THE COTTON TENSOR AND THE RICCI FLOW
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
A BSTRACT. 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.
C ONTENTS
1. Preliminaries
2. The Evolution Equation of the Cotton Tensor in 3D
3. Three–Dimensional Gradient Ricci Solitons
4. The Evolution Equation of the Cotton Tensor in any Dimension
5. The Bach Tensor
5.1. The Evolution Equation of the Bach Tensor in 3D
5.2. The Bach Tensor of Three–Dimensional Gradient Ricci Solitons
References
1
2
10
13
23
24
26
28
1. P RELIMINARIES
The Riemann curvature operator of a Riemannian manifold (M n , g) is defined, as in [6], by
Riem(X, Y )Z = ∇Y ∇X Z − ∇X ∇Y Z + ∇[X,Y ] Z .
In a local coordinate system
curvature tensor are given by
∂ the components of the (3, 1)–Riemann
∂
∂
∂
m
l
Rijk ∂xl = Riem ∂xi , ∂xj ∂xk and we denote by Rijkl = glm Rijk its (4, 0)–version.
With the previous choice, for the sphere Sn we have Riem(v, w, v, w) = Rabcd v i wj v k wl > 0.
In all the paper the Einstein convention of summing over the repeated indices will be adopted.
The Ricci tensor is obtained by the contraction Rik = g jl Rijkl and R = g ik Rik will denote the scalar
curvature.
We recall the interchange of derivative formula,
∇2ij ωk − ∇2ji ωk = Rijkp g pq ωq ,
and Schur lemma, which follows by the second Bianchi identity,
2g pq ∇p Rqi = ∇i R .
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
(Rik gjl − Ril gjk + Rjl gik − Rjk gil ) −
(gik gjl − gil gjk ) + Wijkl .
(1.1)
Rijkl =
n−2
(n − 1)(n − 2)
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,
around every point p ∈ M n there is a conformal deformation geij = ef 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 following Cotton tensor
1
(1.2)
Cijk = ∇k Rij − ∇j Rik −
∇k Rgij − ∇j Rgik ,
2(n − 1)
which expresses the fact that the Schouten tensor
Sij = Rij −
Rgij
2(n − 1)
is a Codazzi tensor (see [1, Chapter 16, Section C]), that is, a symmetric bilinear form Tij such that
∇k Tij = ∇i Tkj .
By means of the second Bianchi identity, one can easily get (see [1]) that
n−3
Cijk .
n−2
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
∇l Wlijk = −
(1.3)
Cijk = −Cikj ,
(1.4)
Cijk + Cjki + Ckij = 0 ,
moreover it is trace–free in any two indices,
g ij Cijk = g ik Cijk = g jk Cijk = 0 ,
(1.5)
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
∂
gij = −2Rij .
∂t
We have then the following evolution equations for the Christoffel symbols, the Ricci tensor and the
scalar curvature (see for instance [7]),
∂ k
Γ
= −g ks ∇i Rjs − g ks ∇j Ris + g ks ∇s Rij
∂t ij
∂
Rij = ∆Rij − 2Rkl Rkijl − 2g pq Rip Rjq
∂t
∂
R = ∆R + 2|Ric|2 .
∂t
(1.6)
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. T HE E VOLUTION E QUATION OF THE C OTTON T ENSOR 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 section 4.
In the special three–dimensional case we have,
R
(gik gjl − gil gjk ) ,
2
1
= ∇k Rij − ∇j Rik − ∇k Rgij − ∇j Rgik ,
4
(2.1)
Rijkl = Rik gjl − Ril gjk + Rjl gik − Rjk gil −
(2.2)
Cijk
THE COTTON TENSOR AND THE RICCI FLOW
3
hence, the evolution equations (1.6) become
∂ k
Γ = − g ks ∇i Rjs − g ks ∇j Ris + g ks ∇s Rij
∂t ij
∂
Rij = ∆Rij − 6g pq Rip Rjq + 3RRij + 2|Ric|2 gij − R2 gij
∂t
∂
R = ∆R + 2|Ric|2 .
∂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,
∂
∇l R = ∇l ∆R + 2∇l |Ric|2 ,
∂t
∂
∇s Rij
∂t
= ∇s ∆Rij − 6∇s Rip Rjp − 6Rip ∇s Rjp + 3∇s RRij + 3R∇s Rij
+2∇s |Ric|2 gij − ∇s R2 gij
+(∇i Rsp + ∇s Rip − ∇p Ris )Rjp
+(∇j Rsp + ∇s Rjp − ∇p Rjs )Rip
= ∇s ∆Rij − 5∇s Rip Rjp − 5Rip ∇s Rjp + 3∇s RRij + 3R∇s Rij
+2∇s |Ric|2 gij − ∇s R2 gij
+(∇i Rsp − ∇p Ris )Rjp + (∇j Rsp − ∇p Rjs )Rip
= ∇s ∆Rij − 5∇s Rip Rjp − 5Rip ∇s Rjp + 3∇s RRij + 3R∇s Rij
+2∇s |Ric|2 gij − ∇s R2 gij + Cspi Rjp + Cspj Rip
+Rjp [∇i Rgsp − ∇p Rgis ]/4 + Rip [∇j Rgsp − ∇p Rgjs ]/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,
∂
∂
∂
∂
Cijk =
∇k Rij − ∇j Rik −
∇k Rgij − ∇j Rgik /4
∂t
∂t
∂t
∂t
= ∇k ∆Rij − 5∇k Rip Rjp − 5Rip ∇k Rjp + 3∇k RRij + 3R∇k Rij
+2∇k |Ric|2 gij − ∇k R2 gij + Ckpi Rjp + Ckpj Rip
+Rjp [∇i Rgkp − ∇p Rgik ]/4 + Rip [∇j Rgkp − ∇p Rgjk ]/4
−∇j ∆Rik + 5∇j Rip Rkp + 5Rip ∇j Rkp − 3∇j RRik − 3R∇j Rik
−2∇j |Ric|2 gik + ∇j R2 gik − Cjpi Rkp − Cjpk Rip
−Rkp [∇i Rgjp − ∇p Rgij ]/4 − Rip [∇k Rgjp − ∇p Rgkj ]/4
+(Rij ∇k R − Rik ∇j R /2
− ∇k ∆R + 2∇k |Ric|2 gij /4 + ∇j ∆R + 2∇j |Ric|2 gik /4
= ∇k ∆Rij − 5∇k Rip Rjp − 5Rip ∇k Rjp + 3∇k RRij + 3R∇k Rij
+3∇k |Ric|2 gij /2 − ∇k R2 gij + Ckpi Rjp + Ckpj Rip
+Rjk ∇i R/4 − Rjp ∇p Rgik /4 + Rik ∇j R/4 − Rip ∇p Rgjk /4
−∇j ∆Rik + 5∇j Rip Rkp + 5Rip ∇j Rkp − 3∇j RRik − 3R∇j Rik
−3∇j |Ric|2 gik /2 + ∇j R2 gik − Cjpi Rkp − Cjpk Rip
−Rkj ∇i R/4 + Rkp ∇p Rgij /4 − Rij ∇k R/4 + Rip ∇p Rgkj /4
+(Rij ∇k R − Rik ∇j R /2
−∇k ∆Rgij /4 + ∇j ∆Rgik /4
= ∇k ∆Rij − 5∇k Rip Rjp − 5Rip ∇k Rjp + 13∇k RRij /4 + 3R∇k Rij
+3∇k |Ric|2 gij /2 − ∇k R2 gij + Ckpi Rjp + Ckpj Rip
−Rjp ∇p Rgik /4
−∇j ∆Rik + 5∇j Rip Rkp + 5Rip ∇j Rkp − 13∇j RRik /4 − 3R∇j Rik
−3∇j |Ric|2 gik /2 + ∇j R2 gik − Cjpi Rkp − Cjpk Rip
+Rkp ∇p Rgij /4
−∇k ∆Rgij /4 + ∇j ∆Rgik /4
and
∆Cijk = ∆∇k Rij − ∆∇j Rik − ∆∇k Rgij /4 + ∆∇j Rgik /4 ,
hence,
∂
Cijk − ∆Cijk = ∇k ∆Rij − ∇j ∆Rik − ∆∇k Rij + ∆∇j Rik
∂t
−∇k ∆Rgij /4 + ∇j ∆Rgik /4 + ∆∇k Rgij /4 − ∆∇j Rgik /4
−5∇k Rip Rjp − 5Rip ∇k Rjp + 13∇k RRij /4 + 3R∇k Rij
+3∇k |Ric|2 gij /2 − ∇k R2 gij + Ckpi Rjp + Ckpj Rip
−Rjp ∇p Rgik /4
+5∇j Rip Rkp + 5Rip ∇j Rkp − 13∇j RRik /4 − 3R∇j Rik
−3∇j |Ric|2 gik /2 + ∇j R2 gik − Cjpi Rkp − Cjpk Rip
+Rkp ∇p Rgij /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 dimension three given by formula (2.1),
∇k ∆Rij − ∆∇k Rij
= ∇3kll Rij − ∇3lkl Rij + ∇3lkl Rij − ∇3llk Rij
= −Rkp ∇p Rij + Rklip ∇l Rjp + Rkljp ∇l Rip
+∇3lkl Rij − ∇3llk Rij
= −Rkp ∇p Rij + Rik ∇j R/2 + Rjk ∇i R/2
−Rkp ∇i Rjp − Rkp ∇j Rip + Rlp ∇l Rjp gik + Rlp ∇l Rip gjk
−Rli ∇l Rjk − Rlj ∇l Rik − R∇j Rgik /4 − R∇i Rgjk /4
+R∇i Rjk /2 + R∇j Rik /2
+∇l Rklip Rpj + Rkljp Rpi
= −Rkp ∇p Rij + Rik ∇j R/2 + Rjk ∇i R/2
−Rkp ∇i Rjp − Rkp ∇j Rip + Rlp ∇l Rjp gik + Rlp ∇l Rip gjk
−Rli ∇l Rjk − Rlj ∇l Rik − R∇j Rgik /4 − R∇i Rgjk /4
+R∇i Rjk /2 + R∇j Rik /2
+∇l Rik Rlj − Ril Rkj + Rpl Rpj gik − Rpk Rpj gil − gik RRlj /2 + gil RRjk /2
+Rjk Rli − Rjl Rki + Rpl Rpi gjk − Rpk Rpi gjl − gjk RRli /2 + gjl RRik /2
= −Rkp ∇p Rij + Rik ∇j R/2 + Rjk ∇i R/2
−Rkp ∇i Rjp − Rkp ∇j Rip + Rlp ∇l Rjp gik + Rlp ∇l Rip gjk
−Rli ∇l Rjk − Rlj ∇l Rik − R∇j Rgik /4 − R∇i Rgjk /4
+R∇i Rjk /2 + R∇j Rik /2
−∇i Rpk Rpj + ∇i RRjk /2 + gik Rpl ∇l Rpj
−Rpk ∇i Rpj − gik R∇j R/4 + R∇i Rjk /2
−∇j Rpk Rpi + ∇j RRik /2 + gjk Rpl ∇l Rpi
−Rpk ∇j Rpi − gjk R∇i R/4 + R∇j Rik /2
= −Rkp ∇p Rij + Rik ∇j R + Rjk ∇i R
−2Rkp ∇i Rjp − 2Rkp ∇j Rip + 2Rlp ∇l Rjp gik + 2Rlp ∇l Rip gjk
−Rli ∇l Rjk − Rlj ∇l Rik − Rpj ∇i Rpk − Rpi ∇j Rpk
−R∇j Rgik /2 − R∇i Rgjk /2 + R∇i Rjk + R∇j Rik
and
∇k ∆R − ∆∇k R = Rkllp ∇p R = −Rkp ∇p R .
6
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
Then, getting back to the main computation, we obtain
∂
Cijk − ∆Cijk = −Rkp ∇p Rij + Rik ∇j R + Rjk ∇i R
∂t
−2Rkp ∇i Rjp − 2Rkp ∇j Rip + 2Rlp ∇l Rjp gik + 2Rlp ∇l Rip gjk
−Rli ∇l Rjk − Rlj ∇l Rik − Rpj ∇i Rpk − Rpi ∇j Rpk
−R∇j Rgik /2 − R∇i Rgjk /2 + R∇i Rjk + R∇j Rik
+Rjp ∇p Rik − Rij ∇k R − Rkj ∇i R
+2Rjp ∇i Rkp + 2Rjp ∇k Rip − 2Rlp ∇l Rkp gij − 2Rlp ∇l Rip gkj
+Rli ∇l Rkj + Rlk ∇l Rij + Rpk ∇i Rpj + Rpi ∇k Rpj
+R∇k Rgij /2 + R∇i Rgkj /2 − R∇i Rkj − R∇k Rij
+Rkp ∇p Rgij /4 − Rjp ∇p Rgik /4
−5∇k Rip Rjp − 5Rip ∇k Rjp + 13∇k RRij /4 + 3R∇k Rij
+3∇k |Ric|2 gij /2 − ∇k R2 gij + Ckpi Rjp + Ckpj Rip
−Rjp ∇p Rgik /4
+5∇j Rip Rkp + 5Rip ∇j Rkp − 13∇j RRik /4 − 3R∇j Rik
−3∇j |Ric|2 gik /2 + ∇j R2 gik − Cjpi Rkp − Cjpk Rip
+Rkp ∇p Rgij /4
= Ckpi Rjp + Ckpj Rip − Cjpi Rkp − Cjpk Rip
+[2Rlp ∇l Rjp + 3R∇j R/2 − Rjp ∇p R/2 − 3∇j |Ric|2 /2]gik
+[−2Rlp ∇l Rkp − 3R∇k R/2 + Rkp ∇p R/2 + 3∇k |Ric|2 /2]gij
−Rkp ∇i Rjp + Rjp ∇i Rkp
−3∇k Rip Rjp − 4Rip ∇k Rjp + 9∇k RRij /4 + 2R∇k Rij
+3∇j Rip Rkp + 4Rip ∇j Rkp − 9∇j RRik /4 − 2R∇j Rik
Now, by means of the very definition of the Cotton tensor in dimension three (2.2) and the identities (1.4), we substitute
Ckpj − Cjpk = − Ckjp − Cjpk = Cpkj
1
∇l Rjp = ∇j Rlp + Cpjl + ∇l Rgpj − ∇j Rgpl
4
1
∇l Rkp = ∇k Rlp + Cpkl + ∇l Rgpk − ∇k Rgpl
4
1
∇i Rjp = ∇j Rip + Cpji + ∇i Rgjp − ∇j Rgip
4
1
∇i Rkp = ∇k Rip + Cpki + ∇i Rgkp − ∇k Rgip
4
THE COTTON TENSOR AND THE RICCI FLOW
7
in the last expression above, getting
∂
Cijk − ∆Cijk = Rjp Ckpi − Rkp Cjpi + Rip Cpkj
∂t
h
+ 2Rlp ∇j Rlp + Cpjl + ∇l Rgpj /4 − ∇j Rgpl /4
i
+ 3R∇j R/2 − Rjp ∇p R/2 − 3∇j |Ric|2 /2 gik
h
+ − 2Rlp ∇k Rlp + Cpkl + ∇l Rgpk /4 − ∇k Rgpl /4
i
− 3R∇k R/2 + Rkp ∇p R/2 + 3∇k |Ric|2 /2 gij
−Rkp ∇j Rip + Cpji + ∇i Rgjp /4 − ∇j Rgip /4
+Rjp ∇k Rip + Cpki + ∇i Rgkp /4 − ∇k Rgip /4
−3∇k Rip Rjp − 4Rip ∇k Rjp + 9∇k RRij /4 + 2R∇k Rij
+3∇j Rip Rkp + 4Rip ∇j Rkp − 9∇j RRik /4 − 2R∇j Rik
= Rjp Ckpi + Cpki − Rkp Cjpi + Cpji + Rip Cpkj
+2Rlp Cpjl gik − 2Rlp Cpkl gij
+ R∇j R − ∇j |Ric|2 /2 gik − R∇k R − ∇k |Ric|2 /2 gij
−2∇k Rip Rjp − 4Rip ∇k Rjp + 2∇k RRij + 2R∇k Rij
+2∇j Rip Rkp + 4Rip ∇j Rkp − 2∇j RRik − 2R∇j Rik .
then, we substitute again
1
4
1
∇j Rkp = ∇p Rjk + Ckpj +
4
1
∇k Rij = ∇i Rkj + Cjik +
4
1
∇j Rik = ∇i Rjk + Ckij +
4
∇k Rjp = ∇p Rkj + Cjpk +
∇k Rgjp − ∇p Rgjk
∇j Rgkp − ∇p Rgkj
∇k Rgij − ∇i Rgjk
∇j Rgik − ∇i Rgkj ,
8
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
finally obtaining
∂
Cijk − ∆Cijk = Rjp Ckpi + Cpki − Rkp Cjpi + Cpji + Rip Cpkj
∂t
+2Rlp Cpjl gik − 2Rlp Cpkl gij
+ R∇j R − ∇j |Ric|2 /2 gik − R∇k R − ∇k |Ric|2 /2 gij
−2∇k Rip Rjp − 4Rip ∇p Rkj + Cjpk + ∇k Rgjp /4 − ∇p Rgjk /4
+2∇k RRij + 2R ∇i Rkj + Cjik + ∇k Rgij /4 − ∇i Rgjk /4
+2∇j Rip Rkp + 4Rip ∇p Rjk + Ckpj + ∇j Rgkp /4 − ∇p Rgkj /4
−2∇j RRik − 2R ∇i Rjk + Ckij + ∇j Rgik /4 − ∇i Rgkj /4
= Rjp Ckpi + Cpki − Rkp Cjpi + Cpji + Rip Cpkj
+4Rip Ckpj − Cjpk + 2R Cjik − Ckij
+2Rlp Cpjl gik − 2Rlp Cpkl gij
+ R∇j R/2 − ∇j |Ric|2 /2 gik − R∇k R/2 − ∇k |Ric|2 /2 gij
−2∇k Rip Rjp + 2∇j Rip Rkp
+∇k RRij − ∇j RRik
= Rjp Ckpi + Cpki − Rkp Cjpi + Cpji + 5Rip Cpkj
+2RCijk + 2Rlp Cpjl gik − 2Rlp Cpkl gij
+ R∇j R/2 − ∇j |Ric|2 /2 gik − R∇k R/2 − ∇k |Ric|2 /2 gij
+2∇j Rip Rkp − 2∇k Rip Rjp
+∇k RRij − ∇j RRik ,
where in the last passage we used again the identities (1.4).
Hence, we can resume this long computation in the following proposition, getting back to a generic
coordinate basis.
Proposition 2.1. During the Ricci flow of a 3–dimensional Riemannian manifold (M 3 , g(t)), the Cotton tensor
satisfies the following evolution equation
(2.3)
∂t − ∆ Cijk = g pq Rpj (Ckqi + Cqki ) + 5g pq Rip Cqkj + g pq Rpk (Cjiq + Cqij )
+2RCijk + 2Rql Cqjl gik − 2Rql Cqkl gij
1
R
R
1
+ ∇k |Ric|2 gij − ∇j |Ric|2 gik + ∇j Rgik − ∇k Rgij
2
2
2
2
+2g pq Rpk ∇j Rqi − 2g pq Rpj ∇k Rqi + Rij ∇k R − Rik ∇j R .
In particular if the Cotton tensor vanishes identically along the flow we obtain,
(2.4)
0 = ∇k |Ric|2 gij − ∇j |Ric|2 gik + R∇j Rgik − R∇k Rgij
+4g pq Rpk ∇j Rqi − 4g pq Rpj ∇k Rqi + 2Rij ∇k R − 2Rik ∇j R .
Corollary 2.2. If the Cotton tensor vanishes identically along the Ricci flow of a 3–dimensional Riemannian
manifold (M 3 , g(t)), the following tensor
7
|Ric|2 gij − 4Rpj Rpi + 3RRij − R2 gij
8
is a Codazzi tensor (see [1, Chapter 16, Section C]).
THE COTTON TENSOR AND THE RICCI FLOW
9
Proof. We compute in an orthonormal basis,
4Rpk ∇j Rpi − 4 Rpj ∇k Rpi + 2Rij ∇k R − 2Rik ∇j R
= 4∇j (Rpk Rpi ) − 4∇k (Rpj Rpi ) − 4Rpi ∇j Rpk + 4Rpi ∇k Rpj
+ 2Rij ∇k R − 2Rik ∇j R
= 4∇j (Rpk Rpi ) − 4∇k (Rpj Rpi ) + Rpi (4Cpjk + ∇k Rgpj − ∇j Rgpk )
+ 2Rij ∇k R − 2Rik ∇j R
= 4∇j (Rpk Rpi ) − 4∇k (Rpj Rpi ) + 3Rij ∇k R − 3Rik ∇j R
= 4∇j (Rpk Rpi ) − 4∇k (Rpj Rpi ) + 3∇k (RRij ) − 3∇j (RRik )
− 3R(∇k Rij − ∇j Rik )
= 4∇j (Rpk Rpi ) − 4∇k (Rpj Rpi ) + 3∇k (RRij ) − 3∇j (RRik )
− 3R(4Cijk + ∇k Rgij − ∇j Rgik )/4
= 4∇j (Rpk Rpi ) − 4∇k (Rpj Rpi ) + 3∇k (RRij ) − 3∇j (RRik )
3
3
− ∇k R2 gij + ∇j R2 gik .
8
8
Hence, we have, by the previous proposition,
0 = ∇k |Ric|2 gij − ∇j |Ric|2 gik + 4∇j (Rpk Rpi ) − 4∇k (Rpj Rpi )
7
7
+ 3∇k (RRij ) − 3∇j (RRik ) − ∇k R2 gij + ∇j R2 gik ,
8
8
which is the thesis of the corollary.
Remark 2.3. All the traces of the 3–tensor in the LHS of equation (2.4) are zero.
Remark 2.4. From the trace–free property (1.5) of the Cotton tensor and the fact that along the Ricci
flow there holds
∂t − ∆ g ij = 2Rij ,
we conclude that the following relations have to hold
g ij (∂t − ∆)Cijk = −2Rij Cijk ,
g ik (∂t − ∆)Cijk = −2Rik Cijk ,
g jk (∂t − ∆)Cijk = −2Rjk Cijk = 0 .
They are easily verified for formula (2.3).
Corollary 2.5. During the Ricci flow of a 3–dimensional Riemannian manifold (M 3 , g(t)), the squared norm
of the Cotton tensor satisfies the following evolution equation, in an orthonormal basis,
∂t − ∆ |Cijk |2 = −2|∇Cijk |2 − 16Cipk Ciqk Rpq + 24Cipk Ckqi Rpq + 4R|Cijk |2
+8Cijk Rpk ∇j Rpi + 4Cijk Rij ∇k R .
10
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
Proof.
∂t − ∆ |Cijk |2 = −2|∇Cijk |2 + 2Cijk Rip g pq Cqjk + 2Cijk Rjp g pq Ciqk + 2Cijk Rkp g pq Cijq
h
+2Cijk g pq Rpj (Ckqi + Cqki ) + 5g pq Rip Cqkj + g pq Rpk (Cjiq + Cqij )
+2RCijk + 2Rql Cqjl gik − 2Rql Cqkl gij
1
1
R
R
+ ∇k |Ric|2 gij − ∇j |Ric|2 gik + ∇j Rgik − ∇k Rgij
2
2
2
2
i
+2g pq Rpk ∇j Rqi − 2g pq Rpj ∇k Rqi + Rij ∇k R − Rik ∇j R
= −2|∇Cijk |2 + 2(Ckij + Cjki )Rip g pq (Ckqj + Cjkq )
+2Cijk Rjp g pq Ciqk + 2Cikj Rkp g pq Ciqj
h
i
+2Cijk 2g pq Rpj (Ckqi + Cqki ) + 5g pq Rip Cqkj
+4R|Cijk |2 + 8g pq Cijk Rpk ∇j Rqi + 4Cijk Rij ∇k R
= −2|∇Cijk |2 − 16Cipk Ciqk Rpq + 24Cipk Ckqi Rpq + 4R|Cijk |2
+8Cijk Rpk ∇j Rpi + 4Cijk Rij ∇k R
where in the last line we assumed to be in a orthonormal basis.
3. T HREE –D IMENSIONAL G RADIENT R ICCI S OLITONS
The structural equation of a gradient Ricci soliton (M n , g, ∇f ) is the following
Rij + ∇i ∇j f = λgij ,
(3.1)
for some λ ∈ R.
The soliton is said to be steady, shrinking or expanding according to the fact that the constant λ is zero,
positive or negative, respectively.
It follows that in dimension three, for (M 3 , g, ∇f ) there holds
(3.2)
∆Rij
= ∇l Rij ∇l f + 2λRij − 2|Ric|2 gij + R2 gij − 3RRij + 4Ris Rsj
∆R = ∇l R∇l f + 2λR − 2|Ric|2
∇i R = 2Rli ∇l f
Rlj gik
Rgij
Rlk gij
Rgik
(3.5)
∇l f −
∇l f + Rij ∇k f − Rik ∇j f +
∇j f −
∇k f
Cijk =
2
2
2
2
∇j R
∇k R
R R =
gij −
gik + Rij − gij ∇k f − Rik − gik ∇j f .
4
4
2
2
In the special case of a steady soliton the first two equations above simplify as follows,
(3.3)
(3.4)
∆Rij
= ∇l Rij ∇l f − 2|Ric|2 gij + R2 gij − 3RRij + 4Ris Rsj
∆R = ∇l R∇l f − 2|Ric|2 .
Remark 3.1. We notice that, by relation (3.5), we have
∇k R∇j f
∇j R∇k f
R
R
−
+ Rij ∇i f ∇k f − ∇j f ∇k f − Rik ∇i f ∇j f + ∇k f ∇j f
4
4
2
2
∇j R∇k f
∇k R∇j f
=
−
,
4
4
where in the last passage we used relation (3.4).
It follows that
h∇f, ∇Ri
|∇f |2
∇k f −
∇k R .
Cijk ∇i f ∇j f =
4
4
Hence, if the Cotton tensor of a three–dimensional gradient Ricci soliton is identically zero, we have
that at every point where ∇R is not zero, ∇f and ∇R are proportional.
This relation is a key step in (yet another) proof of the fact that a three–dimensional, locally conformally flat, steady or shrinking gradient Ricci soliton is locally a warped product of a constant
curvature surface on a interval of R, leading to a full classification, first obtained by H.-D. Cao and
Q. Chen [4] for the steady case and H.-D. Cao, B.-L. Chen and X.-P. Zhu [3] for the shrinking case
Cijk ∇i f
=
THE COTTON TENSOR AND THE RICCI FLOW
11
(actually this is the last paper of a series finally classifying, in full generality, all the three-dimensional
gradient shrinking Ricci solitons, even without the LCF assumption).
Proposition 3.2. Let (M 3 , g, f ) be a three–dimensional gradient Ricci soliton. Then,
∆|Cijk |2 = ∇l |Cijk |2 ∇l f + 2|∇Cijk |2 − 2R|Cijk |2
−6Cijk Rij ∇k R + 8Cjsk Cjik Rsi − 16Cjsk Ckij Rsi − 8Cijk Rlk ∇j Ril .
Proof. First observe that
∆|Cijk |2 = 2Cijk ∆Cijk + 2|∇Cijk |2 .
Using relations (3.5), (3.2) and, repeatedly, the trace–free property (1.5) of the Cotton tensor, we have
that
Cijk ∆Cijk = ∆(Rij ∇k f − Rik ∇j f )Cijk
= (∆Rij ∇k f + Rij ∆∇k f + 2∇l Rij ∇l ∇k f )Cijk
−(∆Rik ∇j f + Rik ∆∇j f + 2∇l Rik ∇l ∇j f )Cijk
= (∇s Rij ∇s f − 3RRij + 4Ris Rsj )∇k f Cijk
+Rij ∆∇k f Cijk + 2∇l Rij ∇l ∇k f Cijk
−(∇s Rik ∇s f − 3RRik + 4Ris Rsk )∇j f Cijk
−Rik ∆∇j f Cijk − 2∇l Rik ∇l ∇j f Cijk
= (∇s Rij ∇k f − ∇s Rik ∇j f )∇s f Cijk
−3R(Rij ∇k f − Rik ∇j f )Cijk
+4Ris (Rsj ∇k f − Rsk ∇j f )Cijk
+(Rij ∇l ∇l ∇k f − Rik ∇l ∇l ∇j f )Cijk
+2(∇l Rij ∇l ∇k f − ∇l Rik ∇l ∇j f )Cijk
= (∇s Rij ∇k f − ∇s Rik ∇j f )∇s f Cijk
+(−3R)|Cijk |2
+4Ris (Rsj ∇k f − Rsk ∇j f )Cijk
+(Rij ∇l ∇l ∇k f − Rik ∇l ∇l ∇j f )Cijk
+2(∇l Rij ∇l ∇k f − ∇l Rik ∇l ∇j f )Cijk ,
where we used the identity
(3.6)
(Rij ∇k f − Rik ∇j f )Cijk = |Cijk |2
12
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
which follows easily by equation (3.5) and the fact that every trace of the Cotton tensor is zero.
Using now equations (3.1), (3.5), (1.5), (1.4), and (3.4), we compute
(∇s Rij ∇k f − ∇s Rik ∇j f )∇s f Cijk = (∇s (Rij ∇k f ) − Rij ∇s ∇k f )∇s f Cijk
−(∇s (Rik ∇j f ) − Rik ∇s ∇j f )∇s f Cijk
= (∇s (Rij ∇k f − Rik ∇j f ) + Rij (Rsk ))∇s f Cijk
−(Rik (Rsj ))∇s f Cijk
= ∇s Cijk Cijk ∇s f + Rij Rsk ∇s f Cijk − Rik Rsj ∇s f Cijk
1
1
1
∇s |Cijk |2 ∇s f + Rij ∇k RCijk − Rik ∇j RCijk
=
2
2
2
1
R
R
1
4Ris (Rsj ∇k f − Rsk ∇j f )Cijk = 4Ris (Csjk − ∇k Rgsj + ∇j Rgsk + ∇k f gsj − ∇j f gsk )Cijk
4
4
2
2
= 4Ris (−Cjks − Cksj )(−Cjki − Ckij ) − Rij ∇k RCijk
+Rik ∇j RCijk + 2RRij ∇k f Cijk − 2RRik ∇j f Cijk
= 8Ris Cjsk Cjik − 8Ris Cjsk Ckij
−Rij ∇k RCijk + Rik ∇j RCijk + 2R|Cijk |2
(Rij ∇l ∇l ∇k f − Rik ∇l ∇l ∇j f )Cijk = (Rij ∇l (−Rlk ) − Rik ∇l (−Rlj ))Cijk
1
1
= − Rij ∇k RCijk + Rik ∇j RCijk
2
2
1
1
2(∇l Rij ∇l ∇k f − ∇l Rik ∇l ∇j f )Cijk = 2((Cijl + ∇j Ril + ∇l Rgij − ∇j Rgil )(−Rlk ))Cijk
4
4
1
1
−2((Cikl + ∇k Ril + ∇l Rgik − ∇k Rgil )(−Rlj ))Cijk
4
4
1
= −2Cijl Cijk Rlk − 2Cijk Rlk ∇j Ril + Cijk Rik ∇j R
2
1
+2Cikl Cijk Rlj + 2Cijk Rlj ∇k Ril − Cijk Rij ∇k R
2
1
= −2Cilj Cikj Rlk − 2Cijk Rlk ∇j Ril + Cijk Rik ∇j R
2
1
−2Cilk Cijk Rlj + 2Cijk Rlj ∇k Ril − Cijk Rij ∇k R.
2
Hence, getting back to the main computation and using again the symmetry relations (1.4), we finally
get
1
∇s |Cijk |2 ∇s f − R|Cijk |2
2
3
3
− Cijk Rij ∇k R + Cijk Rik ∇j R
2
2
+4Cjsk Cjik Rsi − 8Cjsk Ckij Rsi
−2Cijk Rlk ∇j Ril + 2Cijk Rlj ∇k Ril
1
=
∇s |Cijk |2 ∇s f − R|Cijk |2
2
−3Cijk Rij ∇k R + 4Cjsk Cjik Rsi − 8Cjsk Ckij Rsi − 4Cijk Rlk ∇j Ril
Cijk ∆Cijk =
where in the last passage we applied the skew–symmetry of the Cotton tensor in its last two indexes.
The thesis follows.
THE COTTON TENSOR AND THE RICCI FLOW
13
4. T HE E VOLUTION E QUATION OF THE C OTTON T ENSOR IN ANY D IMENSION
In this section we will compute the evolution equation under the Ricci flow of the Cotton tensor
Cijk , for every n–dimensional Riemannian manifold (M n , g(t)) evolving by Ricci flow.
Among the evolution equations (1.6) we expand the one for the Ricci tensor,
2n pq
2n
2
∂
Rij = ∆Rij −
g Rip Rjq +
RRij +
|Ric|2 gij
∂t
n−2
(n − 1)(n − 2)
n−2
2
−
R2 gij − 2Rpq Wpijq .
(n − 1)(n − 2)
Then, we compute the evolution equations of the derivatives of the curvatures assuming, from now
on, to be in normal coordinates,
∂
∇l R = ∇l ∆R + 2∇l |Ric|2 ,
∂t
∂
∇s Rij
∂t
2n
2n
2n
∇s Rip Rjp −
Rip ∇s Rjp +
∇s RRij
n−2
n−2
(n − 1)(n − 2)
2
2
2n
R∇s Rij +
∇s |Ric|2 gij −
∇s R2 gij
+
(n − 1)(n − 2)
n−2
(n − 1)(n − 2)
−2∇s Rkl Wkijl − 2Rkl ∇s Wkijl + (∇i Rsp + ∇s Rip − ∇p Ris )Rjp
+(∇j Rsp + ∇s Rjp − ∇p Rjs )Rip
n+2
n+2
2n
= ∇s ∆Rij −
∇s Rip Rjp −
Rip ∇s Rjp +
∇s RRij
n−2
n−2
(n − 1)(n − 2)
2n
2
2
R∇s Rij +
∇s |Ric|2 gij −
∇s R2 gij
+
(n − 1)(n − 2)
n−2
(n − 1)(n − 2)
−2∇s Rkl Wkijl − 2Rkl ∇s Wkijl + (∇i Rsp − ∇p Ris )Rjp + (∇j Rsp − ∇p Rjs )Rip
n+2
n+2
2n
= ∇s ∆Rij −
∇s Rip Rjp −
Rip ∇s Rjp +
∇s RRij
n−2
n−2
(n − 1)(n − 2)
2n
2
2
+
R∇s Rij +
∇s |Ric|2 gij −
∇s R2 gij
(n − 1)(n − 2)
n−2
(n − 1)(n − 2)
−2∇s Rkl Wkijl − 2Rkl ∇s Wkijl + Cspi Rjp + Cspj Rip
1
1
+
Rjp [∇i Rgsp − ∇p Rgis ] +
Rip [∇j Rgsp − ∇p Rgjs ] ,
2(n − 1)
2(n − 1)
= ∇s ∆Rij −
where in the last passage we substituted the expression of the Cotton tensor.
14
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
We then compute,
∂
Cijk =
∂t
∂
∂
∂
1
∇k Rij − ∇j Rik −
∇k Rgij − ∇j Rgik
∂t
∂t
2(n − 1) ∂t
n+2
2n
n+2
∇k Rip Rjp −
Rip ∇k Rjp +
∇k RRij
= ∇k ∆Rij −
n−2
n−2
(n − 1)(n − 2)
2n
2
2
+
R∇k Rij +
∇k |Ric|2 gij −
∇k R2 gij
(n − 1)(n − 2)
n−2
(n − 1)(n − 2)
−2∇k Rpl Wpijl − 2Rpl ∇k Wpijl + Ckpi Rjp + Ckpj Rip
Rjp
Rip
+
[∇i Rgkp − ∇p Rgik ] +
[∇j Rgkp − ∇p Rgjk ]
2(n − 1)
2(n − 1)
n+2
n+2
2n
−∇j ∆Rik +
∇j Rip Rkp +
Rip ∇j Rkp −
∇j RRik
n−2
n−2
(n − 1)(n − 2)
2
2
2n
R∇j Rik −
∇j |Ric|2 gik +
∇j R2 gik
−
(n − 1)(n − 2)
n−2
n−1
−2∇k Rpl Wpijl − 2Rpl ∇k Wpijl − Cjpi Rkp − Cjpk Rip
Rkp
Rip
−
[∇i Rgjp − ∇p Rgij ] −
[∇k Rgjp − ∇p Rgkj ]
2(n − 1)
2(n − 1)
1
(Rij ∇k R − Rik ∇j R
+
n−1
gij
gik
− ∇k ∆R + 2∇k |Ric|2
+ ∇j ∆R + 2∇j |Ric|2
2(n − 1)
2(n − 1)
n+2
n+2
= ∇k ∆Rij −
∇k Rip Rjp −
Rip ∇k Rjp
n−2
n−2
5n − 2
2n
+
∇k RRij +
R∇k Rij
2(n − 1)(n − 2)
(n − 1)(n − 2)
n
2
+
∇k |Ric|2 gij −
∇k R2 gij
(n − 1)(n − 2)
(n − 1)(n − 2)
+Ckpi Rjp + Ckpj Rip − 2∇k Rpl Wpijl − 2Rpl ∇k Wpijl
1
−
Rpj ∇p Rgik
2(n − 1)
n+2
n+2
−∇j ∆Rik +
∇j Rip Rkp +
Rip ∇j Rkp
n−2
n−2
5n − 2
2n
−
∇j RRik −
R∇j Rik
2(n − 1)(n − 2)
(n − 1)(n − 2)
n
2
−
∇j |Ric|2 gik +
∇j R2 gik
(n − 1)(n − 2)
(n − 1)(n − 2)
−Cjpi Rkp − Cjpk Rip + 2∇j Rpl Wpikl + 2Rpl ∇j Wpikl
1
1
1
+
∇l RRlk gij −
∇k ∆Rgij +
∇j ∆Rgik
2(n − 1)
2(n − 1)
2(n − 1)
and
∆Cijk = ∆∇k Rij − ∆∇j Rik −
1
1
∆∇k Rgij +
∆∇j Rgik ,
2(n − 1)
2(n − 1)
THE COTTON TENSOR AND THE RICCI FLOW
15
hence,
∂
Cijk − ∆Cijk = ∇k ∆Rij − ∇j ∆Rik − ∆∇k Rij + ∆∇j Rik
∂t
1
(∇k ∆Rgij − ∇j ∆Rgik − ∆∇k Rgij + ∆∇j Rgik )
−
2(n − 1)
n+2
5n − 2
(∇k Rip Rjp + Rip ∇k Rjp ) +
∇k RRij
−
n−2
2(n − 1)(n − 2)
2n
R∇k Rij
+
(n − 1)(n − 2)
n
2
+
∇k |Ric|2 gij −
∇k R2 gij
(n − 1)(n − 2)
(n − 1)(n − 2)
+Ckpi Rjp + Ckpj Rip − 2∇k Rpl Wpijl − 2Rpl ∇k Wpijl
1
−
Rjp ∇p Rgik
2(n − 1)
n+2
5n − 2
+
(∇j Rip Rkp + Rip ∇j Rkp ) −
∇j RRik
n−2
2(n − 1)(n − 2)
2n
R∇j Rik
−
(n − 1)(n − 2)
n
2
−
∇j |Ric|2 gik +
∇j R2 gik
(n − 1)(n − 2)
(n − 1)(n − 2)
−Cjpi Rkp − Cjpk Rip + 2∇j Rpl Wpikl + 2Rpl ∇j Wpikl
1
+
Rkp ∇p Rgij
2(n − 1)
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 decomposition formula of the Riemann
16
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
tensor (1.1).
∇k ∆Rij − ∆∇k Rij
= ∇3kll Rij − ∇3lkl Rij + ∇3lkl Rij − ∇3llk Rij
= −Rkp ∇p Rij + Rklip ∇l Rjp + Rkljp ∇l Rip
+∇3lkl Rij − ∇3llk Rij
1
= −Rkp ∇p Rij +
(Rik ∇j R + Rjk ∇i R)
2(n − 2)
1
(Rkp ∇i Rjp + Rkp ∇j Rip − Rlp ∇l Rjp gik − Rlp ∇l Rip gjk )
−
n−2
1
1
(Rli ∇l Rjk + Rlj ∇l Rik ) −
(R∇j Rgik + R∇i Rgjk )
−
n−2
2(n − 1)(n − 2)
1
+
(R∇i Rjk + R∇j Rik )
(n − 1)(n − 2)
+∇l Rklip Rpj + Rkljp Rpi
+Wkljp ∇l Rip + Wklip ∇j Rjp
1
= −Rkp ∇p Rij +
(Rik ∇j R + Rjk ∇i R)
2(n − 2)
1
−
(Rkp ∇i Rjp + Rkp ∇j Rip − Rlp ∇l Rjp gik − Rlp ∇l Rip gjk )
n−2
1
1
(Rli ∇l Rjk + Rlj ∇l Rik ) −
(R∇j Rgik + R∇i Rgjk )
−
n−2
2(n − 1)(n − 2)
1
+
(R∇i Rjk + R∇j Rik )
(n − 1)(n − 2)
1
(Rki gpl Rpj + Rpl gki Rpj − Rli gkp Rpj − Rkp gli Rpj )
+∇l
n−2
1
−
(RRpj gki glp − RRpj gkp gil ) + Wklip Rpj
(n − 1)(n − 2)
1
+
(Rkj glp Rpi + Rlp gkj Rpi − Rlj gkp Rpi − Rkp glj Rpi )
n−2
1
−
(RRpi gkj gpl − RRpi gkp glj ) + Wkljp Rpi
(n − 1)(n − 2)
+Wkljp ∇l Rip + Wklip ∇j Rjp
1
= −Rkp ∇p Rij +
(Rik ∇j R + Rjk ∇i R)
2(n − 2)
1
−
(Rkp ∇i Rjp + Rkp ∇j Rip − Rlp ∇l Rjp gik − Rlp ∇l Rip gjk )
n−2
1
1
−
(Rli ∇l Rjk + Rlj ∇l Rik ) −
(R∇j Rgik + R∇i Rgjk )
n−2
2(n − 1)(n − 2)
1
+
(R∇i Rjk + R∇j Rik )
(n − 1)(n − 2)
1
+
(∇p Rki Rpj + Rki ∇j R/2 + ∇p Rgki Rpj /2
n−2
+Rlp ∇l Rpj gik − ∇i RRjk /2 − Rpi ∇p Rkj − ∇i Rkp Rpj
−Rkp ∇i Rpj )
1
−
(∇p RRpj gik + R∇j Rgik /2 − ∇i RRkj − R∇i Rjk )
(n − 1)(n − 2)
n−3
+
Ckip Rpj + Wklip ∇l Rpj
n−2
THE COTTON TENSOR AND THE RICCI FLOW
1
(∇p Rkj Rpi + Rkj ∇i R/2 + ∇p Rgkj Rpi /2
n−2
+Rlp gkj ∇l Rpi − ∇j RRki /2 − Rpj ∇p Rki − ∇j Rkp Rpi − Rkp ∇j Rpi )
1
−
(∇p RRpi gkj + R∇i Rgjk /2 − ∇j RRki − R∇j Rki )
(n − 1)(n − 2)
n−3
Ckjp Rpi + Wkljp ∇l Rpi
+
n−2
+Wkljp ∇l Rip + Wklip ∇j Rjp
n+1
2
= −Rkp ∇p Rij +
Rkj ∇i R −
Rkp ∇j Rip
2(n − 1)(n − 2)
n−2
1
1
2
Rlp ∇l Rpi gjk −
Rpj ∇p Rik −
R∇i Rgjk
+
n−2
n−2
(n − 1)(n − 2)
2
n+1
2
+
R∇j Rik +
∇j RRki −
Rkp ∇i Rjp
(n − 1)(n − 2)
2(n − 1)(n − 2)
n−2
2
1
1
+
Rlp ∇l Rpj gik −
Rpi ∇p Rjk −
R∇j Rgik
n−2
n−2
(n − 1)(n − 2)
2
+
R∇i Rjk + 2Wkljp ∇l Rpi + 2Wklip ∇l Rpj
(n − 1)(n − 2)
n−3
+
(∇p Rgik Rpj + ∇p Rgjk Rpi )
2(n − 1)(n − 2)
n−3
1
+
(Ckip Rpj + Ckjp Rpi ) −
(∇i Rkp Rpj + ∇j Rkp Rpi )
n−2
n−2
+
and
∇k ∆R − ∆∇k R = Rkllp ∇p R = −Rkp ∇p R .
17
18
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
Then, getting back to the main computation, we obtain
∂
n+1
Cijk − ∆Cijk = −Rkp ∇p Rij +
Rkj ∇i R
∂t
2(n − 1)(n − 2)
2
2
−
Rkp ∇j Rip +
Rlp ∇l Rpi gjk
n−2
n−2
1
1
Rjp ∇p Rik −
R∇i Rgjk
−
n−2
(n − 1)(n − 2)
2
n+1
+
R∇j Rik +
∇j RRki
(n − 1)(n − 2)
2(n − 1)(n − 2)
2
2
−
Rkp ∇i Rpj +
Rlp ∇l Rpj gik
n−2
n−2
1
1
−
Rpi ∇p Rkj −
R∇j Rgik
n−2
(n − 1)(n − 2)
2
R∇i Rjk + 2Wkljp ∇l Rpi + 2Wklip ∇l Rpj
+
(n − 1)(n − 2)
n−3
+
(∇p Rgik Rpj + ∇p Rgjk Rpi )
2(n − 1)(n − 2)
n−3
+
(Ckip Rpj + Ckjp Rpi )
n−2
1
−
(∇i Rkp Rjp + ∇j Rkp Rpi )
n−2
2
n+1
+Rjp ∇p Rik −
Rkj ∇i R +
Rjp ∇k Rip
2(n − 1)(n − 2)
n−2
1
2
Rlp ∇l Rpi gkj +
Rpk ∇p Rij
−
n−2
n−2
1
2
+
R∇i Rgjk −
R∇k Rij
(n − 1)(n − 2)
(n − 1)(n − 2)
n+1
2
−
∇k RRij +
Rjp ∇i Rkp
2(n − 1)(n − 2)
n−2
2
1
−
Rlp ∇p Rpk gij +
Rpi ∇p Rkj
n−2
n−2
1
2
+
R∇k Rgij −
R∇i Rkj
(n − 1)(n − 2)
(n − 1)(n − 2)
−2Wjlkp ∇l Rpi − 2Wjlip ∇l Rpk
n−3
−
(∇p Rgij Rpk + ∇p Rgjk Rpi )
2(n − 2)(n − 2)
n−3
1
(Cjip Rpk + Cjkp Rpi ) +
(∇i Rpj Rpk + ∇k Rjp Rpi )
−
n−2
n−2
1
n+2
+
(Rkp ∇p Rgij − Rjp ∇p Rgki ) −
(∇k Rpi Rpj + Rpi ∇k Rpj )
2(n − 1)
n−2
n
5n − 2
+
∇k |Ric|2 gij +
∇k RRij
(n − 1)(n − 2)
2(n − 1)(n − 2)
2n
2
+
R∇k Rij −
∇k R2 gij
(n − 1)(n − 2)
(n − 1)(n − 2)
−2∇k Rpl Wpijl − 2Rpl ∇k Wpijl
1
+Ckli Rlj −
∇l RRlj gik + Cklj Rli
2(n − 1)
THE COTTON TENSOR AND THE RICCI FLOW
19
n+2
(∇j Rpi Rpk + Rpi ∇j Rpk )
n−2
n
5n − 2
∇j |Ric|2 gki −
∇j RRik
−
(n − 1)(n − 2)
2(n − 1)(n − 2)
2
2n
R∇j Rik +
∇j R2 gik
−
(n − 1)(n − 2)
(n − 1)(n − 2)
+2∇j Rpl Wpikl + 2Rpl ∇j Wpikl
1
−Cjli Rlk +
∇l RRlk gij − Cjlk Rli
2(n − 1)
1
(Rpi Cjkp + Rpk Cjip − Ckip Rpj − Ckjp Rpi )
=
n−2
h 2
3
+
Rlp ∇l Rpj +
∇j R2
n−2
2(n − 1)(n − 2)
i
n
1
∇p RRpj −
∇j |Ric|2 gik
−
2(n − 2)
(n − 1)(n − 2)
h 2
3
−
Rlp ∇l Rpk +
∇k R2
n−2
2(n − 1)(n − 2)
i
1
n
2
−
∇p RRpk −
∇k |Ric| gij
2(n − 2)
(n − 1)(n − 2)
n−3
n−3
−
Rkp ∇p Rij +
Rpj ∇p Rik
n−2
n−2
n
n+1
2
+
Rkp ∇j Rpi +
∇j Rpk Rpi −
R∇j Rik
n−2
n−2
n−2
1
1
4n − 3
∇j RRik −
Rkp ∇i Rpj +
Rpj ∇i Rpk
−
2(n − 1)(n − 2)
n−2
n−2
n
n+1
2
−
Rjp ∇k Rip −
∇k Rjp Rip +
R∇k Rij
n−2
n−2
n−2
4n − 3
+
∇k RRij + 2Wklip ∇l Rpj + 2Wkljp ∇l Rpi − 2Wjlkp ∇l Rpi
2(n − 1)(n − 2)
−2Wjlip ∇l Rpk − 2∇k Rpl Wpijl − 2Rpl ∇k Wpijl + 2∇j Rpl Wpikl + 2Rpl ∇j Wpikl
+
Now, by means of the very definition of the Cotton tensor (1.2), the identities (1.4), and the symmetries of the Weyl tensor, we substitute
Ckpj − Cjpk = − Ckjp − Cjpk = Cpkj
1
∇l Rjp = ∇j Rlp + Cpjl +
2(n − 1)
1
∇l Rkp = ∇k Rlp + Cpkl +
2(n − 1)
1
∇i Rjp = ∇j Rip + Cpji +
2(n − 1)
1
∇i Rkp = ∇k Rip + Cpki +
2(n − 1)
1
∇p Rij = ∇j Rpi + Cijp +
2(n − 1)
1
∇p Rik = ∇k Rpi + Cikp +
2(n − 1)
∇l Rgpj − ∇j Rgpl
∇l Rgpk − ∇k Rgpl
∇i Rgjp − ∇j Rgip
∇i Rgkp − ∇k Rgip
∇p Rgji − ∇j Rgpi
∇p Rgki − ∇k Rgpi
20
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
in the last expression above, getting
∂
Cijk − ∆Cijk =
∂t
1
(Rpi Cpkj + Rpk Cjip − Ckip Rpj )
n−2
h 2
1
Rlp ∇j Rlp + Cpjl +
∇l Rgpj
+
n−2
2(n − 1)
1
3
−
∇j Rgpl ) +
∇j R2
2(n − 1)
2(n − 1)(n − 2)
i
1
n
−
∇p RRpj −
∇j |Ric|2 gik
2(n − 2)
(n − 1)(n − 2)
h 2
1
Rlp ∇k Rpl + Cpkl +
∇l Rgpk
−
n−2
2(n − 1)
1
3
−
∇k Rgpl +
∇k R2
2(n − 1)
2(n − 1)(n − 2)
i
1
n
−
∇p RRpk −
∇k |Ric|2 gij
2(n − 2)
(n − 1)(n − 2)
1
n−3
Rkp Cijp + ∇j Rip +
(∇p Rgij − ∇j Rgip )
−
n−2
2(n − 1)
n−3
1
+
Rpj Cikp + ∇k Rip +
(∇p Rgik − ∇k Rgip )
n−2
2(n − 1)
n
n+1
2
+
Rkp ∇j Rpi +
∇j Rpk Rpi −
R∇j Rik
n−2
n−2
n−2
4n − 3
−
∇j RRik
2(n − 1)(n − 2)
1
1
−
Rkp ∇j Rip + Cpji +
(∇i Rgjp − ∇j Rgip )
n−2
2(n − 1)
1
1
Rpj ∇k Rip + Ckpi +
(∇i Rgkp − ∇k Rgip )
+
n−2
2(n − 1)
n
n+1
2
−
Rjp ∇k Rip −
∇k Rjp Rip +
R∇k Rij
n−2
n−2
n−2
4n − 3
+
∇k RRij
2(n − 1)(n − 2)
+2Cplj Wpikl − 2Cplk Wpijl − 2Cpil Wjklp
−2Wjklp ∇i Rpl − 2Rpl ∇k Wpijl + 2Rpl ∇j Wpikl
1
=
(Rpi Cpkj + Rpk (Cjip − Cpji − (n − 3)Cijp ) + Rpj (Cpki − Ckip + (n − 3)Cikp ))
n−2
2
2
+
Cpjl Rpl gik −
Cpkl Rpl gij − 2Cpjl Wpikl + 2Cpkl Wpijl − 2Cpil Wjklp
n−2
n−2
h
i
∇j R2
1
+gik
−
∇j |Ric|2
(n − 1)(n − 2) (n − 1)(n − 2)
h
i
∇k R 2
1
−gij
−
∇k |Ric|2
(n − 1)(n − 2) (n − 1)(n − 2)
2
n+1
3n − 1
2
−
Rjp ∇k Rip −
∇k Rjp Rip +
∇k RRij +
R∇k Rij
n−2
n−2
2(n − 1)(n − 2)
n−2
2
n+1
3n − 1
2
+
Rkp ∇j Rip +
∇j Rkp Rip −
∇j RRik −
R∇j Rik
n−2
n−2
2(n − 1)(n − 2)
n−2
−2Wjklp ∇i Rlp − 2Rlp ∇k Wpijl + 2Rpl ∇j Wpikl .
THE COTTON TENSOR AND THE RICCI FLOW
21
then, we substitute again
1
2(n − 1)
1
∇j Rkp = ∇p Rjk + Ckpj +
2(n − 1)
1
∇k Rij = ∇i Rkj + Cjik +
2(n − 1)
1
∇j Rik = ∇i Rjk + Ckij +
2(n − 1)
∇k Rjp = ∇p Rkj + Cjpk +
∇k Rgjp − ∇p Rgjk
∇j Rgkp − ∇p Rgkj
∇k Rgij − ∇i Rgjk
∇j Rgik − ∇i Rgkj ,
finally obtaining
∂
Cijk − ∆Cijk =
∂t
1
(Rpi Cpkj + Rpk (Cjip − Cpji − (n − 3)Cijp ) + Rpj (Cpki − Ckip + (n − 3)Cikp ))
n−2
2
2
+
Cpjl Rpl gik −
Cpkl Rpl gij − 2Cpjl Wpikl + 2Cpkl Wpijl − 2Cpil Wjklp
n−2
n−2
i
h
∇j R2
1
−
∇j |Ric|2
+gik
(n − 1)(n − 2) (n − 1)(n − 2)
h
i
∇k R 2
1
−gij
−
∇k |Ric|2
(n − 1)(n − 2) (n − 1)(n − 2)
2
n+1
n+1
−
Rjp ∇k Rip −
Rip ∇p Rkj −
Rip Cjpk
n−2
n−2
n−2
n+1
n+1
Rij ∇k R +
Rip ∇p Rgjk
−
2(n − 1)(n − 2)
2(n − 1)(n − 2)
3n − 1
2
1
+
∇k RRij +
R(∇i Rjk + Cjik +
(∇k Rgij − ∇i Rgjk ))
2(n − 1)(n − 2)
n−2
2(n − 1)
2
n+1
n+1
n+1
+
Rkp ∇j Rip +
Rip ∇p Rkj +
Rip Ckpj +
∇j RRik
n−2
n−2
n−2
2(n − 1)(n − 2)
n+1
3n − 1
−
Rip ∇p Rgjk −
∇j RRik
2(n − 1)(n − 2)
2(n − 1)(n − 2)
2
1
−
R(∇i Rjk + Ckij +
(∇j Rgik − ∇i Rgjk ))
n−2
2(n − 1)
−2Wjklp ∇i Rlp − 2Rlp ∇k Wpijl + 2Rpl ∇j Wpikl
1
=
(Rpk (Cjip − Cpji − (n − 3)Cijp ) − Rpj (Ckip − Cpki − (n − 3)Cikp )
n−2
2
2
+(n + 2)Rpi Cpkj ) +
(Cpjl Rpl gik − Cpkl Rpl gij ) +
RCijk
n−2
n−2
−2Wpikl Cpjl + 2Wpijl Cpkl − 2Cpil Wjklp
h
i
∇j R 2
1
+gik
−
∇j |Ric|2
2(n − 1)(n − 2) (n − 1)(n − 2)
h
i
∇k R2
1
−gij
−
∇k |Ric|2
2(n − 1)(n − 2) (n − 1)(n − 2)
2
1
−
Rjp ∇k Rip +
∇k RRij
n−2
n−2
2
1
+
Rkp ∇j Rip −
∇j RRik
n−2
n−2
+2Rlp ∇j Wpikl − 2Rlp ∇k Wpijl ,
where in the last passage we used again the identities (1.4) and the fact that
Wjklp ∇i Rlp = Wjkpl ∇i Rpl = Wjkpl ∇i Rlp = −Wjklp ∇i Rlp .
Hence, we can resume this long computation in the following proposition, getting back to a generic
coordinate basis.
22
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
Proposition 4.1. During the Ricci flow of a n–dimensional Riemannian manifold (M n , g(t)), the Cotton
tensor satisfies the following evolution equation
∂t − ∆ Cijk =
1
[g pq Rpj (Ckqi + Cqki + (n − 3)Cikq )
n−2
+(n + 2)g pq Rip Cqkj − g pq Rpk (Cjqi + Cqji + (n − 3)Cijq )]
2
2
2
RCijk +
Rql Cqjl gik −
Rql Cqkl gij
+
n−2
n−2
n−2
1
1
+
∇k |Ric|2 gij −
∇j |Ric|2 gik
(n − 1)(n − 2)
(n − 1)(n − 2)
R
R
+
∇j Rgik −
∇k Rgij
(n − 1)(n − 2)
(n − 1)(n − 2)
2
2
1
1
+
g pq Rpk ∇j Rqi −
g pq Rpj ∇k Rqi +
Rij ∇k R −
Rik ∇j R
n−2
n−2
n−2
n−2
−2g pq Wpikl Cqjl + 2g pq Wpijl Cqkl − 2g pq Wjklp Cqil + 2g pq Rpl ∇j Wqikl − 2g pq Rpl ∇k Wqijl .
In particular if the Cotton tensor vanishes identically along the flow we obtain,
0 =
1
1
∇k |Ric|2 gij −
∇j |Ric|2 gik
(n − 1)(n − 2)
(n − 1)(n − 2)
R
R
∇j Rgik −
∇k Rgij
+
(n − 1)(n − 2)
(n − 1)(n − 2)
2
2
1
1
+
g pq Rpk ∇j Rqi −
g pq Rpj ∇k Rqi +
Rij ∇k R −
Rik ∇j R
n−2
n−2
n−2
n−2
+2g pq Rpl ∇j Wqikl − 2g pq Rpl ∇k Wqijl ,
while, in virtue of relation (1.3), if the Weyl tensor vanishes along the flow we obtain (compare with [5, Proposition 1.1 and Corollary 1.2])
0 =
1
1
∇k |Ric|2 gij −
∇j |Ric|2 gik
(n − 1)(n − 2)
(n − 1)(n − 2)
R
R
+
∇j Rgik −
∇k Rgij
(n − 1)(n − 2)
(n − 1)(n − 2)
2
2
1
1
+
g pq Rpk ∇j Rqi −
g pq Rpj ∇k Rqi +
Rij ∇k R −
Rik ∇j R .
n−2
n−2
n−2
n−2
Corollary 4.2. During the Ricci flow of a n–dimensional Riemannian manifold (M n , g(t)), the squared norm
of the Cotton tensor satisfies the following evolution equation, in an orthonormal basis,
16
24
∂t − ∆ |Cijk |2 = −2|∇Cijk |2 −
Cipk Ciqk Rpq +
Cipk Ckqi Rpq
n−2
n−2
4
8
4
+
R|Cijk |2 +
Cijk Rpk ∇j Rpi +
Cijk Rij ∇k R
n−2
n−2
n−2
+8Cijk Rlp ∇j Wpikl − 8Cijk Cpjl Wpikl − 4Cjpi Cljk Wpikl .
THE COTTON TENSOR AND THE RICCI FLOW
23
∂t − ∆ |Cijk |2 = −2|∇Cijk |2 + 2Cijk Rip g pq Cqjk + 2Cijk Rjp g pq Ciqk + 2Cijk Rkp g pq Ciqk
h 1
[(Rpj (Ckpi + Cpki + (n − 3)Cikp )
+2Cijk
n−2
+(n + 2)Rpi Cpkj − Rpk (Cjpi + Cpji + (n − 3)Cijp ))
2
2
2
+
RCijk +
Rql Cqjl gik −
Rql Cqkl gij
n−2
n−2
n−2
1
1
+
∇k |Ric|2 gij −
∇j |Ric|2 gik
(n − 1)(n − 2)
(n − 1)(n − 2)
R
R
+
∇j Rgik −
∇k Rgij
(n − 1)(n − 2)
(n − 1)(n − 2)
2
2
1
1
+
Rqk ∇j Rqi −
Rqj ∇k Rqi +
Rij ∇k R −
Rik ∇j R
n−2
n−2
n−2
n−2
−2Wpikl Cpjl + 2Wpijl Cpkl − 2Wjklp Cpil + 2Rpl ∇j Wpikl − 2Rpl ∇k Wpikl
i
16
24
Cipk Ciqk Rpq +
Cipk Ckqi Rpq
n−2
n−2
4
8
4
+
R|Cijk |2 +
Cijk Rpk ∇j Rpi +
Cijk Rij ∇k R
n−2
n−2
n−2
+8Cijk Rlp ∇j Wpikl − 8Cijk Cpjl Wpikl − 4Cjpi Cljk Wpikl .
= −2|∇Cijk |2 −
Remark 4.3. Notice that if n = 3 the two formulas in Proposition 4.1 and Corollary 4.2 become the ones
in Proposition 2.1 and Corollary 2.5.
5. T HE B ACH T ENSOR
The Bach tensor in dimension three is given by
Bik = ∇j Cijk .
Let Sij = Rij −
(5.1)
1
2(n−1) Rgij
be the Schouten tensor, then
Bik = ∇j Cijk = ∇j (∇k Sij − ∇j Sik ) = ∇j ∇k Sij − ∆Sik .
We compute, in generic dimension n,
∇j Cijk = ∇j ∇k Rij −
1
∇j ∇k Rgij − ∆Sik
2(n − 1)
1
= +Rjkil Rjl + Rjkjl Ril + ∇k ∇j Rij −
∇k ∇j Rgij − ∆Sik
2(n − 1)
1
R
= +
Rij gkl − Rjl gki + Rkl gij − Rki gjl −
(gij gkl − gjl gki ) Rjl + Wjkil Rjl
n−2
(n − 1)
1
1
+Rkl Ril + ∇k ∇i R −
∇k ∇i R − ∆Sik
2
2(n − 1)
1
R
R2
= +
(Rji Rjk − |Ric|2 gik + Rkl Ril − RRik ) −
Rik +
gik
n−2
(n − 1)(n − 2)
(n − 1)(n − 2)
n−2
+Wjkil Rjl + Rkl Ril +
∇k ∇i R − ∆Sik
2(n − 1)
n
n
1
R2
=
Rij Rkj −
RRik −
|Ric|2 gik +
gik
n−2
(n − 1)(n − 2)
n−2
(n − 1)(n − 2)
n−2
+Wjkil Rjl +
∇k ∇i R − ∆Sik .
2(n − 1)
From this last expression, it is easy to see that the Bach tensor in dimension 3 is symmetric, i.e. Bik =
Bki . Moreover, it is trace–free, that is, g ik Bik = 0 as g ik ∇Cijk = 0.
24
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
Remark 5.1. In higher dimension, the Bach tensor is given by
Bik =
1
(∇j Cijk − Rjl Wijkl ) .
n−2
We note that, since Rjl Wijkl = Rjl Wklij = Rjl Wkjil , from the above computation we get that the Bach
tensor is symmetric in any dimension; finally, as the Weyl tensor is trace-free in every pair of indexes,
there holds g ik Bik = 0.
We recall that Schur lemma yields the following equation for the divergence of the Schouten tensor
∇j Sij =
(5.2)
n−2
∇i R .
2(n − 1)
We write
∇k ∇j Cijk = ∇k ∇j ∇k Sij − ∇k ∇j ∇j Sik = [∇j , ∇k ]∇j Sik ,
therefore,
∇k ∇j Cijk = Rjkjl ∇l Sik + Rjkil ∇j Slk + Rjkkl ∇j Sli
= Rkl ∇l Sik + Rjkil ∇j Slk − Rjl ∇j Sli
1
1
=
(Rij gkl − Rjl gik + Rkl gij − Rik gjl ) −
R(gij gkl − gik gjl ) + Wjkil ∇j Slk
n−2
(n − 1)(n − 2)
1
=
(−Rjl ∇j Sil + Rkl ∇i Skl ) + Wjkil ∇j Slk
n−2
1
Rjl (∇i Slj − ∇j Sil ) + Wjkil ∇j Rkl
=
n−2
1
=
Rjl Clji + Wiljk ∇j Rkl ,
n−2
where we repeatedly used equation (5.2), the trace–free property of the Weyl tensor and the definition
of the Cotton tensor.
Recalling that
n−3
n−3
∇k Wijkl = ∇k Wklij = −
Clij =
Clji ,
n−2
n−2
the divergence of the Bach tensor is given by
1
1
n−3
∇k (∇j Cijk − Rjl Wijkl ) =
Rjl Cjli −
Cjli Rjl
2
n−2
(n − 2)
(n − 2)2
n−4
= −
Cjli Rjl .
(n − 2)2
∇k Bik =
In particular, for n = 3, we obtain ∇k Bik = ∇k Bki = Rjl Cjli and, for n = 4, we get the classical result
∇k Bik = ∇k Bki = 0.
5.1. The Evolution Equation of the Bach Tensor in 3D.
We turn now our attention to the evolution of the Bach tensor along the Ricci flow in dimension
three. In order to obtain its evolution equation, instead of calculating directly the time derivative and
the Laplacian of the Bach tensor, we employ the following equation
(5.3)
(∂t − ∆)Bik = ∇j (∂t − ∆)Cijk − [∆, ∇j ]Cijk + 2Rpj ∇p Cijk + [∂t , ∇j ]Cijk ,
which relates the quantity we want to compute with the evolution of the Cotton tensor, the evolution
of the Christoffel symbols and the formulas for the exchange of covariant derivatives. We will work
on the various terms separately.
By the commutations formulas for derivatives, we have
∇l ∇l ∇q Cijk − ∇l ∇q ∇l Cijk = ∇l (Rlqip Cpjk + Rlqjp Cipk + Rlqkp Cijp )
∇l ∇q ∇s Cijk − ∇q ∇l ∇s Cijk = Rlqsp ∇p Cijk + Rlqip ∇s Cpjk + Rlqjp ∇s Cipk + Rlqkp ∇s Cijp ,
THE COTTON TENSOR AND THE RICCI FLOW
25
and putting these together with q = j and l = s, we get
[∆, ∇j ]Cijk = ∇l (Rljip Cpjk − Rlp Cipk + Rljkp Cijp )
+Rjp ∇p Cijk + Rljip ∇l Cpjk − Rlp ∇l Cipk + Rljkp ∇l Cijp
R
= ∇l Rli gjp − Rlp gji + Rjp gli − Rji glp − (gli gjp − glp gji ) Cpjk
2
R
−Rlp Cipk + Rlk gjp − Rlp gjk + Rjp glk − Rjk glp − (glk gjp − glp gjk ) Cijp
2
+Rjp ∇p Cijk + Rljip ∇l Cpjk − Rlp ∇l Cipk + Rljkp ∇l Cijp
1
= − ∇p RCpik − Rlp ∇l Cpik + ∇i Rjp Cpjk + Rjp ∇i Cpjk − ∇p Rji Cpjk − Rji ∇p Cpjk
2
1
R
1
1
+ ∇p RCpik + ∇p Cpik − ∇p RCipk − Rlp ∇l Cipk − ∇p RCikp − Rlp ∇l Cikp
2
2
2
2
R
1
+∇k Rjp Cijp + Rjp ∇k Cijp − ∇p Rjk Cijp − Rjk ∇p Cijp + ∇p RCikp + ∇p Cikp
2
2
R
+Rjp ∇p Cijk − Rlp ∇l Cpik + Rjp ∇i Cpjk − Rji ∇p Cpjk + ∇p Cpik
2
R
−Rlp ∇l Cipk − Rlp ∇l Cikp + Rjp ∇k Cijp − Rjk ∇p Cijp + ∇p Cikp
2
= ∇i Rjp Cpjk − ∇p Rji Cijp − ∇p Rjk Cijp − ∇p Rjk Cijp − 2Rlp ∇l Cpik
1
+2Rlp ∇i Cplk − 2Rji ∇p Cpjk + R∇p Cpik + ∇p RCikp + 2Rjp ∇k Cijp
2
−2Rjk ∇p Cijp + R∇p Cikp + Rjp ∇p Cijk
= ∇i Rlp Cplk − ∇p Rli Cplk + ∇k Rlp Cilp − ∇p Rlk Cilp
−2Rlp ∇l Cpik + 2Rlp ∇i Cplk + 2Rli Bkl − 2Rli Blk + 2Rlp ∇k Cilp
1
+2Rlk Bil + Rlp ∇p Cilk − RBik + ∇p RCikp + RBik − RBik
2
= ∇i Rlp Cplk − ∇p Rli Cplk + ∇k Rlp Cilp − ∇p Rlk Cilp
+Rlp ∇p Cilk + 2Rlp ∇i Cplk + 2Rlp ∇k Cilp − 2Rlp ∇l Cipk
1
+ ∇p RCikp + 2Rlk Bil − RBik .
2
The covariant derivative of the evolution of the Cotton tensor is given by
∇j (∂t − ∆)Cijk =
5
∇p RCipk + ∇p RCpki + Rlp ∇p Ckli + Rlp ∇p Clki − ∇p Rkl Cpli
2
−∇p Rkl Clpi − Rkp Bpi + 5∇p Ril Clkp − 5Rip Bpk + 2RBik
+2∇s Rpl Cpsl gik + 2Rpl Bpl gik − 2∇i Rpl Cpkl − 2Rpl ∇i Cpkl
1
1
+ (|∇R|2 + R∆R − ∆|Ric|2 )gik − (∇i R∇k R + R∇i ∇k R − ∇i ∇k |Ric|2 )
2
2
+2∆Rip Rkp + 2∇l Rip ∇l Rkp − 2∇l ∇k Rip Rlp − ∇k Rip ∇p R
1
+∇l ∇k RRil + ∇k R∇i R − ∆RRik − ∇l R∇l Rik .
2
Finally, the commutator between the covariant derivative and the time derivative can be expressed in
terms of the time derivatives of the Christoffel symbols, as follows
[∂t , ∇j ]Cijk = −∂t Γpij Cpjk − ∂t Γpjk Cijp
=
=
=
=
∇i Rjp Cpjk + ∇j Rip Cpjk − ∇p Rij Cpjk + ∇j Rkp Cijp + ∇k Rjp Cijp − ∇p Rjk Cijp
∇i Rjp Cpjk + ∇p Rij Cjpk + ∇p Rij Cpkj + ∇p Rkj Cipj + ∇k Rjp Cijp + ∇p Rjk Cipj
∇i Rjp Cpjk − ∇p Rij Cpkj − ∇p Rij Ckjp + ∇p Rij Cpkj + 2∇p Rkj Cipj
∇i Rjp Cpjk − ∇p Rij Ckjp + 2∇p Rkj Cipj .
Substituting into (5.3), and making some computations, we obtain the evolution equation
26
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
Proposition 5.2. During the Ricci flow of a 3–dimensional Riemannian manifold (M 3 , g(t)) the Bach tensor
satisfies the following evolution equation
(∂t − ∆)Bik =
[3∇p RCipk + ∇p RCpki − ∇p R∇k Rip ]
+ [−2Rpl ∇p Cikl − 3Rpk Bpi − 5Rpi Bpk + 2∆Rip Rkp
−2∇l ∇k Rpi Rpl + ∇l ∇k RRli − ∆RRik ]
+ [−2∇p Rkl Clpi − 2∇p Rkl Cilp − 4∇p Ril Clpk − 2∇i Rpl Cpkl ]
+ [3RBik + 2∇s Rpl Cpsl gik + 2Rpl Bpl gik
1
1
+ (|∇R|2 + R∆R − ∆|Ric|2 )gik − (R∇i ∇k R − ∇i ∇k |Ric|2 )
2
2
+2∇l Rip ∇l Rkp − ∇l R∇l Rik ] .
Hence, if the Bach tensor vanishes identically along the flow, we have
0 = 3∇p RCipk + ∇p RCpki − ∇p R∇k Rip − 2Rpl ∇p Cikl
+2∆Rip Rkp − 2∇l ∇k Rpi Rpl + ∇l ∇k RRli − ∆RRik
−2∇p Rkl Clpi − 2∇p Rkl Cilp − 4∇p Ril Clpk − 2∇i Rpl Cpkl
1
+2∇s Rpl Cpsl gik + (|∇R|2 + R∆R − ∆|Ric|2 )gik
2
1
− (R∇i ∇k R − ∇i ∇k |Ric|2 ) + 2∇l Rip ∇l Rkp − ∇l R∇l Rik .
2
Remark 5.3. Note that, from the symmetry property of the Bach tensor, we have that the RHS in the
evolution equation of the Bach tensor should be symmetric in the two indices. It is not so difficult
to check that this property is verified for the formula in Proposition 5.2. Indeed, each of the terms in
between square brackets is symmetric in the two indices.
As a consequence of Proposition 5.2, we get that during the Ricci flow of a 3–dimensional Riemannian manifold the squared norm of the Bach tensor satisfies
(∂t − ∆)|Bik |2 = −2|∇Bik |2 − 12Bik Biq Rqk + 6Bik ∇p R − 4Bik Rpl ∇p Cikl
+4Bik ∇p Rkl Cpil − 8Bik ∇p Rkl Clpi − 4Bik ∇i Rpl Cpkl + 6R|Bik |2
−2Bik ∇p R∇k Rip + 4Bik ∆Rip Rkp − 4Bik ∇l ∇k Rpi Rpl + 2Bik ∇l ∇k RRli
−2Bik ∆RRik − Bik R∇i ∇k R + Bik ∇i ∇k |Ric|2 − 2Bik ∇l R∇l Rik
+4Bik ∇l Rip ∇l Rkp .
5.2. The Bach Tensor of Three–Dimensional Gradient Ricci Solitons.
In what follows, we will use formulas (3.1)–(3.5) to derive an expression of the Bach tensor and of
its divergence in the particular case of a gradient Ricci soliton in dimension three.
THE COTTON TENSOR AND THE RICCI FLOW
27
By straightforward computations, we obtain
Bik = ∇j Cijk
∇i ∇k R ∆R
gik
=
−
gik − ∇j Rik ∇j f +
∇j R∇j f
4
4
2
R
R
+ Rij − gij ∇j ∇k f − Rik − gik ∆f
2
2
1
1
1
=
∇i ∇k R − ∆Rgik − ∇j Rik ∇j f + ∇j R∇j f gik − Rij Rjk + λRik
4
4
2
λ
3
1
1
+ RRik − Rgik − 3λRik + RRik + λRgik − R2 gik
2
2
2
2
1
1
λ
1
λ
=
∇i Rlk ∇l f − Rlk Rli + Rik − ∇l R∇l f gik − Rgik
2
2
2
4
2
1
1
1
2
+ |Ric| gik − ∇j Rik ∇j f + ∇j R∇j f gik − Rij Rjk + λRik + RRik
2
2
2
λ
3
1 2
− Rgik − 3λRik + RRik + λRgik − R gik
2
2
2
1
3
3
1
∇i Rlk ∇l f + ∇j R∇j f gik − ∇j Rik ∇j f − Rij Rjk − λRik
=
2
4
2
2
3
λ
1
1
+ RRik + Rgik + |Ric|2 gik − R2 gik .
2
2
2
2
A more compact formulation, employing equations (3.2) and (3.3), is given by
1
1
1
λ
1
1
Bik = ∇i Rlk ∇l f + ∆Rgik − ∆Rik − ∇j Rik ∇j f − Rik + Rij Rjk .
2
4
2
2
2
2
Moreover, as we know that ∇k Bik = Clji Rlj , we have
1
1
1
1
R∇i R − Rij ∇j R + |Ric|2 ∇i f − R2 ∇i f − Ril ∇j f Rlj + RRij ∇j f
4
4
2
2
1
3
1
R∇i R − Ril ∇l R + |Ric|2 ∇i f − R2 ∇i f .
=
2
4
2
Therefore, if the divergence of the Bach tensor vanishes, we conclude
∇k Bik =
1
3
1
R∇i R − Rik ∇k R + |Ric|2 ∇i f − R2 ∇i f = 0 .
2
4
2
Taking the scalar product with ∇f in both sides of this equation, we obtain
1
3
1
0 = Rh∇R, ∇f i − |∇R|2 + |Ric|2 |∇f |2 − R2 |∇f |2
2
8
2
and, from formulas (3.5) and (3.6), we compute
∇j R
R
R
∇k R
2
|Cijk | = (Rij ∇k f − Rik ∇j f )
gij −
gik + Rij − gij ∇k f − Rik − gik ∇j f
4
4
2
2
2
R
1
R
R
=
∇k R∇k f − Rkj ∇j R∇k f + |Ric|2 |∇f |2 −
|∇f |2 − Rij ∇j f Rik ∇k f + Rkj ∇k f ∇j f
4
4
2
2
1
R
R
R2
− Rjk ∇j f ∇k R + ∇j R∇j f − Rik ∇k f Rij ∇j f + Rjk ∇j f ∇k f + |Ric|2 |∇f |2 −
|∇f |2
4
4
2
2
3
= 2|Ric|2 |∇f |2 − R2 |∇f |2 + R∇k R∇k f − |∇R|2 ,
4
where we repeatedly used equation (3.4).
Therefore, we obtain
1
∇k Bik ∇i f = |Cijk |2 ,
2
so, if the divergence of the Bach tensor vanishes then the Cotton tensor vanishes as well (this was
already obtained in [2]). As a consequence, getting back to Section 3, the soliton is locally a warped
product of a constant curvature surface on a interval of R.
28
CARLO MANTEGAZZA, SAMUELE MONGODI, AND MICHELE RIMOLDI
R EFERENCES
1. A. L. Besse, Einstein manifolds, Springer–Verlag, Berlin, 2008.
2. H.-D. Cao, G. Catino, Q. Chen, C. Mantegazza, and L. Mazzieri, Bach–flat gradient steady Ricci solitons, Calc. Var. Partial
Differential Equations 49 (2014), no. 1-2, 125–138.
3. H.-D. Cao, B.-L. Chen, and X.-P. Zhu, Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol.
XII. Geometric flows, vol. 12, Int. Press, Somerville, MA, 2008, pp. 47–112.
4. H.-D. Cao and Q. Chen, On locally conformally flat gradient steady Ricci solitons, Trans. Amer. Math. Soc. 364 (2012), 2377–
2391.
5. G. Catino and C. Mantegazza, Evolution of the Weyl tensor under the Ricci flow, Ann. Inst. Fourier (2011), 1407–1435.
6. S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Springer–Verlag, 1990.
7. R. S. Hamilton, Three–manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), no. 2, 255–306.
(Carlo Mantegazza) S CUOLA N ORMALE S UPERIORE , P IAZZA C AVALIERI 7, P ISA , I TALY, 56126
E-mail address, C. Mantegazza: c.mantegazza@sns.it
(Samuele Mongodi) S CUOLA N ORMALE S UPERIORE , P IAZZA C AVALIERI 7, P ISA , I TALY, 56126
E-mail address, S. Mongodi: samuele.mongodi@gmail.com
(Michele Rimoldi) D IPARTIMENTO DI M ATEMATICA E A PPLICAZIONI , U NIVERSIT A´ DEGLI S TUDI DI M ILANO –B ICOCCA ,
V IA C OZZI 55, M ILANO , I TALY, 20125
E-mail address, M. Rimoldi: michele.rimoldi@gmail.com