Stability and moduli space of generalized Ricci solitons

The generalized Einstein Hilbert action is an extension of the classic scalar curvature energy and Perelman F functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. Through a delicate analysis of the group of generalized gauge transformations, and implementing a novel connection, we give a simple formula for the second variation of this energy which generalizes the Lichnerowicz operator in the Einstein case. As an application, we show that all Bismut flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso and Podesta, Spiro, Kr\"oncke to the moduli space of generalized Ricci solitons. To finish we classify deformations of the Bismut-flat structure on S3 and show that some are integrable while others are not.


Introduction
Let M be a smooth manifold and fix a closed 3-form H 0 .Given a smooth family of Riemannian metrics and 2-forms (g t , b t ), we say that (g t , b t ) is a generalized Ricci flow (we will abbreviate it as GRF later on) if This parabolic flow was written in [21], [27] and it can be viewed as the Ricci flow using the Bismut connections ∇ ± = ∇ ± 1 2 g −1 H.The generalized Ricci flow arises naturally in complex geometry [30,32], mathematical physics [25] and generalized geometry [8], [28].We define the generalized Einstein-Hilbert functional and One can see that λ(g, b) can be achieved by some f uniquely, i.e., λ(g, b) = F (g, b, f ) and λ is the first eigenvalue of the Schrödinger operator −4△ + R − 1  12 |H 0 + db| 2 .In [21], it was shown that λ is monotone increasing under the generalized Ricci flow and critical points of λ are steady gradient generalized Ricci solitons.
We say that a pair G(g, b) is a steady generalized Ricci soliton if there exists a smooth vector field X such that where H = H 0 + db.In this work, we only focus on the case when X = ∇f for some smooth function f and we say that G is a steady gradient generalized Ricci soliton.In fact, the first variation of λ implies that if G(g, b) is a compact steady gradient generalized Ricci soliton, then (g, b) satisfies (4) with X = ∇f and f is the minimizer of λ(g, b).
The first goal of this work is to understand the variational structure of λ.In [19,21] second variation formulas were derived which employ the Levi-Civita connection and are difficult to understand geometrically due to the presence of many torsion terms.Here we provide a conceptually distinct formulation which is the foundation of the results to follow.The first point is to address the invariance of λ under the group of generalized gauge transformations, which is the semidirect product of the group of diffeomorphisms with the space of B-field transformations.To address this we employ a slice theorem shown in [26,19] to reduce to certain nontrivial deformations.On this restricted space of deformations we were able to discern a subtle structure in the second variation which leads to many applications.The key point is the introduction of a modified connection ∇ on the variational space T * M ⊗ T * M which employs both Bismut connections ∇ ± .In particular, ∇ is defined by Using the connection ∇ we are lead to the following conceptually clear formulation of the second variation of λ, which forms the foundation of the results to follow.
Theorem 1.1.Given a compact steady gradient generalized Ricci soliton G(g, b) on a smooth manifold M .Suppose G(g t , b t ) is a one-parameter family of generalized metrics such that Let u be the unique solution of where the definition of div f is given in (14) and (15).The second variation of λ on G(g, b) is given by where div * f is the formal adjoint of div f with respect to (5), △ f is defined in (16), R+ (γ), γ = R + iklj γ ij γ kl and R + is the Bismut curvature given in Proposition 2.9.
Clearly, we have the following corollary.
Corollary 1.2.Every compact, Bismut-flat manifold (M, G) is linearly stable.The kernel of the second variation on a compact, Bismut-flat manifold consists of non-trivial 2-tensors γ which are parallel with respect to ∇.
Note that in [19], the author proved that linear stability and dynamical stability are equivalent so we also have the following corollary.
Corollary 1.3.Every compact, Bismut-flat manifold (M, G) is dynamically stable, i.e., for any neighborhood U of G, there exists a smaller neighborhood V such that the generalized Ricci flow starting in V will stay in U for all t ≥ 0 and converge to a critical point of λ.
In the second part of this work, we study the moduli space of generalized Ricci solitons.In a series of papers of Koiso [10,11,12] and [24,16,17], authors discuss the moduli space of Einstein metrics and Ricci solitons.We extend their work to more general setting.Define an operator where f is the minimizer of λ(g, b).So the space of steady gradient generalized Ricci solitons GRS can be viewed as Using the generalized slice theorem, we will say that the premoduli space of steady gradient generalized Ricci solitons at G is the set where S f G is the generalized slice constructed by Theorem 2.7.
Definition 1.4.Let G be a steady gradient generalized Ricci soliton and let S f G denote the generalized slice of G.
• A steady gradient generalized Ricci soliton G is called rigid if there exists a neighborhood U in the space of generalized metrics GM such that G is the only element in U ∩ S f G .
• A 2-tensor γ ∈ T G S f G is called an essential infinitesimal generalized solitonic deformation of G if R ′ G (γ) = 0 where T G S f G denotes the tangent space of the generalized slice S f G at G. • An essential infinitesimal generalized solitonic deformation γ is integrable if there exists a curve of steady gradient generalized Ricci solitons G(t) with G(0) = G and d dt | t=0 G(t) = γ.In the following, we denote the set of essential infinitesimal generalized solitonic deformation to be IGSD.We furthermore obtain results on the rigidity of the steady gradient generalized Ricci soliton.The first step to approach the rigidity question is to discuss the existence of essential infinitesimal generalized solitonic deformations.We prove that Theorem 1.5.Given any simply connected, compact, Bismut-flat manifold (M, G).There exist essential infinitesimal generalized solitonic deformations of G, and its dimension is no less than n 2 .
Our second step is to discuss the integrability of IGSD.We prove the following theorem which shows that it is equivalent to computing the k-th derivative of R G to check the integrability.
Theorem 1.6.Let G be a steady gradient generalized Ricci soliton.There exists a neighborhood U of G in the generalized slice S f G and a finite-dimensional real analytic submanifold Z ⊂ U such that • Z contains the premoduli space P G as a real analytic subset.
In the last part of this work, we focus on the Bismut-flat case.The existence of the non-trivial IGSD is proved in Theorem 1.5; however, it is hard to check its integrability.In a 3-dimensional case, we can explicitly construct the essential infinitesimal generalized solitonic deformations and prove that Theorem 1.7.Suppose (M, g, H) is a 3-dimensional Bismut-flat, Einstein manifold with positive Einstein constant µ.Then, any essential infinitesimal generalized solitonic deformation is of the form where u is an eigenfunction with eigenvalue 4µ.Moreover, γ is not integrable up to the second order if ˆM µu 2 wdV g = 0 for some eigenfunctions w with eigenvalue 4µ.
In summary, in the 3-dimensional, Bismut-flat case, the dimension of the space of essential infinitesimal generalized solitonic deformation is 9.Some essential infinitesimal generalized solitonic deformations are integrable up to the second order while some are not.In [29] Corollary 1.4, Streets showed that there exists a non-trivial steady gradient generalized Ricci soliton in any dimension n ≥ 3.In particular, his proof showed that the Bismut-flat metric on S 3 is not rigid.Therefore, there exist an integrable essential infinitesimal generalized solitonic deformations.
The layout of this paper is as follows: in Section 2, we will mention some preliminaries regarding Courant algebroids, the generalized slice theorem, and the generalized Ricci solitons.In Section 3, we analyze the second variation formula of λ.In Section 4, we will discuss the essential infinitesimal generalized solitonic deformation and integrability properties.In Section 5, we will focus on the Bismut-flat case and provide some examples of essential infinitesimal generalized solitonic deformations.

Notations
In this work, we will use the following notation.Suppose h ∈ Γ(S 2 M ) and where R + denotes the Bismut curvature which is defined in Proposition 3.1.Besides, we will consider the f -twisted L 2 inner product where , g denotes the standard inner product induced by a Riemannian metric g and γ 1 , γ 2 are tensors of same order.In particular, we mainly focus on the case when γ is a 2-tensor.We then require the notation Later, we will use the different connection ∇ ± which is defined in (10).Thus, we will denote div ± as the divergence operator computed by using connction ∇ ± and div ±, * as their formal adjoint.

Generalized Geometry
In this section, we review some basic definitions and properties of generalized geometry.More details can be found in [9].
We say a Courant algebroid E is exact if we have the following exact sequence of vector bundles 0 Definition 2.3.Given a smooth manifold M and an exact Courant algebroid E over M , a generalized metric on E is a bundle endomorphism G ∈ Γ(End(E)) satisfying • Ga, b is symmetric and positive definite for any a, b ∈ E.
Example 2.4.The most common and important example of Courant algebroids is T M ⊕ T * M .In this case, we define a nondegenerate bilinear form •, • and a bracket [ where X, Y ∈ T M , ξ, η ∈ T * M and H is a 3-form.Define π to be the standard projection, one can check that (T M ⊕ •] H , π) satisfies the Courant algebroid conditions.Moreover, its automorphism groups are given as follows. GDiff where The product of automorphisms is given by In the following, we will denote GDiff H to be the automorphism group of (T M ⊕ T * M ) H , GM to be the space of all generalized metrics, and M to be the space of all Riemannian metrics.
Recall that in [9] Proposition 2.10, we see that for any exact Courant Courant algebroid E with a isotropic splitting Therefore, we see that Moreover, we have the following proposition.Proposition 2.5 ([9] Proposition 2.38 and 2.40).Let E be an exact Courant algebroid.The space of all generalized metrics GM on E is isomorphic to M × Ω 2 .Remark 2.6.Fix a background 3-form H 0 such that E ∼ = (T M ⊕ T * M ) H0 , the proof of Proposition 2.5 implies that the 3-form H of any generalized metric G = G(g, b) is induced by an isotropic splitting σ(X) = X + i X b and then we have H = H 0 + db.(See [19] Remark 2.7 for more details.)

Generalized Slice Theorem
Recall that on M, we have a natural group action which is given by The quotient of M in terms of the action ρ M is called the moduli space of Riemannian metrics.In order to study the moduli space of Riemannian metrics, Ebin proposed his slice theorem [7] which proved the existence of a slice for the diffeomorphism group of a compact manifold acting on M.
In the generalized geometry, we define the GDiff H action on generalized metrics by Here, we note that GM ∼ = M × Ω 2 so in the following, we will always denote a generalized metric G by G(g, b) for some (g, b) ∈ M × Ω 2 .Therefore, Naturally, we have a L 2 inner product on T G GM defined by where In [26], Rubio and Tipler proposed the generalized Ebin's slice theorem based on the inner product (8).In [19], we proved the generalized slice theorem based on the f -twisted inner product (5).The precise statement is as follows.
Theorem 2.7 ([19] Theorem 2.14).Let G be a generalized metric on an exact Courant algebroid E and f be Isom(G) invariant, then there exists an submanifold S f G of GM such that • There exists a local cross section χ of the map where Isom H (G) is the isotropy group of G under the GDiff H -action and it is called the group of generalized isometries of G ∈ GM.Moreover, the tangent space of the generalized slice on a generalized metric G(g, b) is given by

Bismut connection and curvature
In this subsection, we aim to discuss the generalized Ricci flow and generalized Ricci solitons.Most of the contents can be found in [9] Section 4. Definition 2.8.Let (M, g, H) be a Riemannian manifold and H ∈ Ω 3 .The Bismut connections ∇ ± associated to (g, H) are defined as Here, ∇ is the Levi-Civita connection associated with g, i.e., ∇ ± are the unique compatible connections with torsion ±H.Later, we will mainly use ∇ + .
Following the definitions, we are able to compute the curvature tensor of the Bismut connection.Proposition 2.9 ([9] Proposition 3.18).Let (M n , g, H) be a Riemannian manifold with H ∈ Ω 3 and dH = 0, then for any vector fields X, Y, Z, W we have where H 2 (X, Y ) = i X H, i Y H .Here Rm + , Rc + , R + denote the Riemannian curvature, Ricci curvature, and scalar curvature with respect to the Bismut connection ∇ + .In particular, if (M, g, H) is Bismut-flat, then H(Y, W ), H(X, Z) and ∇H = 0 for any vector fields X, Y, Z, W .
In [31], the author defined general tensors regarding the Bismut connection.Later, we will see that these quantities are related to the generalized Einstein-Hilbert functional.Definition 2.10.Given a metric g, closed three-form H, and smooth function f , a triple (g, H, f ) determines a twisted Bakry-Emery curvature and a generalized scalar curvature As we mentioned in the introduction, we will consider a special connection ∇ on 2-tensors.Definition 2.11.Let (M, g, H) be a Riemannian manifold, H ∈ Ω 3 and ∇ ± be the Bismut connections given in (10).The mixed Bismut connection is a connection ∇ on 2-tensors defined as follows.For any γ ∈ ⊗ 2 T * M and tangent vectors X, Y, Z, we have Definition 2.12.Let ∇ be the mixed connection.
In the following, we denote div * f to be the formal adjoint of the f -twisted divergence operator div f with respect to f -twisted L 2 inner product (5).
Remark 2.13.Due to (13), it is not hard to see that Definition 2.14.The Laplace operator of the mixed Bismut connection △ f is defined by where ∇ * f is the formal adjoint of ∇ with respect to f -twisted L 2 inner product (5).
The following two lemmas provide us a detail information about div * f and △ f .Lemma 2.15.Given any Proof.From the definition, we compute Proof.For any 2-tensor γ ∈ ⊗ 2 T * M , using normal coordinates we have Then, Therefore, the result follows.
Proposition 2.17.Let G be a generalized metric on an exact Courant algebroid E and f be Isom(G) invariant.The tangent space of the generalized slice on a generalized metric G is given by In the following, we say that a 2-tensor γ is non-trivial if γ ∈ T G S f G .Proof.By (10), we observe that Then, Therefore, our result follows by Theorem 2.7.

Generalized Ricci Solitons
In this subsection, we discuss the generalized Ricci flow and generalized Ricci solitons.Most of the contents can be found in [9] Section 4.
Definition 2.18.Let E be an exact Courant algebroid over a smooth manifold M and H 0 is a background closed 3-form.
A one-parameter family of generalized metrics Equivalently, the generalized Ricci flow can also be expressed as where Rc + denotes the Ricci curvature with respect to the Bismut connection ∇ + .
Motivated by the Ricci flow, we define the stationary points to be the steady generalized Ricci solitons.(For more details about motivations, readers can consult with [9] and [19].)Definition 2.19.Given a Riemannian metric g, a closed 3-form H, and a smooth function f on a smooth manifold M .We say (M, g, H, f ) is a steady gradient generalized Ricci soliton if Example 2.20.The most basic example of the steady generalized Ricci soliton is the work on S 3 .Given a standard unit sphere metric g S 3 .By taking H S 3 = 2dV g S 3 , we get ) is a generalized Einstein metric.Moreover, any 3-dimensional compact generalized Einstein manifold is a quotient of S 3 .(See [19], Corollary 2.22 for more detail.) Example 2.21.Suppose G is a compact Lie group.By [20], we know that G possesses a bi-invariant metric •, • , and its corresponding connection, Riemann curvatures, sectional curvatures are given by By direct computation, we know that connections ∇ ± are flat.Thus, any compact Lie group admits a Bismut-flat structure.
On the other hand, a famous result of Cartan Schouter (See [9] Theorem 3.54) shows that if (M, g, H) is complete, simply connected and Bismut-flat then (M, g) is isometric to a product of simple Lie groups with bi-invariant metrics g, and g −1 H(X, Y ) = ±[X, Y ] on left-invariant vector fields.

Generalized Einstein-Hilbert functional
In this subsection, we will see that generalized Ricci solitons are related to the generalized Einstein-Hilbert functional which are defined below.Definition 2.22.Given a smooth manifold M and a background closed 3-form H 0 , the generalized Einstein-Hilbert functional where R H,f is the generalized scalar curvature given in ( 12) and H = H 0 + db.Also, we define Following the same argument in the Ricci flow case (see [5] for more details), we can deduce that for any (g, b), the minimizer f is always achieved.Moreover, λ satisfies that and it is the lowest eigenvalue of the Schrödinger operator −4△ + R − 1 12 |H 0 + db| 2 .Let G t (g t , b t ) be a smooth family of generalized metrics on a smooth compact manifold M .Assume The first variation formula of the generalized Einstein-Hilbert functional is given by where Rc H,f is the twisted Bakry-Emery curvature given in (11) and γ = h − K. Based on the first variation formula, we conclude that Corollary 2.23.The generalized metric G(g, b) is a critical point of λ if and only if (g, b, f ) is a steady gradient generalized Ricci soliton with f realizing the infinmum in the definition of λ.
Remark 2.24.Due to Corollary 2.23, we say that a generalized metric G(g, b) is a steady gradient generalized Ricci soliton if (g, b, f ) satisfies the equation (19) where f is the minimizer of λ.
3 Linear Stability

Analytic Properties of Generalized Einstein-Hilbert Functional
Given a compact steady gradient generalized Ricci soliton G).The main goal of this section is to compute the second variation formula.First, we recall that in [19] we have an analyticity property.  .Suppose G t (g t , b t ) is a one-parameter family of generalized metrics and f t is the minimizer of λ(g t , b t ) such that We have dK, H .
Remark 3.3.Recall in the proof of Proposition 2.17, we have Thus, Then, the variation φ can be viewed as a function of γ which satisfies In particular, if γ ∈ T G S f G , by (18) we have tr g γ = 2φ.

Variation of Generalized Metrics
In this subsection, we compute some variation formulas.
Lemma 3.4.Given a compact steady gradient generalized Ricci soliton G(g, b).Suppose G(g t , b t ) is a one-parameter family of generalized metrics and f t is the minimizer of λ(g t , b t ) such that We have In particular, if Proof.Recall that in [19] Lemma 3.5 we computed the derivative in the general case.Here, we use Proposition 2.9 to derive that Replace R by R+ , we get our first result.Now, we consider γ = h − K ∈ T G S f G .Using (18) , we have Therefore, we see that Here, we note that tr g h = 2φ.
Lemma 3.5.Given a compact steady gradient generalized Ricci soliton G(g, b).Suppose G(g t , b t ) is a one-parameter family of generalized metrics and f t is the minimizer of λ(g t , b t ) such that We have In particular, if Proof.First, we note that the variation of d * H and i ∇f H are given by Therefore, Then, where we use the fact that d * f K = d * K + i ∇f K. Similarly, we can replace the Riemann curvature with the Bismut Riemann curvature and deduce that Using the fact that dH = 0, we have Thus, we get our first result.If we consider γ = h − K ∈ T G S f G , our result follows by (18).
Proposition 3.6.Given a compact steady gradient generalized Ricci soliton G(g, b).Suppose G(g t , b t ) is a one-parameter family of generalized metrics and f t is the minimizer of λ(g t , b t ) such that We have where u = tr g h − 2φ.In particular, if γ ∈ T G S f G , we have ). Lemma 3.4 and Lemma 3.5 imply that where Recall that in Proposition 2.17, we deduced that By Lemma 2.15, we conclude that If γ ∈ T G S f G , then div f γ vanish, giving the final claim.

Second variation formula
Next, we analyze the second variation formula of the generalized Einstein-Hilbert functional.
Theorem 3.7.Given a compact steady gradient generalized Ricci soliton G(g, b) on a smooth manifold M .Suppose G(g t , b t ) is a one-parameter family of generalized metrics such that The second variation of λ on G(g, b) is given by where u is the unique solution of Proof.The first variation formula (20) suggests that By Proposition 3.6 and Lemma 3.2, our result follows.
Corollary 3.8.Every Bismut-flat, compact manifolds are linearly stable.The kernel of the second variation on a Bismutflat, compact manifold is non-trivial 2-tensor γ which are parallel with respect to the connection ∇.
Proof.Due to the fact that λ is diffeomorphism invariant and the generalized slice theorem, it suffices to show that Therefore, the result follows by Theorem 3.7.
Recall that in Einstein manifold case, it suffices to consider variations which lie in T T g = {h ∈ Γ(S 2 M ), div h = 0, tr g h = 0} to discuss the linear stability.Therefore, our result matches our expectation.

The kernel variation of the second variation
In this subsection, we assume that (M, g, H) is a compact, Bismut-flat manifold.Corollary 3.8 implies that γ is a kernel of the second variation if γ is non-trivial and parallel to the connection ∇.The lemma below suggests an idea to construct parallel variations.
Proof.By assumption, we see that Then, which shows that ∇γ = 0. Also, we have Therefore, i.e., γ is non-trivial.
Corollary 3.11.Any compact Lie group is linearly stable and admits a non-trivial variation γ such that its second variation vanishes.Moreover, the dimension of the kernel of the second variation is no less than n 2 .
Proof.In Example 2.21, we know that any compact Lie group G with a bi-invariant metric g admits a Bismut-flat structure when we define a 3-form H by g −1 H(X, Y ) = [X, Y ] where X, Y are left-invariant vector fields.Then, we consider the left-invariant coframe {ω L 1 , ω L 2 , ..., ω L n } and the right-invariant coframe {ω R 1 , ω R 2 , ..., ω R n }.By definition, we see that each left-invariant one form is ∇ − -parallel and right-invariant one form is ∇ + -parallel.Therefore, corollary follows by Lemma 3.10.

Infinitesimal Deformation
In this section, we aim to discuss the infinitesimal deformations of gradient generalized Ricci solitons.Before we get started, let us review the results of Einstein manifolds.Most materials can be found in [2] and a series of papers from Koiso [10,11,12,13].One can also consult with the work in the Ricci soliton case done by Kröncke Podestà,and Spiro [16,17,18,24].

Infinitesimal Einstein Deformation
Fix an Einstein metric g on a manifold M .Let M 1 denote the space of smooth metrics with unit volume.The moduli space of Einstein structures is the coset space M 1 under the action of ρ M endowed with the quotient topology.Naturally, we have a decomposition Ebin's slice theorem suggests that there exists an analytic submanifold S g ⊂ M 1 with T g S g = ker div.Then, we call the subset of Einstein metrics in S g to be the premoduli space of Einstein structure around g.
Define the Einstein operator E by where S(g) is the total scalar curvature functional which is defined as S(g) = ´M R g dV g .In other words, Einstein metrics are the set E −1 (0).Then, we say h ∈ Γ(S 2 M ) is an essential infinitesimal Einstein deformation of an Einstein metric g if In the following, we denote the space of all infinitesimal Einstein deformation by ǫ(g) and the direct computation shows that ǫ(g) = {h : h ∈ ker div, tr g h = 0, △h + 2 R(h) = 0}.
An infinitesimal Einstein deformation h is said integrable if there exists a C 1 curve of Einstein metrics g(t) through g = g 0 such that d dt | t=0 g(t) = h.Given h ∈ ǫ(g), one of our questions is whether h is integrable.To answer this question, [2] Corollary 12.50 found that it is equivalent to check whether h is formally integrable, i.e., if there exists h 2 , h 3 , ... such that E(g(t)) ≡ 0 where Define the Bianchi operator β g by β g (h) = div h− 1 2 d tr g h where h ∈ Γ(S 2 M ).It is not hard to see that β g (E(g)) ≡ 0 so the formal integrability is closely related to the Bianchi operator.In fact, it depends on the obstruction space ker β g / Im E ′ g .In [2] Theorem 12.45, we have Therefore, one sees that some infinitesimal deformations are not formally integrable.where ρ GM is given in (7).

Infinitesimal Generalized Solitonic Deformation
In other words, we say G 0 is rigid if there exists some neighborhood U such that the generalized slice S f0 G0 only contains one element G 0 .To study the local behavior, we define an operator where f is the minimizer of λ(g, b).Let's denote the space of steady gradient generalized Ricci solitons by GRS and we see that Definition 4.2.Let G 0 be a steady gradient generalized Ricci soliton.The premoduli space of steady gradient generalized Ricci soliton at G 0 is the set where S f0 G0 is the generalized slice constructed in Theorem 2.7.Locally, the map R is analytic and we are able to compute its derivative.By Proposition 3.6, we have the following results.
Lemma 4.3.Let A denote the derivative of the operator R.Then, A is given by where u = trg h 2 − φ and φ is the derivative of f which is a function depending on γ.Moreover, A is a self-adjoint operator with respect to the inner product (5).
Notice that A is not an elliptic operator.Motivated by Proposition 3.6, we then define an elliptic operator B by Definition 4.4.Let G be a steady gradient generalized Ricci soliton.
In the following, we denote the set of all essential infinitesimal generalized solitonic deformations by IGSD, i.e.
Next, we define the Bianchi operator β G by Recall the proof of Proposition 2.17, if γ = h − K, we could also write Lemma 4.5.For any generalized metric G, Using the fact that dH = 0, we have Our result follows by the fact Proposition 4.6.For any steady gradient generalized Ricci soliton G, we have a decomposition Proof.Since γ is non-trivial if γ ∈ IGSD, it is obvious that IGSD ⊂ ker β G .Also, Lemma 4.5 implies that Im A ⊂ ker β G since R(G) ≡ 0. On the other hand, because A is self adjoint, we have For any γ ∈ ker β G , we write γ = A(γ 1 ) + γ 2 where γ 2 ∈ ker A. Then, which implies that γ 2 is non-trivial.Therefore, we finish the proof.
Lemma 4.7.Define an elliptic operator For any steady gradient generalized Ricci soliton G and γ ∈ ⊗ 2 T * M , we have Proof.By Lemma 4.5, we see that if G ∈ GRS, for any γ ∈ ⊗ 2 T * M .From the definition, we see that where Therefore, We compute Also, Furthermore, Here, we note that by Lemma 3.2 In addition, where we use the fact that d * f ζ = 0. Therefore, we compute that By definition, u l = α l + ζ l and v l = α l − ζ l .We note that so the lemma follows.

Integrability
Notations: In the following, we write the formal power series expansion of generalized metrics by It means that (g(t), b(t)) is expanded by where h l ∈ Γ(S 2 M ), K l ∈ Ω 2 and γ l = h l − K l for l = 1, 2, 3, ....In other words, this notation means that if we denote Definition 4.8.Let G be a steady gradient generalized Ricci soliton.
• An essential infinitesimal generalized solitonic deformation γ is integrable if there exists a curve of steady gradient generalized Ricci solitons G(t) with G(0) = G and d dt | t=0 G(t) = γ.Theorem 4.9.Let G(g, b) be a steady gradient generalized Ricci soliton.There exists a neighborhood U of G(g, b) in the slice S f G and a finite-dimensional real analytic submanifold Z ⊂ U such that • T G (Z) = IGSD • Z contains the premoduli space P G as a real analytic subset.
Proof.Due to [10] Lemma 13.6, it suffices to show that We aim to check that the set {B(γ) : γ ∈ T G S f G } is closed.By Lemma 4.7, we see that Note that B is an elliptic operator then ker β ∩ Im B is a closed subset in ker β.On the other hand, for any B(γ) ∈ ker β, we can decompose γ = γ 1 + γ 2 where γ 1 ∈ T G S f G and γ 2 ∈ T G O G by the generalized slice theorem.Then, Since Φ is elliptic, it implies that {B(γ) : γ ∈ T G S f G } has finite codimension in the closed subset ker β ∩ Im B. Follow the argument in [22] page 119, one can prove that {B(γ) Corollary 4.10.Let G be a steady gradient generalized Ricci soliton.An essential infinitesimal generalized solitonic deformation is integrable if and only if it is formally integrable.
Corollary 4.11.Let G be a steady gradient generalized Ricci soliton.G is rigid if every essential infinitesimal solitonic deformation at G is integrable up to finite order.
In general, it is hard to check whether γ ∈ IGSD is integrable or not.The following result provides us a condition to check whether γ is integrable up to the second order.Lemma 4.12.Suppose γ is an infinitesimal generalized Ricci solitonic deformation on a steady gradient generalized Ricci soliton G.Then, γ is integrable up to the second order if and only if R ′′ (γ, γ) is orthogonal to IGSD.

Proof. Consider a curve of generalized metrics
Therefore, γ is integrable up to second order if and only if The result is followed by Proposition 4.6.

Infinitesimal generalized Einstein deformations
Similar to the steady gradient generalized Ricci soliton case, we define we define an operator We denote the space of generalized Einstein metrics with unit volume to be GE and then Let G 0 be a generalized Einstein metric.The premoduli space of generalized Einstein metrics at 0 is where S 0 G0 is the generalized slice (Note that the minimizer is 0 in this case.).Similarly, we can take the derivative of E. By (21), we see that the deformation γ satisfies dK, H , ˆM tr g hdV g = 0. ( 23) Therefore, we have the following definition.Definition 4.13.Let G be a generalized Einstein metric (f = 0).
G with tr g h = 0.In the following, we denote the set of all essential infinitesimal generalized Einstein deformations by IGED, i.e.
Remark 4.14.In short, we see that IGED is a subset of IGSD.More precisely, where φ is the function of γ given by equation ( 23)}.

Deformations of Bismut-flat structure
In this section, we aim to discuss essential infinitesimal generalized solitonic deformations on a Bismut-flat manifold.

Equivalent conditions
Proposition 5.1.Let G be a Bismut-flat metric.The following statements are equivalent.
(a) γ is an essential infinitesimal generalized solitonic deformation of G.
We recall that the second variation of generalized Einstein-Hilbert functional is given by By using Proposition 5.1 part (c), we compute On the other hand, Note that where we use the fact that △h we have Lemma 5.6.Suppose (M n , g, H) is a compact Bismut-flat manifold.If γ = h − K is an essential infinitesimal generalized solitonic deformation on (M n , g, H), then Proof.By definition, Therefore, It also implies Theorem 5.7.Suppose (M n , g, H) is a compact Bismut-flat manifold with positive sectional curvature.Then, there does not exist any essential infinitesimal generalized Einstein deformation.
Proof.Suppose γ = h − K is an essential infinitesimal general Einstein deformation.Taking the trace of Proposition 5.1 part (c), we get From the definition, we have div h = 0.Then, Lemma 5.5 and Lemma 5.6 imply and sec(e i , e j )(K ij ) 2 = 0.
We can conclude that K = h = 0 since Ricci curvatures are positive.
Remark 5.8.Suppose (M n , g, H) is a compact Bismut-flat, Ricci flat manifold.If γ = h − K is an essential infinitesimal generalized Einstein deformation, then the proof in Theorem 5.7 reduces to say that ∇h = 0 and tr g h = 0.
In fact, (M n , g, H) is a flat manifold (H = 0) and we observe that h is also an essential infinitesimal Einstein deformation that matches our expectation.
Example 5.9.Suppose (M, g, H) is a compact semisimple Lie group with bi-invariant metric g and 3-form H is constructed by g −1 H(X, Y ) = [X, Y ] where X, Y are left-invariant vector fields.In fact, the Killing form is negative definite thus (M, g, H) is a compact, Bismut-flat, Einstein manifold with Einstein constant µ.Recall that in the proof of Corollary 3.11 we constructed essential infinitesimal generalized solitonic deformations which are defined by where α is a left-invariant 1-form and β is a right-invariant 1-form.Since γ is ∇-parallel, we compute that In other words, △γ + 2 R(γ) + 2µγ = 0.By taking the trace, we see that △(tr g γ) + 4µ(tr g γ) = 0.
In this example, it is clear to see that tr g γ can not be 0 i.e., γ must be an essential infinitesimal generalized solitonic deformation and tr g γ is an eigenfunction of △ with eigenvalue 4µ.In S 3 case, we will see that all infinitesimal solitonic deformations are constructed by using left and right invariant one forms.The arguments above shows that IGED = ∅ which matches the result of Theorem 5.7.

Dimension 3 Case.
In this section, we focus on the 3-dimensional case.Recall that in [19] Corollary 2.22, we show that any compact, 3-dimensional generalized Einstein manifold has constant nonnegative sectional curvatures.In this section, we assume (M, g, H) is a 3-dimensional, compact, Bismut-flat manifold with positive Einstein constant µ and reader should note that it must be a quotient of S 3 .
Proposition 5.10.Suppose (M, g, H) is a 3-dimensional, compact, Bismut-flat manifold with positive Einstein constant µ.Then, any essential infinitesimal generalized solitonic deformation is of the form where u is an eigenfunction with eigenvalue 4µ.
Remark 5.11.Let (M n , g, H) be a Bismut-flat, Einstein manifold with Einstein constant µ and n ≥ 4. Suppose u is an eigenfunction with eigenvalue 4µ.By direct computation, we can pick nonzero constants a, c such that γ = aµug + c∇ 2 u − d * (uH) satisfy (24) and γ is non-trivial.Since λ is diffeomorphism invariant, the variation of the form L X g − i X H is trivial for any vector field X.In particular, we pick X = − c 2 ∇u.Then, If γ ∈ IGSD, then there exists a conformal variation in the kernel of second variation.This result contradicts with Lemma 4.9 in [19].Therefore, the argument in the proof of Proposition 5.10 only works in 3-dimensional case.
Recall that in the unit sphere S 3 with Einstein constant µ = 2, the eigenvalues of Laplacian are k(k + 2) where k = 1, 2, ... and their corresponding eigenfunctions are homogenous harmonic polynomials of degree k in R 4 .Therefore, we get the following corollary.
Corollary 5.12.Let {x 1 , x 2 , x 3 , x 4 } be a coordinate of R 4 .Any essential infinitesimal generalized solitonic deformation γ on the unit sphere S 3 is of the form where u is spanned by functions on the unit sphere |x| = 1.The dimension of essential infinitesimal generalized solitonic deformations on S 3 is 9.

Second order Integrability
In the last part of this section, we would like to see that not all of these essential infinitesimal generalized solitonic deformations are integrable up to second order.Let (M, g, H) be a 3-dimensional, compact, Bismut-flat manifold with positive Einstein constant µ.Due to Lemma 4.12, γ = 2µug + ∇ 2 u − 1 2 d * (uH) ∈ IGSD is integrable up to second order if and only if for any w satisfying △w + 4µw = 0. Since λ is diffeomorphism invariant, we can normalize and consider In the following, we fix a background 3-form H and consider a family of smooth metrics and 2-forms (g t , b t ) with Then, In addition, we take f t to be the minimizer of λ(g t , b t ).For convenience, we denote i.e., u t = u 1+tu and ũt = − △u 2µ−(△u)t .Therefore, By using above notation, we can derive the evolution formulas which we record in appendix B. We then derive the following theorem. Theorem where we use integration by parts to see that ˆM u ∇u, ∇w dV g = − ˆM u 2 △wdV g − ˆM u ∇u, ∇w dV g = 4 ˆM u 2 wµdV g − ˆM u ∇u, ∇w dV g .
By this theorem, we see that not all γ ∈ IGSD are integrable up to second order.For example, But, there are some examples that satisfy the criterion.For instance, Then,

A Variation formulas
In this appendix, let (M, g, H) be a 3-dimensional, compact, Bismut-flat manifold with positive Einstein constant µ and u is an eigenfunction of Laplacian with eigenvalue 4µ.Consider a family (g t , H t , f t ) ∈ Γ(S 2 M ) × Ω 3 × C ∞ (M ) with g t = (1 + tu)g, H t = (1 + 2tu)H, and f t is the minimizer of λ(g t , H t ).We have the following results.
Besides, we know that a Bismut-flat manifold is a critical point of λ so we are able to compute the derivative of f .Lemma A.2. Let (M, g, H) be a 3-dimensional, compact, Bismut-flat manifold with positive Einstein constant µ and u is an eigenfunction of Laplacian with eigenvalue 4µ.Suppose (g t , H t , f t ) ∈ Γ(S 2 M ) × Ω 3 × C ∞ (M ) with g t = (1 + tu)g, H t = (1 + 2tu)H, and f t is the minimizer of λ(g t , H t ).Then, Proof.We recall that the derivative of f satisfies the equation ( 21) so in this case 0 = 2△( ∂ ∂t t=0 f ) − △u, ˆM 3u − 2( ∂ ∂t t=0 f )dV g = 0.
Since ´M udV g = 0, we can conclude that ∂ ∂t t=0 f = u 2 .For the second derivative, we observe that We compute Proof.Following Lemma A.1, we compute Proposition 3.1 ([19] Proposition 5.5).Let G(g 0 , b 0 ) be a compact steady gradient generalized Ricci soliton.There exists a C 2,α -neighborhood U of (g 0 , b 0 ) such that the minimizers f (g,b) depends analytically on (g, b) and λ(g, b) is an analytic function in U.The proof of Proposition 3.1 is based on the implicit function theorem.Since compact steady gradient generalized Ricci solitons are critical points of λ-functional, we have the following results.