated with g and the fixed orientation. Notice how, up to scaling, the pluriclosed flow introduced in [23] is equivalent to a particular case of the generalized Ricci flow, as is proven in [24, Propositions 6.3 and 6.4]. By [24, Theorem 6.5], a solution to (2.7) can be pulled back to a solution of

(2.8){g˙⁢(t)=-2⁢Rcg⁢(t)+∘g⁢(t)⁡12⁢H⁢(t)H⁢(t),H˙⁢(t)=-Δg⁢(t)⁢H⁢(t),

via the one-parameter family of diffeomorphisms generated by 14⁢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 Lauret (see, for example, [15]).

Every simply connected nilpotent Lie group G is diffeomorphic to its Lie algebra of left-invariant fields 𝔤 via the exponential map. Identifying 𝔤 with ℝn via the choice of a basis, denote by μ∈Λ2⁢(ℝn)*⊗ℝn the induced Lie bracket. Now, exploiting the Campbell–Baker–Hausdorff formula, we get

exp⁡(X)⋅exp⁡(Y)=exp⁡(X+Y+pμ⁢(X,Y)),

X,Y∈𝔤≅ℝn, where pμ is an ℝn-valued polynomial in the variables X,Y, and one can endow ℝn with the operation ⋅μ, i.e.

X⋅μY=X+Y+pμ⁢(X,Y),

so that exp:(ℝn,⋅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 ℝn: these form an algebraic subset of the vector space of skew-symmetric bilinear forms on ℝn, i.e.

𝒱n≔Λ2⁢(ℝn)*⊗ℝn,

which parametrizes all skew-symmetric algebra structures on ℝn. Coordinates for 𝒱n can be obtained by fixing a basis {ei}i=1n for ℝn: this allows to determine the so-called structure constants of any fixed μ∈𝒱n as the real numbers {μi⁢jk:i,j,k=1,…,n} given by

μ⁢(ei,ej)=μi⁢jk⁢ek.

One can then consider

ℒn≔{μ∈𝒱n:μ⁢ satisfies the Jacobi identity},

the algebraic subset of 𝒱n consisting of Lie brackets on ℝn, and

𝒩n≔{μ∈ℒn:μ⁢ is nilpotent},

which parametrizes all nilpotent Lie algebra structures on ℝn. By the previous remarks, 𝒩n parametrizes all n-dimensional simply connected nilpotent Lie groups, up to isomorphism.

Let us consider the following family of Riemannian metrics on ℝn:

(2.9){gμ,q:μ∈𝒩n,q⁢ is a positive definite bilinear form on ⁢ℝn},

where gμ,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 ℝn which are invariant by some transitive action of a nilpotent Lie group. By [25, Theorem 3], the Riemannian manifolds (ℝn,gμ,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 [25, Theorem 3] that gμ,q is isometric to gμ′,q′ if and only if there exists h∈GLn such that μ′=h*⁢μ and q′=h*⁢q. By convention we shall set gμ≔gμ,〈⋅,⋅〉, where 〈⋅,⋅〉 denotes the standard scalar product.

Since the Riemannian metrics gμ,q are completely determined by their value at 0 and by the Lie bracket μ, so will be all curvature quantities related to gμ,q. In particular, we are interested in Riemannian metrics gμ and their Ricci tensor, which we shall encounter in two guises, which we denote by

Rcμ≔Rcgμ⁡(0)∈S2⁢(ℝn)*⊂(ℝn)*⊗(ℝn)*,

Ricμ≔Ricgμ⁡(0)∈(ℝn)*⊗ℝn=𝔤⁢𝔩n,

with Rcμ⁡(X,Y)=〈Ricμ⁡(X),Y〉, X,Y∈ℝn.

For these, explicit formulas can be computed [14]. Let {ei}i=1n be the standard basis of ℝn, which, in particular, is orthonormal with respect to 〈⋅,⋅〉: one has

(2.10)Rcμ⁡(X,Y)=-12⁢〈μ⁢(X,ek),el〉⁢〈μ⁢(Y,ek),el〉+14⁢〈μ⁢(ek,el),X〉⁢〈μ⁢(ek,el),Y〉,

so that, if Rcμ=(Rcμ)i⁢j⁢ei⊗ej and Ricμ=(Ricμ)ij⁢ei⊗ej, one has

(2.11)(Rcμ)i⁢j=(Ricμ)ij=-12⁢μi⁢kl⁢μj⁢kl+14⁢μk⁢li⁢μk⁢lj.

Notice that one can use formulas (2.10) and (2.11) to define Rcμ∈S2⁢(ℝn)* and Ricμ∈𝔤⁢𝔩n for any μ∈𝒱n.

3 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.

Let E be a Courant algebroid over an oriented smooth manifold M with Ševera class [H0]∈H3⁢(M). Fixing an isotropic splitting EH0=𝕋⁢M, we consider a smooth self-similar pair of families (𝒢⁢(t),div⁡(t))t∈I, 0∈I, on E of the form

𝒢⁢(t)=eB⁢(t)⁢(0(c⁢(t)⁢φt*⁢g0)-1c⁢(t)⁢φt*⁢g00)⁢e-B⁢(t),

div⁡(t)=divg⁢(t),θ⁢(t),

where g0∈Γ⁢(S+2⁢(T*⁢M)) is a Riemannian metric, c:I→ℝ is smooth and positive, c⁢(0)=1, (φt) is a one-parameter family of diffeomorphisms of M, (B⁢(t))⊂Γ⁢(Λ2⁢T*⁢M), B⁢(0)=0, (θ⁢(t))⊂Γ⁢(T*⁢M), θ⁢(0)=θ0∈Γ⁢(T*⁢M), and g⁢(t)=c⁢(t)⁢φt*⁢g0.

By Proposition 2.11, such (𝒢⁢(t),div⁡(t))t∈I is a solution of the generalized Ricci flow if and only if

{c˙⁢(t)⁢φt*⁢g0+c⁢(t)⁢φt*⁢ℒYt⁢g0=-2⁢Rcg⁢(t)+∘g⁢(t)⁡12⁢H⁢(t)H⁢(t)+14⁢ℒg⁢(t)-1⁢θ⁢(t)⁢g⁢(t),B˙⁢(t)=-dg⁢(t)*⁢H⁢(t)-14⁢d⁢θ⁢(t)+14⁢ιg⁢(t)-1⁢θ⁢(t)⁢H⁢(t),

where H⁢(t)=H0+d⁢B⁢(t) and (Yt)t∈I⊂Γ⁢(T⁢M) is such that

dd⁢t⁢φt⁢(x)=Yt⁢(φt⁢(x))

for all t∈I, x∈M.

Setting t=0 and rearranging the terms, we obtain

(3.1){Rcg0=λ⁢g0+ℒX⁢g0+∘g0⁡14⁢H0H0-14⁢ℒg0-1⁢θ0⁢g0,ω=-dg0*⁢H0+12⁢d⁢θ0-12⁢ιg0-1⁢θ0⁢H0,

where -2⁢λ=c˙⁢(0)∈ℝ, -2⁢X=Y0∈Γ⁢(T⁢M) and ω=B˙⁢(0)∈Γ⁢(Λ2⁢T*⁢M). Summing together the two equations of (3.1), which involve symmetric and skew-symmetric tensor fields, respectively, one has

(3.2)Rcg0=λ⁢g0+ℒX⁢g0+∘g0⁡14⁢H0H0-12⁢∇g0,H0+⁡θ0+12⁢dg0*⁢H0+12⁢ω,

which is therefore equivalent to (3.1). We can now introduce the following definition, which generalizes the notion of Ricci soliton.

Definition 3.1.

A Riemannian metric g0 on M is called a generalized Ricci soliton if there exist λ∈ℝ, X∈Γ⁢(T⁢M), H0∈Γ⁢(Λ3⁢T*⁢M) closed, θ0∈Γ⁢(T*⁢M), and ω∈Γ⁢(Λ2⁢T*⁢M) such that (3.2), or equivalently (3.1), holds.

When working on a Lie group G, for simplicity one can assume all structures to be left-invariant, so that the generalized Ricci soliton condition reduces to an algebraic condition on structures on the Lie algebra of G, i.e. (𝔤,μ).

In the context of semi-algebraic Ricci solitons, it was proven in [12, Theorem 1.5] that, if g0 is a left-invariant Riemannian metric on G, the Lie derivative of g0 with respect to a left-invariant vector field X can be written as

ℒXg0=g0(12(D+Dt))=g0(12(D+Dt)⋅,⋅)

for some D=DX∈Der⁡(𝔤), where Der⁡(𝔤) denotes the algebra of derivations of 𝔤. It was then shown in [11, Theorem 1] (generalizing the already known fact for the simply connected nilpotent case in [14, Proposition 1.1]) that D can be chosen to be symmetric with respect to g0, so that one always has

ℒXg0=g0(D)=g0(D⋅,⋅)

for some D=DX∈Der⁡(𝔤)∩Sym⁡(𝔤,g0). Then (3.2) becomes

(3.3)Rcg0=λ⁢g0+g0⁢(D)+∘g0⁡14⁢H0H0+12⁢dg0*⁢H0-12⁢∇g0,H0+⁡θ0+12⁢ω∈S2⁢𝔤*

for g0∈S+2⁢𝔤*, λ∈ℝ, D∈Der⁡(𝔤)∩Sym⁡(𝔤,g0), H0∈Λ3⁢𝔤* (with dμ⁢H0=0, and dμ:Λ3⁢𝔤*→Λ4⁢𝔤* denoting the Chevalley–Eilenberg differential of the Lie algebra (𝔤,μ)), θ0∈𝔤*, and ω∈Λ2⁢𝔤*, or equivalently

(3.4){Rcg0=λ⁢g0+g0⁢(D)+∘g0⁡14⁢H0H0-14⁢ℒg0-1⁢θ0⁢g0,ω=-dg0*⁢H0+12⁢d⁢θ0-12⁢ιg0-1⁢θ0⁢H0.

Notice that dg0*⁢H0 is still a left-invariant form since the Hodge star operator commutes with pull-backs via orientation-preserving isometries of g0, such as left translations Lg, g∈G, by left-invariance of g0.

4 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 Lauret to study the Ricci flow on nilpotent Lie groups [15]. In particular, Lauret proved that the Ricci flow on an n-dimensional simply connected nilpotent Lie group G starting from a left-invariant Riemannian metric g0 is equivalent to an ODE system defined on the variety of nilpotent Lie algebras 𝒩n:

{μ˙⁢(t)=-π⁢(Ricμ⁢(t))⁢μ⁢(t),μ⁢(0)=μ0,

where μ0 is the nilpotent Lie bracket associated with a fixed g0-orthonormal left-invariant frame and π:𝔤⁢𝔩n→𝔤⁢𝔩⁢(𝒱n), given by

(π⁢(ϕ)⁢μ)⁢(X,Y)=ϕ⁢μ⁢(X,Y)-μ⁢(ϕ⁢X,Y)-μ⁢(X,ϕ⁢Y),ϕ∈𝔤⁢𝔩n,μ∈𝒱n,X,Y∈ℝn,

is the differential of the standard GLn-action on 𝒱n:

(A⋅μ)⁢(X,Y)=A⁢μ⁢(A-1⁢X,A-1⁢Y),A∈GLn,μ∈𝒱n,X,Y∈ℝn.

More generally, in literature many other bracket flows have been considered (see, for example, [1, 4, 16, 19, 17, 18, 2]): these can be written in the form

(4.1){μ˙⁢(t)=-π⁢(ϕ⁢(μ⁢(t)))⁢μ⁢(t),μ⁢(0)=μ0

for some smooth function ϕ:𝒱n→𝔤⁢𝔩n.

4.1 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 (ℝn,⋅μ) for some Lie bracket μ∈𝒩n.

As we have recalled, there exists a unique cohomology class [H]∈H3⁢(ℝn) such that, for any H∈[H], E is isomorphic to EH=T⁢ℝn⊕T*⁢ℝn, 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 ℝn⊕(ℝn)*, if and only if the 3-form H is left-invariant. Explicitly, when X+ξ,Y+η∈ℝn⊕(ℝn)*, the Dorfman bracket [⋅,⋅]H reduces to the operator

MH⁢(X+ξ,Y+η)=μ⁢(X,Y)-η∘adμ⁡(X)+ξ∘adμ⁡(Y)+ιY⁢ιX⁢H

=μ⁢(X,Y)-η⁢μ⁢(X,⋅)+ξ⁢μ⁢(Y,⋅)+H⁢(X,Y,⋅).

We call such a bilinear operator a (nilpotent) left-invariant Dorfman bracket.

As one can check directly and also deduce from the axioms of Courant algebroids, a left-invariant Dorfman bracket is totally skew-symmetric, namely

〈MH⁢(⋅,⋅),⋅〉∈Λ3⁢(ℝn⊕(ℝn)*)*.

By a little abuse, we can say MH∈Λ3⁢(ℝn⊕(ℝn)*)*, by identifying Λ3⁢(ℝn⊕(ℝn)*)* with a subset of

𝓥n≔Λ2⁢(ℝn⊕(ℝn)*)*⊗(ℝn⊕(ℝn)*).

We shall denote the set of left-invariant Dorfman brackets on ℝn by 𝓒n. By definition, it is clear that

𝓒n↔1:1{(μ,H)∈ℒn×Λ3(ℝn)*,dμH=0}.

Equivalently, a quick analysis using the axioms of Courant algebroids and the previous remarks shows that 𝓒n can be identified with the algebraic subset of 𝓥n consisting of all brackets M∈𝓥n such that:

M∈Λ3⁢(ℝn⊕(ℝn)*)*,
M⁢((ℝn)*,(ℝn)*)=0,
M satisfies the Jacobi identity.

Given any M∈𝓥n, one can define the structure constants with respect to the standard basis of ℝn as the (2⁢n)3=8⁢n3 real numbers Mi¯¯⁢j¯¯⁢k¯¯, i,j,k=1,…,n, given by

M⁢(ei,ej)=Mi¯⁢j¯⁢k¯⁢ek+Mi¯⁢j¯⁢k¯⁢ek,

M⁢(ei,ej)=Mi¯⁢j¯⁢k¯⁢ek+Mi¯⁢j¯⁢k¯⁢ek.

Taking MH∈𝓒n, 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,

(MH)i¯⁢j¯⁢k¯=μi⁢jk,(MH)i¯⁢j¯⁢k¯=Hi⁢j⁢k.

The set of nilpotent left-invariant Dorfman brackets on ℝn, denoted by 𝓝n, is an algebraic subset of 𝓥n contained in 𝓒n. It is easy to see that its elements are exactly those Dorfman brackets MH for which μ∈𝒩n.

4.2 Generalized bracket flows

To introduce classical bracket flows, one uses the differential of the GLn-action on 𝒱n. In the same spirit, one can consider the natural GL⁡(ℝn⊕(ℝn)*) on 𝓥n:

(F⋅M)⁢(z1,z2)=F⁢M⁢(F-1⁢z1,F-1⁢z2),F∈GL⁡(ℝn⊕(ℝn)*),M∈𝓥n,z1,z2∈ℝn⊕(ℝn)*,

which induces an action of GLn≅GLn⊂SO⁡(ℝn⊕(ℝn)*) on 𝓥𝒏, preserving both 𝓒n and 𝓝n.

Now, identifying M∈𝓒n with (μ,H)∈𝒱n×Λ3⁢(ℝn)*, it is evident that this action distributes as

A⋅(μ,H)=(A⋅μ,A⋅H),

where A∈GLn and A⋅H≔(A-1)*⁢H.

We denote the differential of this action again by π:𝔤⁢𝔩n→𝔤⁢𝔩⁢(𝓥n): for M∈𝓥n and ϕ∈𝔤⁢𝔩n, one has

π(ϕ)M=dd⁢s|s=0(es⁢ϕ⋅M)∈TM𝓥n≅𝓥n.

Since the curve s↦es⁢ϕ⋅M is contained in the orbit GLn⋅M, in this interpretation one has

(4.2)π⁢(ϕ)⁢M∈TM⁢(GLn⋅M).

Following the ideas in the work of Lauret (see [15]), these remarks suggest the idea of defining a flow, which we shall refer to as generalized bracket flow, on the vector space 𝓥n, of the form

(4.3){M˙⁢(t)=-π⁢(ϕ⁢(M⁢(t)))⁢M⁢(t),M⁢(0)=M0

for some smooth function ϕ:𝓥n→𝔤⁢𝔩n and some M0∈𝓝n. By (4.2), a solution M⁢(t) to (4.3) satisfies

M˙⁢(t)∈TM⁢(t)⁢(GLn⋅M⁢(t))⊂TM⁢(t)⁢𝓝n for all ⁢t,

so that the curve M⁢(t) is entirely contained in 𝓝n. For this reason, the function ϕ may also be defined on 𝓝n only.

System (4.3) may be rewritten as the following ODE system on 𝒩n×Λ3⁢(ℝn)*:

(4.4){μ˙⁢(t)=-π⁢(ϕ⁢(μ⁢(t),H⁢(t)))⁢μ⁢(t),H˙⁢(t)=-π⁢(ϕ⁢(μ⁢(t),H⁢(t)))⁢H⁢(t),μ⁢(0)=μ0∈𝒩n,H⁢(0)=H0∈Λ3⁢(ℝn)*,dμ0⁢H0=0,

where π denotes the differential of the GLn-action on 𝒱n or Λ3⁢(ℝn)*.

In what follows, we shall omit the time dependencies of the quantities involved. Fixing the standard basis {ei}i=1n for ℝn, we shall denote by ϕij, i,j=1,…,n, the entries of the generic ϕ∈GLn with respect to it, such that ϕ⁢(ei)=ϕij⁢ej for all i=1,…,n. One can then compute the coordinate expression for the evolution equations (4.4), obtaining

(4.5)μ˙i⁢jk=ϕil⁢μl⁢jk+ϕjl⁢μi⁢lk-ϕlk⁢μi⁢jl,

(4.6)H˙i⁢j⁢k=ϕil⁢Hl⁢j⁢k+ϕjl⁢Hi⁢l⁢k+ϕkl⁢Hi⁢j⁢l

for i,j,k=1,…,n.

Special generalized bracket flows are obtained when the 𝔤⁢𝔩n-valued smooth function ϕ only depends on μ, i.e. ϕ=ϕ⁢(μ): when this happens, the first equation of (4.4) is independent from the second one and corresponds to a usual bracket flow (4.1) on 𝒩n.

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.

5 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 H3 is a three-dimensional simply connected Lie group, which can be defined as a closed subgroup of GL3:

H3={(1ac01b001)∈GL3,a,b,c∈ℝ}.

Via the exponential map, H3 is diffeomorphic to its Lie algebra

𝔥3={(0ac00b000)∈𝔤𝔩3,a,b,c∈ℝ}.

By fixing the basis

(5.1)e1=(010000000),e2=(000001000),e3=(001000000)

for 𝔥3, the induced bracket μ∈𝒩3 is μ=e1∧e2⊗e3 since [e1,e2]=e3 and [e1,e3]=[e2,e3]=0.

5.1 Generalized Ricci solitons on the Heisenberg group

Let H3 be the Heisenberg group, and fix the basis (5.1) for its Lie algebra 𝔥3.

In order to find generalized Ricci solitons on H3, we first notice that the codifferential dg0* is the null map for every g0∈S+2⁢𝔥3* since *g0 sends Λ3⁢𝔥3* to ℝ and d:ℝ→𝔥3* is the null map. With respect to the basis {e1,e2,e3} in (5.1), the generic derivation D of 𝔥3 can be written in matrix form as

D=(a1a20a3a40a5a6a1+a4),

with ai∈ℝ, i=1,…,6.

Let g0 be the standard metric

g0=e1⊗e1+e2⊗e2+e3⊗e3

such that {e1,e2,e3} is an orthonormal basis. Now, symmetric derivations with respect to g0 are simply represented by symmetric matrices with respect to this basis:

(5.2)D=(a1a20a2a3000a1+a3),

ai∈ℝ, i=1,2,3. In what follows, assume

H0=a⁢e123=a6⁢εi⁢j⁢k⁢ei⁢j⁢k,θ0=θi⁢ei,ω=12⁢ωi⁢j⁢ei⁢j,

where a,θi,ωi⁢j∈ℝ, ωi⁢j=-ωj⁢i, i,j=1,2,3, ei1⁢⋯⁢ik≔ei1∧⋯∧eik, and εi⁢j⁢k 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):

Rcg0: from (2.11), since the basis {e1,e2,e3} is orthonormal, by a direct computation, we get
Rcg0=(-12000-1200012)
in the fixed basis.
g0⁢(D): it is simply represented by the matrix (5.2) with respect to the orthonormal basis.
∘g0⁡H0H0: one has
∘g0⁡H0H0⁢(ei,ej)=g⁢(ιei⁢H0,ιej⁢H0)=g0r⁢l⁢g0s⁢t⁢(H0)i⁢r⁢s⁢(H0)j⁢l⁢t=a2⁢εi⁢s⁢t⁢εj⁢s⁢t,
so that, in matrix form, we get
∘g0⁡H0H0=(2⁢a20002⁢a20002⁢a2).
∇g0,H0+⁡θ0: writing ∇+ instead of ∇g0,H0+ and by the left-invariance of the quantities involved, one has
∇+⁡θ0⁢(ei,ej)=-θ0⁢(∇ei+⁡ej).
Now, ∇+=∇g0+12⁢g0-1⁢H0 and, letting ∇eig0⁡ej=Γi⁢jk⁢ek and recalling the Koszul formula, one computes
Γi⁢jk=-12⁢(μj⁢ki+μi⁢kj+μj⁢ik),12⁢g0-1⁢H0⁢(ei,ej)=12⁢a⁢εi⁢j⁢k⁢ek,
so that
∇+⁡θ0⁢(ei,ej)=12⁢θk⁢(μj⁢ki+μi⁢kj+μj⁢ik-a⁢εi⁢j⁢k).
The corresponding matrix with respect to the orthonormal basis is thus
∇+⁡θ0=(0-12⁢θ3⁢(1+a)12⁢θ2⁢(1+a)12⁢θ3⁢(1+a)0-12⁢θ1⁢(1+a)12⁢θ2⁢(1-a)-12⁢θ1⁢(1-a)0),
so that its symmetric and skew-symmetric parts are
S⁢(∇+⁡θ0)=(0012⁢θ200-12⁢θ112⁢θ2-12⁢θ10),
A⁢(∇+⁡θ0)=(0-12⁢θ3⁢(1+a)12⁢a⁢θ212⁢θ3⁢(1+a)0-12⁢a⁢θ1-12⁢a⁢θ212⁢a⁢θ10).

The first equation of (3.4) gives now rise to a system of six equations in the unknowns λ, a1, a2, a3, θ1, θ2, θ3:

{12+λ+a1+12⁢a2=0,a2=0,θ2=0,12+λ+a3+12⁢a2=0,θ1=0,-12+λ+a1+a3+12⁢a2=0,

which is equivalent to

{λ=-12⁢(3+a2),a1=a3=1,a2=θ1=θ2=0.

The second equation of (3.4) now implies

ω12=-ω21=-12⁢θ3⁢(1+a),

while all the other ωi⁢j’s vanish.

We thus obtain generalized Ricci solitons with the data

{g0=e1⊗e1+e2⊗e2+e3⊗e3,λ=-12⁢(3+a2),D=e1⊗e1+e2⊗e2+2⁢e3⊗e3,H0=a⁢e123,θ0=θ3⁢e3,ω=-12⁢θ3⁢(1+a)⁢e12.

Remark 5.1.

The metric g0 above is actually also a Ricci soliton in the classical sense since, by setting a=θ3=0, H0, θ0 and ω vanish, leaving g0 satisfying Rcg0=λ⁢g0+g0⁢(D), or equivalently, applying g0-1, Ricg0=λ⁢Id+D for λ=-32 and D as above. By [14, Theorem 3.5], g0 is the only left-invariant Ricci soliton on H3, up to isometry and rescaling.

5.2 A generalized bracket flow on the Heisenberg group

The definition of the gauge-corrected generalized Ricci flow (2.8) suggest the generalized bracket flow

(5.3){M˙⁢(t)=-π⁢(Ricμ⁢(t)-14⁢H⁢(t)2)⁢M⁢(t),M⁢(0)=M0∈𝓝n.

Here, for every H∈Λ3⁢(ℝn)*, we set H2≔〈⋅,⋅〉-1⁢(H∘H), where ∘ is meant with respect to 〈⋅,⋅〉. Recalling (2.11), the whole endomorphism ϕ⁢(M)=ϕ⁢(μ,H)=Ricμ-14⁢H2 can then be written in coordinates as

ϕij=-12⁢μi⁢kl⁢μj⁢kl+14⁢μk⁢li⁢μk⁢lj-14⁢Hi⁢k⁢l⁢Hj⁢k⁢l,

with respect to the standard basis of ℝn.

Now, let n=3 and let μ0 be the Heisenberg Lie bracket μ0=e1∧e2⊗e3. Let H0 be the generic (trivially dμ0-closed) 3-form H0=c⁢e123, c∈ℝ. Then, using (2.11), (4.5) and (4.6), it is easy to compute that the solution to (5.3) is of the form M⁢(t)=(x⁢(t)⁢μ0,y⁢(t)⁢e123), with x⁢(t) and y⁢(t) satisfying the ODE system

(5.4){x˙=-32⁢x3-12⁢x⁢y2,y˙=-32⁢y3-12⁢x2⁢y,x⁢(0)=1,y⁢(0)=a.

It is easy to see that the solution (x⁢(t),y⁢(t)) is defined for all positive times and converges to (0,0) since

dd⁢t⁢(x⁢(t)2+y⁢(t)2)=2⁢(x⁢(t)⁢x˙⁢(t)+y⁢(t)⁢y˙⁢(t))≤-(x⁢(t)2+y⁢(t)2)2,

so that, by comparison, we get

x⁢(t)2+y⁢(t)2≤1+a21+(1+a2)⁢t

for all t≥0. For a=1, the explicit solution to (5.4) is given by

x⁢(t)=y⁢(t)=(1+4⁢t)-12.

defined for t∈(-14,∞).

6 Generalized Ricci flow on the Heisenberg group

Let us consider the gauge-corrected generalized Ricci flow (2.8) on the three-dimensional Heisenberg group H3, with initial data

g0=e1⊗e1+e2⊗e2+e3⊗e3,H0=a⁢e123,a∈ℝ.

Denoting by (g⁢(t),H⁢(t)) the solution at time t, we adopt the ansatz

g⁢(t)=g1⁢(t)⁢e1⊗e1+g2⁢(t)⁢e2⊗e2+g3⁢(t)⁢e3⊗e3,gi⁢(0)=1,i=1,2,3,

while H⁢(t)=H0 is necessarily constant since dg* and Δg are null maps for every left-invariant Riemannian metric g, as remarked in Section 5.1.

An explicit computation yields

Rcg⁢(t)=(-12⁢g3g2000-12⁢g3g100012⁢g32g1⁢g2),H0∘H0=(2⁢a2g2⁢g30002⁢a2g1⁢g30002⁢a2g1⁢g2),

so that (2.8) reduces to the ODE system

{g˙1=g3g2+a2g2⁢g3,g˙2=g3g1+a2g1⁢g3,g˙3=-g32g1⁢g2+a2g1⁢g2,gi⁢(0)=1,i=1,2,3.

By uniqueness, we thus have g1⁢(t)=g2⁢(t) for all t and we obtain

(6.1){g˙1=a2+g32g1⁢g3,g˙3=a2-g32g12,g1⁢(0)=g3⁢(0)=1.

Special cases are given by

a=0: the generalized Ricci flow reduces to the classical Ricci flow and an explicit solution to (6.1) is given by
g1⁢(t)=(1+3⁢t)13,g3⁢(t)=(1+3⁢t)-13,
defined on the maximal definition interval I=(-13,∞) (cf. [10]).
a=±1: the system reduces to
{g˙1=2g1,g˙3=0,g1⁢(0)=g3⁢(0)=1,
with solution
g1⁢(t)=(1+4⁢t)12,g3⁢(t)=1
for t∈I=(-14,∞).

A quick qualitative analysis of (6.1) shows that, for all a∈ℝ, the solution to (6.1) exists for all positive times, with

limt→∞⁡g1⁢(t)=∞,limt→∞⁡g3⁢(t)=|a|.

The maximal definition interval is always of the form Ia=(Tmin⁢(a),∞), where Tmin:ℝ→ℝ<0 is an even function, with Tmin⁢(0)=-13 and monotonically converging to 0 as a goes to infinity (see Figure 1). We also have

limt→Tmin⁢(a)+⁡g1⁢(t)=0,limt→Tmin⁢(a)+⁡g3⁢(t)={∞,|a|<1,1,a=±1,0,|a|>1.

In Figure 2, we show some solutions of (6.1), sampled for a=k4, k=0,…,9, and viewed as curves in the phase plane (g1,g3). The red and blue curves correspond to a=0 and a=1, respectively.

$Figure 1 Behavior of the map Tmin{T_{\min}}.$

Figure 1

Behavior of the map Tmin.

$Figure 2 Examples of solutions to (6.1) viewed in the phase plane (g1,g3){(g_{1},g_{3})}.$

Figure 2

Examples of solutions to (6.1) viewed in the phase plane (g1,g3).

Communicated by Jan Frahm

Funding statement: The author was supported by GNSAGA of INdAM.

Acknowledgements

This paper is an adaptation of the author’s master’s thesis, written under 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 [22]. 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.

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Received: 2020-06-26

Revised: 2021-05-11

Published Online: 2021-06-30

Published in Print: 2021-07-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/forum-2020-0171

Keywords for this article

Generalized geometry; generalized Ricci flow; nilpotent Lie groups

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