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THE COTTON TENSOR and the RICCI FLOW 1. Preliminaries 1 2

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THE COTTON TENSOR and the RICCI FLOW 1. Preliminaries 1 2 THE COTTON TENSOR AND THE RICCI FLOW CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI ABSTRACT. We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications. CONTENTS 1. Preliminaries 1 2. The Evolution Equation of the Cotton Tensor in 3D2 3. Three–Dimensional Gradient Ricci Solitons 10 4. The Evolution Equation of the Cotton Tensor in any Dimension 13 5. The Bach Tensor 23 5.1. The Evolution Equation of the Bach Tensor in 3D 24 5.2. The Bach Tensor of Three–Dimensional Gradient Ricci Solitons 26 References 28 1. PRELIMINARIES The Riemann curvature operator of a Riemannian manifold (M n; g) is defined, as in [6], by Riem(X; Y )Z = rY rX Z − rX rY Z + r[X;Y ]Z: In a local coordinate system the components of the (3; 1)–Riemann curvature tensor are given by l @ @ @ @ m Rijk @xl = Riem @xi ; @xj @xk and we denote by Rijkl = glmRijk its (4; 0)–version. n i j k l With the previous choice, for the sphere S we have Riem(v; w; v; w) = Rabcdv w v w > 0. In all the paper the Einstein convention of summing over the repeated indices will be adopted. jl ik The Ricci tensor is obtained by the contraction Rik = g Rijkl and R = g Rik will denote the scalar curvature. We recall the interchange of derivative formula, 2 2 pq rij!k − rji!k = Rijkpg !q ; and Schur lemma, which follows by the second Bianchi identity, pq 2g rpRqi = riR : They both will be used extensively in the computations that follows. The so called Weyl tensor is then defined by the following decomposition formula (see [6, Chapter 3, Section K]) in dimension n ≥ 3, 1 R (1.1) R = (R g − R g + R g − R g ) − (g g − g g ) + W : ijkl n − 2 ik jl il jk jl ik jk il (n − 1)(n − 2) ik jl il jk ijkl The Weyl tensor satisfies all the symmetries of the curvature tensor, moreover, all its traces with the metric are zero, as it can be easily seen by the above formula. In dimension three W is identically zero for every Riemannian manifold. It becomes relevant instead when n ≥ 4 since its vanishing is a condition equivalent for (M n; g) to be locally conformally flat, that is, n f around every point p 2 M there is a conformal deformation geij = e gij of the original metric g, such that the new metric is flat, namely, the Riemann tensor associated to ge is zero in Up (here f : Up ! R is a smooth function defined in a open neighborhood Up of p). Date: March 4, 2014. 1 2 CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI In dimension n = 3, instead, locally conformally flatness is equivalent to the vanishing of the follow- ing Cotton tensor 1 (1.2) C = r R − r R − r Rg − r Rg ; ijk k ij j ik 2(n − 1) k ij j ik which expresses the fact that the Schouten tensor Rg S = R − ij ij ij 2(n − 1) is a Codazzi tensor (see [1, Chapter 16, Section C]), that is, a symmetric bilinear form Tij such that rkTij = riTkj. By means of the second Bianchi identity, one can easily get (see [1]) that n − 3 (1.3) rlW = − C : lijk n − 2 ijk Hence, when n ≥ 4, if we assume that the manifold is locally conformally flat (that is, W = 0), the Cotton tensor is identically zero also in this case, but this is only a necessary condition. By direct computation, we can see that the tensor Cijk satisfies the following symmetries (1.4) Cijk = −Cikj; Cijk + Cjki + Ckij = 0 ; moreover it is trace–free in any two indices, ij ik jk (1.5) g Cijk = g Cijk = g Cijk = 0 ; by its skew–symmetry and Schur lemma. We suppose now that (M n; g(t)) is a Ricci flow in some time interval, that is, the time–dependent metric g(t) satisfies @ g = −2R : @t ij ij We have then the following evolution equations for the Christoffel symbols, the Ricci tensor and the scalar curvature (see for instance [7]), @ Γk = −gksr R − gksr R + gksr R @t ij i js j is s ij @ (1.6) R = ∆R − 2RklR − 2gpqR R @t ij ij kijl ip jq @ R = ∆R + 2jRicj2 : @t All the computations which follow will be done in a fixed local frame, not in a moving frame. Acknowledgments. The first and second authors are partially supported by the Italian FIRB Ideas “Analysis and Beyond”. Note. We remark that Huai-Dong Cao also, independently by us, worked out the computation of the evolution of the Cotton tensor in dimension three, in an unpublished note. 2. THE EVOLUTION EQUATION OF THE COTTON TENSOR IN 3D The goal of this section is to compute the evolution equation under the Ricci flow of the Cotton tensor Cijk in dimension three (see [5] for the evolution of the Weyl tensor), the general computation in any dimension is postponed to section4. In the special three–dimensional case we have, R (2.1) R = R g − R g + R g − R g − (g g − g g ) ; ijkl ik jl il jk jl ik jk il 2 ik jl il jk 1 (2.2) C = r R − r R − r Rg − r Rg ; ijk k ij j ik 4 k ij j ik THE COTTON TENSOR AND THE RICCI FLOW 3 hence, the evolution equations (1.6) become @ Γk = − gksr R − gksr R + gksr R @t ij i js j is s ij @ R = ∆R − 6gpqR R + 3RR + 2jRicj2g − R2g @t ij ij ip jq ij ij ij @ R = ∆R + 2jRicj2 : @t From these formulas we can compute the evolution equations of the derivatives of the curvatures assuming, from now on, to be in normal coordinates, @ r R = r ∆R + 2r jRicj2 ; @t l l l @ r R = r ∆R − 6r R R − 6R r R + 3r RR + 3Rr R @t s ij s ij s ip jp ip s jp s ij s ij 2 2 +2rsjRicj gij − rsR gij +(riRsp + rsRip − rpRis)Rjp +(rjRsp + rsRjp − rpRjs)Rip = rs∆Rij − 5rsRipRjp − 5RiprsRjp + 3rsRRij + 3RrsRij 2 2 +2rsjRicj gij − rsR gij +(riRsp − rpRis)Rjp + (rjRsp − rpRjs)Rip = rs∆Rij − 5rsRipRjp − 5RiprsRjp + 3rsRRij + 3RrsRij 2 2 +2rsjRicj gij − rsR gij + CspiRjp + CspjRip +Rjp[riRgsp − rpRgis]=4 + Rip[rjRgsp − rpRgjs]=4 ; where in the last passage we substituted the expression of the Cotton tensor. 4 CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI We then compute, @ @ @ @ C = r R − r R − r Rg − r Rg =4 @t ijk @t k ij @t j ik @t k ij j ik = rk∆Rij − 5rkRipRjp − 5RiprkRjp + 3rkRRij + 3RrkRij 2 2 +2rkjRicj gij − rkR gij + CkpiRjp + CkpjRip +Rjp[riRgkp − rpRgik]=4 + Rip[rjRgkp − rpRgjk]=4 −∇j∆Rik + 5rjRipRkp + 5RiprjRkp − 3rjRRik − 3RrjRik 2 2 −2rjjRicj gik + rjR gik − CjpiRkp − CjpkRip −Rkp[riRgjp − rpRgij]=4 − Rip[rkRgjp − rpRgkj]=4 +(RijrkR − RikrjR =2 2 2 − rk∆R + 2rkjRicj gij=4 + rj∆R + 2rjjRicj gik=4 = rk∆Rij − 5rkRipRjp − 5RiprkRjp + 3rkRRij + 3RrkRij 2 2 +3rkjRicj gij=2 − rkR gij + CkpiRjp + CkpjRip +RjkriR=4 − RjprpRgik=4 + RikrjR=4 − RiprpRgjk=4 −∇j∆Rik + 5rjRipRkp + 5RiprjRkp − 3rjRRik − 3RrjRik 2 2 −3rjjRicj gik=2 + rjR gik − CjpiRkp − CjpkRip −RkjriR=4 + RkprpRgij=4 − RijrkR=4 + RiprpRgkj=4 +(RijrkR − RikrjR =2 −∇k∆Rgij=4 + rj∆Rgik=4 = rk∆Rij − 5rkRipRjp − 5RiprkRjp + 13rkRRij=4 + 3RrkRij 2 2 +3rkjRicj gij=2 − rkR gij + CkpiRjp + CkpjRip −RjprpRgik=4 −∇j∆Rik + 5rjRipRkp + 5RiprjRkp − 13rjRRik=4 − 3RrjRik 2 2 −3rjjRicj gik=2 + rjR gik − CjpiRkp − CjpkRip +RkprpRgij=4 −∇k∆Rgij=4 + rj∆Rgik=4 and ∆Cijk = ∆rkRij − ∆rjRik − ∆rkRgij=4 + ∆rjRgik=4 ; hence, @ C − ∆C = r ∆R − r ∆R − ∆r R + ∆r R @t ijk ijk k ij j ik k ij j ik −∇k∆Rgij=4 + rj∆Rgik=4 + ∆rkRgij=4 − ∆rjRgik=4 −5rkRipRjp − 5RiprkRjp + 13rkRRij=4 + 3RrkRij 2 2 +3rkjRicj gij=2 − rkR gij + CkpiRjp + CkpjRip −RjprpRgik=4 +5rjRipRkp + 5RiprjRkp − 13rjRRik=4 − 3RrjRik 2 2 −3rjjRicj gik=2 + rjR gik − CjpiRkp − CjpkRip +RkprpRgij=4 THE COTTON TENSOR AND THE RICCI FLOW 5 Now to proceed, we need the following commutation rules for the derivatives of the Ricci tensor and of the scalar curvature, where we will employ the special form of the Riemann tensor in dimen- sion three given by formula (2.1), 3 3 3 3 rk∆Rij − ∆rkRij = rkllRij − rlklRij + rlklRij − rllkRij = −RkprpRij + RkliprlRjp + RkljprlRip 3 3 +rlklRij − rllkRij = −RkprpRij + RikrjR=2 + RjkriR=2 −RkpriRjp − RkprjRip + RlprlRjpgik + RlprlRipgjk −RlirlRjk − RljrlRik − RrjRgik=4 − RriRgjk=4 +RriRjk=2 + RrjRik=2 +rl RklipRpj + RkljpRpi = −RkprpRij + RikrjR=2 + RjkriR=2 −RkpriRjp − RkprjRip + RlprlRjpgik + RlprlRipgjk −RlirlRjk − RljrlRik − RrjRgik=4 − RriRgjk=4 +RriRjk=2 + RrjRik=2 +rl RikRlj − RilRkj + RplRpjgik − RpkRpjgil − gikRRlj=2 + gilRRjk=2 +RjkRli − RjlRki + RplRpigjk − RpkRpigjl − gjkRRli=2 + gjlRRik=2 = −RkprpRij + RikrjR=2 + RjkriR=2 −RkpriRjp − RkprjRip + RlprlRjpgik + RlprlRipgjk −RlirlRjk − RljrlRik − RrjRgik=4 − RriRgjk=4 +RriRjk=2 + RrjRik=2 −∇iRpkRpj + riRRjk=2 + gikRplrlRpj −RpkriRpj − gikRrjR=4 + RriRjk=2 −∇jRpkRpi + rjRRik=2 + gjkRplrlRpi −RpkrjRpi − gjkRriR=4 + RrjRik=2 = −RkprpRij + RikrjR + RjkriR −2RkpriRjp − 2RkprjRip + 2RlprlRjpgik + 2RlprlRipgjk −RlirlRjk − RljrlRik − RpjriRpk − RpirjRpk −RrjRgik=2 − RriRgjk=2 + RriRjk + RrjRik and rk∆R − ∆rkR = RkllprpR = −RkprpR : 6 CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI Then, getting back to the main computation, we obtain @ C − ∆C = −R r R + R r R + R r R @t ijk ijk kp p ij ik j jk i −2RkpriRjp − 2RkprjRip + 2RlprlRjpgik + 2RlprlRipgjk −RlirlRjk − RljrlRik − RpjriRpk − RpirjRpk −RrjRgik=2 − RriRgjk=2 + RriRjk + RrjRik +RjprpRik − RijrkR − RkjriR +2RjpriRkp + 2RjprkRip − 2RlprlRkpgij − 2RlprlRipgkj +RlirlRkj + RlkrlRij + RpkriRpj + RpirkRpj +RrkRgij=2 + RriRgkj=2 − RriRkj − RrkRij +RkprpRgij=4 − RjprpRgik=4 −5rkRipRjp − 5RiprkRjp + 13rkRRij=4 + 3RrkRij 2 2 +3rkjRicj gij=2 − rkR gij + CkpiRjp + CkpjRip −RjprpRgik=4 +5rjRipRkp + 5RiprjRkp − 13rjRRik=4 − 3RrjRik 2 2 −3rjjRicj gik=2 + rjR gik − CjpiRkp − CjpkRip +RkprpRgij=4 = CkpiRjp + CkpjRip − CjpiRkp − CjpkRip 2 +[2RlprlRjp + 3RrjR=2 − RjprpR=2 − 3rjjRicj =2]gik 2 +[−2RlprlRkp − 3RrkR=2 + RkprpR=2 + 3rkjRicj =2]gij −RkpriRjp + RjpriRkp −3rkRipRjp − 4RiprkRjp + 9rkRRij=4 + 2RrkRij +3rjRipRkp + 4RiprjRkp − 9rjRRik=4 − 2RrjRik Now, by means of the
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