Conformal Ricci Solitons and related Integrability Conditions
Abstract.
In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of some appropriate and highly nontrivial tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor recently introduced by Cao and Chen. A significant part of our investigation, which has independent interest, is the derivation of a number of commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.
Key words and phrases:
integrability conditions, conformal Ricci soliton, Ricci soliton, conformal Einstein manifold, commutation rules, conformal change of the metric2010 Mathematics Subject Classification:
53C20, 53C25.G. Catino^{1}^{1}1Politecnico di Milano, Italy.
Email: giovanni.catino@polimi.it. Supported
by GNAMPA projects “Equazioni differenziali con invarianze in
analisi globale” and “Equazioni di evoluzione geometriche e strutture di tipo Einstein”., P.
Mastrolia^{2}^{2}2Università degli Studi di Milano, Italy.
Email: paolo.mastrolia@gmail.com. Partially supported by FSE,
Regione Lombardia and GNAMPA project “Analisi Globale ed Operatori Degeneri”., D. D. Monticelli^{3}^{3}3Università degli
Studi di Milano, Italy. Email: dario.monticelli@unimi.it. Supported
by GNAMPA projects “Equazioni differenziali con invarianze in
analisi globale” and “Analisi Globale ed Operatori Degeneri”. and M.
Rigoli^{4}^{4}4Università degli Studi di Milano, Italy. Email:
marco.rigoli@unimi.it.
The first, the second and the third authors are members of the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA)
of the Istituto Nazionale di Alta Matematica (INdAM).
,
1. Introduction
In recent years the pioneering works of R. Hamilton ([18]) and G. Perelman ([30]) towards the solution of the Poincaré conjecture in dimension have produced a flourishing activity in the research of self similar solutions, or solitons, of the Ricci flow. The study of the geometry of solitons, in particular their classification in dimension , has been essential in providing a positive answer to the conjecture; however, in higher dimension and in the complete, possibly noncompact case, the understanding of the geometry and the classification of solitons seems to remain a desired goal for a not too proximate future. In the generic case a soliton structure on the Riemannian manifold is the choice (if any) of a smooth vector field on and a real constant satisfying the structural requirement
(1.1) 
where is the Ricci tensor of the metric and is the Lie derivative of this latter in the direction of . In what follows we shall refer to as to the soliton constant. The soliton is called expanding, steady or shrinking if, respectively, , or . When is the gradient of a potential , the soliton is called a gradient Ricci soliton and the previous equation (1.1) takes the form
(1.2) 
Both equations (1.1) and (1.2) can be considered as perturbations of the Einstein equation
(1.3) 
and reduce to this latter in case or are Killing vector fields. When or is constant we call the underlying Einstein manifold a trivial Ricci soliton. The great interest raised by these structures is also shown by the rapidly increasing number of works devoted to their study; for instance, just to cite a few of them, we mention in particular [19], [29], [16], [28], [36], [3], [27], [31], [9], [6], [12], [32], [10], [11], [8], [5], [25], [13] (and references therein) on Ricci solitons and [20], [2], [33], [34], [24] (and references therein) on Einstein manifolds.
A natural question, which arises for instance in conformal geometry, is to construct conformally Einstein manifolds, i.e. Riemannian manifolds for which there exists a pointwise conformal deformation , , such that the new metric is Einstein. This problem has received a considerable amount of attention by mathematicians and physicists in the last decades: just to mention some old and recent papers we cite the pioneering work of Brinkmann, [4], Yano and Nagano, [35], Gover and Nurowski, [17], Kapadia and Sparling, [21], Derdzisnki and Maschler, [15], and references therein. In particular in [17] the authors describe two necessary integrability conditions for the existence of the conformal deformation realizing the Einstein metric. They are of course related to the system
(1.4) 
where tilded quantities refer to the metric , and they are expressed in terms of the Cotton, Weyl and Bach tensors and the gradient of in the background metric (see Section 2 for precise definitions); precisely, performing a computation in some sense reminiscent of the classical Cartan’s approach to the treatment of differential systems, Gover and Nurowski show that if is a conformally Einstein Riemannian manifold, then the Cotton tensor, the Weyl tensor, the Bach tensor and the exponent of the stretching factor satisfy the conditions (see also Proposition 6.4)
(1.5)  
(1.6) 
On the other hand, Cao and Chen in [7] and [8] study the geometry of Bach flat gradient solitons, introducing a tensor related to the geometry of the level surfaces of the potential that generates the soliton structure. The vanishing of , obtained via the vanishing of the Bach tensor, is a crucial ingredient in their classification of a wide family of complete gradient Ricci solitons; in particular in their proof they show that every gradient Ricci soliton satisfies the two conditions
(1.7)  
(1.8) 
The above equations must be intended as integrability conditions for solitons, in the same way as (1.5) and (1.6) are related to conformally Einstein manifolds. We observe that the aforementioned classification result has been recently generalized by the present authors in [14] to a new general structure (which includes Ricci solitons, Yamabe solitons, quasiEinstein manifolds and almost Ricci solitons), called (gradient) Einsteintype manifold, for which the corresponding integrability conditions have also been computed.
In the present work we introduce for the first time the counterpart of the tensor in the case of generic Ricci solitons: we call it and we show that in this setting the integrability conditions take the form
(1.9)  
(1.10) 
(see Theorem 8.2). We explicitly note that, if for some , then and the two previous equations become, as one should expect, (1.7) and (1.8) respectively.
Since Einstein metrics are trivial solitons, it is now natural to study conformal Ricci solitons, i.e. to search for pointwise conformal transformations of the metric as above, such that the manifold is a gradient Ricci soliton, that is for some and we have the structural relation
(1.11) 
One of the main aims of the paper is to produce integrability conditions corresponding to these structures; in their study we introduce here for the first time a natural tensor, which we denote by (see (7.8)) and which allows to interpret the corresponding integrability conditions as interpolations between those associated to conformally Einstein manifolds (1.5) and (1.6), and those related to gradient Ricci solitons (1.7) and (1.8). Moreover, vanishes identically in the case of a conformally Einstein manifold, while it reduces to the tensor on a gradient Ricci soliton. More precisely, in Section 7 we obtain two integrability conditions for (1.11) (see Theorems 7.6 and 7.10), which tell us that if is a conformal gradient Ricci soliton then
(1.12) 
and
(1.13) 
In Section 9 we further extend our results to the very general case of a conformal generic Ricci soliton, that is a Riemannian manifold such that, for a conformal change of the metric with , there exist a smooth vector field , not necessarily a gradient, and a constant such that
In this case the integrability conditions that we produce (see Theorems 9.6 and 9.8) involve the construction of the appropriate generalization of both the tensors and , that we call and which reduces to the previous ones in the corresponding cases. As one can expect, these new conditions capture all those appearing in the aforementioned settings.
As it will become apparent to the reader, the analysis carried out in this paper is very heavy from the computational point of view; in order to ease the comprehension and also to provide help for future investigations, another aim of this paper is to present, in a organized way, a number of useful formulas ranging from transformation laws for certain tensors to commutation rules for covariant derivatives that, to the best of our knowledge, are either difficult to find or not even present in the literature. In performing our calculations we exploit the moving frame formalism, that turns out to be particularly appropriate for very long and involved computations like those appearing in our work.
The paper is organized as follows. In Section 2 we recall the relevant definitions and notation; in Section 3 we compute the transformations laws of the previously introduced geometric objects under a conformal change of the underlying metric, while in Section 4 we provide (and prove, in some particular cases) a number of useful commutation rules of covariant derivatives of functions, vector fields and geometric tensors. Sections 5 and 6 are brief reviews of results related to Ricci solitons and conformally Einstein manifolds, respectively. In Section 7 we study conformal gradient Ricci solitons, introducing the tensor and the related integrability conditions. The subsequent Sections 8 and 9 are devoted to the analysis of generic Ricci solitons and their conformal counterparts, involving the tensors and . In Section 10 we come back to the case of gradient Ricci solitons and we deduce the third and fourth integrability conditions. We end the paper with a final section in which we describe some interesting open problems, which  we hope  will inspire further investigations in these challenging but stimulating lines of research.
2. Definitions and notation
We begin by introducing some classical notions and objects we will be dealing with in the sequel (see also [26] and [14]).
To perform computations, we use the moving frame notation with respect to a local orthonormal coframe. Thus we fix the index range and recall that the Einstein summation convention will be in force throughout.
We denote by the Riemann curvature tensor (of type ) associated to the metric , and by and the corresponding Ricci tensor and scalar curvature, respectively. The versions of the Riemann curvature tensor and of the Weyl tensor are related in the following way:
(2.1) 
and they satisfy the symmetry relations:
(2.2) 
(2.3) 
A simple checking shows that the Weyl tensor is also totally tracefree and that it vanishes if .
According to the above the (components of the) Ricci tensor and the scalar curvature are given by
(2.4) 
and
(2.5) 
The Schouten tensor is defined as
(2.6) 
so that its trace is
(2.7) 
In terms of the Schouten tensor the decomposition of the Riemann curvature tensor reads as
(2.8) 
where is the KulkarniNomizu product; in components,
(2.9) 
We note that W (more precisely, its version) is a conformal invariant (see e.g. [26]), hence the above decomposition shows that the Schouten tensor is crucial in the study of conformal transformations.
The Cotton tensor can be introduced as the obstruction for the Schouten tensor to be Codazzi, that is,
(2.10) 
We recall that, for , the Cotton tensor can also be defined as one of the possible divergences of the Weyl tensor:
(2.11) 
A computation shows that the two definitions coincide (see again [26]). The Cotton tensor enjoys skewsymmetry in the second and third indices (i.e. ) and furthermore is totally tracefree (i.e. ).
In what follows a relevant role will be played by the Bach tensor, first introduced in general relativity by Bach, [1]. Its componentwise definition is
(2.12) 
A computation using the commutation rules for the second covariant derivative of the Weyl tensor or of the Schouten tensor (see the next section for both) shows that the Bach tensor is symmetric (i.e. ); it is also evidently tracefree (i.e. ). As a consequence we observe that we can write
It is worth reporting here the following interesting formula for the divergence of the Bach tensor (see e. g. [8] for its proof)
(2.13) 
We also recall the definition of the Einstein tensor, which in components is given by
(2.14) 
One of the main objects of our investigation are Ricci solitons, which are defined through equation (1.1); we explicitly note that in components this latter becomes
(2.15) 
and, in the gradient case,
(2.16) 
The tensor , introduced by Cao and Chen in [6], turns out to be a fundamental tool in the study of the geometry of gradient Ricci solitons and, more in general, of gradient Einsteintype manifolds, as observed in [14]; in components it is defined as
(2.17) 
The tensor is skewsymmetric in the second and third indices (i.e. ) and totally tracefree (i.e. ). Note that our convention for the tensor differs from that in [8]. A simple computation, using the definitions of the tensors involved, equation (2.16) and the fact that, for gradient Ricci solitons, the fundamental identity
holds (see Section 4), shows that the tensor can be written in four equivalent ways:
(2.18)  
3. Transformation laws under a conformal change of the metric
Let be a Riemannian manifold of dimension . The moving frame formalism is extremely useful in the calculation of the transformation laws of geometric tensors under a conformal change of the metric and in the derivation of commutation rules, as we shall see in the next section. For the sake of completeness (see [26] for details) we recall that, having fixed a (local) orthonormal coframe , with dual frame , , the corresponding LeviCivita connection forms , are the unique forms satisfying
(3.1)  
(3.2) 
The curvature forms , , associated to the coframe are the forms defined through the second structure equations
(3.3) 
They are skewsymmetric (i.e. ) and they can be written as
(3.4) 
where are precisely the coefficients of the (version of the) Riemann curvature tensor.
The covariant derivative of a vector field is defined as
while the covariant derivative of a form is defined as
The divergence of the vector field is the trace of , that is,
(3.5) 
For a function we can write
(3.6) 
for some smooth coefficients . The Hessian of , , is the tensor defined as
(3.7) 
with
(3.8) 
and
(see also next section). The Laplacian of is the trace of the Hessian, that is
Now we are ready to recall (and prove, in some cases) the transformation laws that will be useful in our computations.
We consider the conformal change of the metric (written in “exponential form”)
(3.9) 
is called the stretching factor of the conformal change. We use the superscript to denote quantities related to the metric .
It is obvious that in the new metric the forms
(3.10) 
give a local orthonormal coframe. It is easy to deduce that, if , the forms
(3.11) 
are skewsymmetric and satisfy the first structure equation. Hence, they are the connection forms relative to the coframe defined in (3.10). A straightforward computation using the structure equations and (3.8) shows that the curvature forms relative to the coframe (3.10) are
(3.12) 
Equation (3.12) is the starting point for the next transformation laws that we list without further comments.

Riemann curvature tensor:
(3.13) Proof.
Tracing (3.13) we get

Scalar curvature:
(3.16) Next we derive the transformation laws for the

Covariant derivative of the Ricci tensor:
(3.17) 
Differential of the scalar curvature:
(3.20) Proof.
It follows from the fact that and from (3.16). ∎

Hessian of the scalar curvature:
(3.21) Proof.

Laplacian of the scalar curvature:
(3.22) 
the Hessian of a function :
(3.23) which in components reads as
(3.24) Proof.
Tracing (3.24) we get

the Laplacian of a function :
(3.26) 
the third derivative of a function :
(3.27) in particular,
(3.28)