Canad. Math. Bull. Vol. 64 (3), 2021, pp. 591–604
http://dx.doi.org/10.4153/S0008439520000673
© Canadian Mathematical Society 2020
Some results on Ricci-Bourguignon solitons
and almost solitons
Shubham Dwivedi
Abstract. We prove some results for the solitons of the Ricci–Bourguignon flow, generalizing the
corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then
introduce the notion of Ricci–Bourguignon almost solitons and prove some results about them that
generalize previous results for Ricci almost solitons. We also derive integral formulas for compact
gradient Ricci–Bourguignon solitons and compact gradient Ricci–Bourguignon almost solitons.
Finally, using the integral formula, we show that a compact gradient Ricci–Bourguignon almost
soliton is isometric to a Euclidean sphere if it has constant scalar curvature or its associated vector
field is conformal.
1 Introduction
Ricci solitons play a major role in Ricci flow where they correspond to self-similar
solutions of the flow. Thus, given a geometric flow, it is natural to study the solitons
associated with that flow. A family of metrics g(t) on an n-dimensional Riemannian
manifold (M n , g) is said to evolve by the Ricci-Bourguignon flow (RB flow for short)
if g(t) satisfies the following evolution equation
∂g
= −2(Ric − ρRg),
∂t
where Ric is the Ricci tensor of the metric, R is the scalar curvature, and ρ ∈ R is a
constant. The flow in equation (1.1) was first introduced by Jean-Pierre Bourguignon
[Bou81], building on some unpublished work of Lichnerowicz and a paper by Aubin
[Aub70]. We note that (1.1) is precisely the Ricci flow for ρ = 0. In particular, the righthand side of the evolution equation (1.1) is of special interest for different values of ρ,
for example,
(1.1)
•
•
•
•
ρ = 21 , the Einstein tensor Ric − R2 g.
ρ = n1 , the traceless Ricci tensor Ric − Rn g.
1
R
ρ = 2(n−1)
, the Schouten tensor Ric − 2(n−1)
g.
ρ = 0, the Ricci tensor Ric.
A systematic study of the parabolic theory of the RB flow was initiated in
[CCD+17]. In that paper, the authors proved, along with many other results, the
short time existence of the flow (1.1) on any closed n-dimensional manifold starting
Received by the editors March 12, 2020; revised August 8, 2020.
Published online on Cambridge Core August 20, 2020.
AMS subject classification: 53CXX, 53C20, 53C21, 53C44, 53E20.
Keywords: Ricci–Bourguignon flow, solitons, almost solitons, self-similar solutions.
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
592
S. Dwivedi
with an arbitrary initial metric g 0 for ρ <
the following definition.
1
.
2(n−1)
As in the Ricci flow case, we make
Definition 1.1 A Ricci–Bourguignon soliton (RB soliton for short) is a Riemannian
manifold (M n , g) endowed with a vector field X on M that satisfies
(1.2)
1
R i j + (L X g) i j = λg i j + ρRg i j ,
2
where L X g denotes the Lie derivative of the metric g with respect to the vector field
X and λ ∈ R is a constant.
When X = ∇ f for some smooth f ∶ M → R, then (M, g) is called a gradient RB
soliton. The soliton is called
(1) expanding when λ < 0,
(2) steady when λ = 0,
(3) shrinking when λ > 0.
RB solitons correspond to self-similar solutions of the RB flow. An RB soliton is
called trivial if X is a Killing vector field, i.e., L X g = 0. We remark that even though the
1
, any value of ρ is possible
short time existence result for the flow (1.1) is for ρ < 2(n−1)
for the considerations of self-similar solutions of the flow.
Gradient RB solitons were studied in detail, for example, in [CM16] and [CMM15],
where the authors called them gradient ρ-Einstein solitons. Various classification and
rigidity results about gradient RB solitons were proved in those papers and we refer
the reader to those papers for precise statements and proofs of the results.
The notion of Ricci almost solitons was introduced in [PRRS11], where the authors
modified the definition of a Ricci soliton by considering the parameter λ in the
definition of a Ricci soliton to be a function rather than a constant. Motivated by the
Ricci flow case, we make the following definition.
Definition 1.2 A Riemannian manifold (M n , g) is a Ricci–Bourguignon almost
soliton (RB almost soliton for short) if there is a vector field X and a soliton function
λ ∶ M → R satisfying
(1.3)
1
Ric + L X g = λg + ρRg.
2
An RB almost soliton is called a gradient RB almost soliton if X = ∇ f for some
smooth function f on M and is expanding, steady, or shrinking if λ < 0, λ = 0, or
λ > 0, respectively. We note that if X is a Killing vector field, then a RB almost soliton
is just a RB soliton, as it forces λ to be a constant.
Recall that a vector field Y on a Riemannian manifold (M, g) is called a conformal
vector field if there exists a function ψ ∶ M → R such that
LY g = 2ψg.
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
Some results on Ricci-Bourguignon solitons and almost solitons
593
The function ψ is sometimes referred as the potential of the vector field Y. The
conformal vector field is nontrivial if ψ ≠ 0. If ψ = 0, then Y is called a Killing vector
field.
Some characterization results for compact Ricci and Ricci almost solitons were
obtained in [ABR11] and [BR12], respectively. The goal of this paper is to generalize
the results obtained in those papers to RB and RB almost solitons. More precisely, in
Section 3 we prove the following theorems.
Theorem 1.1 Let (M n , g, X, λ, ρ), n ≥ 3, be a RB soliton and suppose that the vector
field X is a conformal vector field.
If M is compact, then X is a Killing vector field, and hence (M n , g, X, λ, ρ) is a
trivial RB soliton.
(ii) There is no nontrivial, complete noncompact RB soliton (M n , g, X, λ, ρ) with
conformal vector field X.
(i)
This generalizes [ABR11, Theorem 3] and characterizes compact RB solitons when
X is a conformal vector field. The following corollary gives a lower bound for the first
eigenvalue of the Laplacian on a compact RB soliton when X is a conformal vector
field and generalizes [ABR11, Theorem 4].
Corollary 1.1 Let (M n , g, X, λ, ρ) be a compact RB soliton with X a conformal vector
field. If n ≥ 3 and λ + ρR > 0, then the first eigenvalue λ 1 of the Laplacian satisfies λ 1 ≥
n
. Moreover, equality occurs if and only if M n is isometric to a Euclidean
(λ + ρR) n−1
sphere.
The next theorem characterizes compact RB almost solitons with X a conformal
vector field and generalizes [BR12, Theorem 2].
Theorem 1.2 Let (M n , g, X, λ, ρ), n ≥ 3, be a compact RB almost soliton. If X is a
nontrivial conformal vector field, then M n is isometric to a Euclidean sphere.
Example 1.3 Consider the Euclidean sphere (S n , ground , X, λ), where ground is the
round metric on S n , X is the projection of a non-zero constant vector field X̄ on Rn+1 ,
over S n , and λ = 1 − ρ(n − 1) div X, with ρ ∈ R a constant. Then X is a conformal
vector field on S n and is not a Killing vector field. Since Ric = (n − 1)ground for S n ,
(S n , ground , X, λ) is a RB almost soliton.
Example 1.4 Let M = I × h Σ be the h-warped product of the real interval I ⊂ R with
0 ∈ I, and the Riemannian manifold (Σ, g Σ ) with dim Σ = n. That is, M is an (n + 1)dimensional product manifold I × Σ with the metric
g M = dt 2 + h 2 g Σ ,
where t is a global parameter on I and h is a positive function on I. Suppose
Σ is an Einstein manifold with RicΣ = −(n − 1)a with a < 0. Following [PRRS11,
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
594
S. Dwivedi
Example 2.5], define a function f (x, t) = f (t) by
(1.4)
f (t) = B + ∫
t
0
h(s)[A + (n − 1) ∫
s
0
h ′′ h − (h ′ )2 − a
dx]ds
h3
for some constants A, B ∈ R. Also define λ(x, t) = λ(t) by
(h ′ )2 + a
h ′′
−
anρ]
−
h2
h
s h ′′ h − (h ′ )2 − a
′
dx],
+ h [A + (n − 1) ∫
h3
0
λ(t) = −(n − 1)[
(1.5)
where ρ ∈ R is a constant. It follows from [PRRS11] that (M, g M , f ′ ∂t∂ , λ, ρ) is a RB
almost soliton, where f is given by (1.4), and λ is given by (1.5).
The next theorem generalizes in [BR12, Theorem 3] obtained for compact Ricci
almost solitons, which is the case when ρ = 0.
Let (M n , g, X, λ, ρ), n ≥ 3, be a compact RB almost soliton. If ρ ≠
Theorem 1.5
1
n
and
nρ
∇ X div X − 2ρg(∇R, X)
nρ − 1
(n(2ρ + 1) − 2)
−
g(∇λ, X)] dv ≤ 0,
nρ − 1
∫ [Ric(X, X) +
M
(1.6)
then X is a Killing vector field and M n is a trivial RB soliton.
Since every RB almost soliton is also a RB soliton for constant λ, using ∇λ = 0, we
get the following corollary for compact RB solitons.
Corollary 1.2
(1.7)
Let (M n , g, X, λ, ρ), n ≥ 3, be a compact RB soliton. If ρ ≠
∫ [Ric(X, X) +
M
1
n
and
nρ
∇ X div X − 2ρg(∇R, X)]dv ≤ 0,
(nρ − 1)
then X is a Killing vector field and M n is a trivial RB soliton.
Remark 1.6 Corollary 1.2 is an analog of [PW09, Theorem 1.1], which was for the
case of compact Ricci solitons. We obtain Petersen–Wylie’s result from our result by
taking ρ = 0. In fact, the condition in (1.7) is analogous to the condition in [PW09,
Theorem 1.1], which is obtained when ρ = 0 in (1.7).
Finally, we obtain integral formulas for compact gradient RB almost solitons,
generalizing a corresponding result for compact gradient Ricci almost solitons from
[BR12].
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
Some results on Ricci-Bourguignon solitons and almost solitons
Theorem 1.7
(1.8)
595
Let (M n , g, ∇ f , λ, ρ) be a compact gradient RB almost soliton. Then
(n − 2)
∆f 2
g∣ dv =
∫ g(∇R, ∇ f )dv,
n
2n
M
M
R 2
(n − 2)
∫ ∣Ric − g∣ dv =
∫ g(∇R, ∇ f )dv.
n
2n
M
M
2
∫ ∣∇ f −
(1.9)
As an application of the previous theorem, we provide some conditions for a
compact gradient RB almost soliton to be isometric to a Euclidean sphere.
Corollary 1.3 A nontrivial compact gradient RB almost soliton (M n , g, ∇ f , λ, ρ), n ≥
3, is isometric to a Euclidean sphere if any of the following assertions hold:
(i) M n has constant scalar curvature;
(ii) ∫ M g(∇R, ∇ f )dv ≤ 0;
(iii) M n is a homogenous manifold.
The paper is organized as follows. In Section 2, we state and prove some identities
for RB solitons and RB almost solitons that will be used to prove the main results. In
Section 3, we prove the main theorems and their corollaries.
2 Preliminaries
In this section, we prove some general results about RB solitons and almost solitons.
The proofs of some of these results in the compact gradient case can also be found in
[CM16] or [CMM15]. Let us first recall the Ricci identity for a (0, 2)-tensor α:
∇ i ∇ j α k l − ∇ j ∇ i α k l = −R i jkm α ml − R i jl m α km ,
where R i jk l is the Riemann curvature tensor. The Ricci curvature is obtained from the
Riemann curvature tensor by contracting on the first and last index
R i j = g k l R k i jl ,
and the contracted second Bianchi identity is
1
∇ i R i j = ∇ j R.
2
We start with the following proposition.
Proposition 2.1 Let (M n , g, ∇ f , λ, ρ) be a gradient RB almost soliton. Then the
following identities hold:
(2.1)
(1 − nρ)R + ∆ f = nλ.
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
596
S. Dwivedi
(1 − 2ρ(n − 1))∇ i R = 2R i l ∇ l f + 2(n − 1)∇ i λ.
(2.2)
∇ j R i k − ∇ k R i j = R jk i l ∇ l f + ρ(∇ j Rg i k − ∇ k Rg i j ) + (∇ j λg i k − ∇ k λg i j ).
(2.3)
∇ i [(1 − 2ρ(n − 1))R + ∣∇ f ∣2 − 2(n − 1)λ] = (2ρR + 2λ)∇ i f .
(2.4)
For a gradient RB almost soliton, we have
Proof
R i j + ∇ i ∇ j f = λg i j + ρRg i j .
(2.5)
Taking the trace of the above equation gives (2.1). Next, taking the covariant derivative
of (2.1) with respect to an orthonormal frame gives
(1 − nρ)∇ i R + ∇ i ∇ j ∇ j f = n∇ i λ.
Commuting the covariant derivatives and using the contracted second Bianchi identity, we obtain
(1 − nρ)∇ i R = −∇ j ∇ i ∇ j f + R i l ∇ l f + n∇ i λ
= −∇ j (−R i j + λg i j + ρRg i j ) + R i l ∇ l f + n∇ i λ
1
= ∇ i R − ρ∇ i R − ∇ i λ + R i l ∇ l f + n∇ i λ,
2
and hence
1
( − ρ(n − 1))∇ i R = R i l ∇ l f + (n − 1)∇ i λ,
2
(2.6)
which proves (2.2).
For proving (2.3), we use (2.5) and commute the covariant derivatives to get
∇ j R i k − ∇ k R i j = (∇ k ∇ i ∇ j f − ∇ j ∇ i ∇ k f ) + ρ(∇ j Rg i k − ∇ k Rg i j )
+ (∇ j λg i k − ∇ k λg i j )
= (∇ k ∇ j ∇ i f − ∇ j ∇ k ∇ i f ) + ρ(∇ j Rg i k − ∇ k Rg i j )
+ (∇ j λg i k − ∇ k λg i j )
= R jk i l ∇ l f + ρ(∇ j Rg i k − ∇ k Rg i j ) + (∇ j λg i k − ∇ k λg i j ).
(2.7)
Finally, from (2.2), we get
(1 − 2ρ(n − 1))∇ i R = 2∇ l f (−∇ i ∇ l f + λg i l + ρRg i l ) + 2(n − 1)∇ i λ
= −2∇ l f ∇ i ∇ l f + 2λ∇ i f + 2ρR∇ i f + 2(n − 1)∇ i λ
= −∇ i ∣∇ l f ∣2 + 2λ∇ i f + 2ρR∇ i f + 2(n − 1)∇ i λ,
so we get
(2.8)
∇ i [(1 − 2ρ(n − 1))R + ∣∇ f ∣2 − 2(n − 1)λ] = (2ρR + 2λ)∇ i f ,
which proves (2.4).
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
∎
Some results on Ricci-Bourguignon solitons and almost solitons
Remark 2.2
The analogous identities for gradient RB solitons (M n , g, ∇ f , λ, ρ) are
(2.9)
(1 − nρ)R + ∆ f = nλ.
(2.10)
(1 − 2ρ(n − 1))∇ i R = 2R i l ∇ l f .
(2.11)
∇ j R i k − ∇ k R i j = R jk i l ∇ l f + ρ(∇ j Rg i k − ∇ k Rg i j ).
(2.12)
∇ i [(1 − 2ρ(n − 1))R + ∣∇ f ∣2 − 2λ f ] = 2ρR∇ i f .
The proofs of these identities are special cases of the previous result as ∇λ = 0.
We recall the following lemma from [PW09, Lemma 2.1].
Lemma 2.3
Let X be a vector field on a Riemannian manifold (M n , g). Then
1
div(L X g)(X) = ∆∣X∣2 − ∣∇X∣2 + Ric(X, X) + ∇ X div X.
2
(2.13)
When X = ∇ f and Z is any vector field,
div(L∇ f g)(Z) = 2 Ric(Z, ∇ f ) + 2∇ Z div ∇ f .
(2.14)
We use the preceding lemma to prove the following lemma.
Lemma 2.4
(2.15)
Let (M n , g, X, λ, ρ) be a RB almost soliton. Then
(1 − nρ)
∆∣X∣2 = (1 − nρ)∣∇X∣2 + (nρ − 1) Ric(X, X) + nρ∇ X div X
2
+ 2ρ(1 − nρ)g(∇R, X) − (n(2ρ + 1) − 2)g(∇λ, X)
and
(1 − nρ)
(∆ − ∇ X )∣X∣2 = (1 − nρ)∣∇X∣2 + λ(nρ − 1)∣X∣2 + ρ(nρ − 1)R∣X∣2
2
+ nρ∇ X div X + 2ρ(1 − nρ)g(∇R, X)
(2.16)
− (n(2ρ + 1) − 2)g(∇λ, X).
Proof
(2.17)
We first notice that (1.3) gives
2 div Ric + div(L X g) = 2∇λ + 2ρ∇R.
Taking the trace of (1.3) gives (1 − nρ)R + div X = nλ, and thus
(2.18)
597
(1 − nρ)∇ X R + ∇ X (div X) = n∇ X λ.
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
598
S. Dwivedi
So, using (2.13), (2.17), (2.18), and the contracted second Bianchi identity, we get
∇ X (div X) = (nρ − 1)∇ X R + ng(∇λ, X)
= 2(nρ − 1) div Ric(X) + ng(∇λ, X)
= −(nρ − 1) div(L X g)(X) + 2ρ(nρ − 1)g(∇R, X) + 2(nρ − 1)g(∇λ, X)
+ ng(∇λ, X)
1
= (1 − nρ)( ∆∣X∣2 − ∣∇X∣2 + Ric(X, X) + ∇ X div X)
2
+ 2ρ(nρ − 1)g(∇R, X) + (n(2ρ + 1) − 2)g(∇λ, X)
=
(1 − nρ)
∆∣X∣2 − (1 − nρ)∣∇X∣2 + (1 − nρ) Ric(X, X)
2
+ (1 − nρ)∇ X div X + 2ρ(nρ − 1)g(∇R, X)
+ (n(2ρ + 1) − 2)g(∇λ, X),
which gives
(1 − nρ)
∆∣X∣2 = (1 − nρ)∣∇X∣2 + (nρ − 1) Ric(X, X) + nρ∇ X div X
2
(2.19)
+ 2ρ(1 − nρ)g(∇R, X) − (n(2ρ + 1) − 2)g(∇λ, X),
thus proving (2.15).
Using (1.3) to write Ric(X, X) = − 21 (L X g)(X, X) + λ∣X∣2 + ρR∣X∣2 in (2.15), we
get
1
(1 − nρ)
∆∣X∣2 = (1 − nρ)∣∇X∣2 + (nρ − 1)( − (L X g)(X, X) + λ∣X∣2 + ρR∣X∣2 )
2
2
+ nρ∇ X div X + 2ρ(1 − nρ)g(∇R, X) − (n(2ρ + 1) − 2)g(∇λ, X)
(1 − nρ)
∇ X ∣X∣2 + λ(nρ − 1)∣X∣2 + ρ(nρ − 1)R∣X∣2
2
+ nρ∇ X div X + 2ρ(1 − nρ)g(∇R, X) − (n(2ρ + 1) − 2)g(∇λ, X),
= (1 − nρ)∣∇X∣2 +
which gives
(1 − nρ)
(∆ − ∇ X )∣X∣2 = (1 − nρ)∣∇X∣2 + λ(nρ − 1)∣X∣2 + ρ(nρ − 1)R∣X∣2
2
+ nρ∇ X div X + 2ρ(1 − nρ)g(∇R, X)
(2.20)
− (n(2ρ + 1) − 2)g(∇λ, X),
proving (2.16).
∎
If we consider the diffusion operator ∆ X = ∆ − ∇ X , then from the previous lemma
with X = ∇ f and ∆ f = ∆ − ∇∇ f , we obtain the following corollary.
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
Some results on Ricci-Bourguignon solitons and almost solitons
Corollary 2.1
599
For a gradient RB almost soliton (M n , g, ∇ f , λ, ρ), we have
(1 − nρ)
∆ f ∣∇ f ∣2 = (1 − nρ)∣∇2 f ∣2 + λ(nρ − 1)∣∇ f ∣2 + ρ(nρ − 1)R∣∇ f ∣2
2
+ nρ∇∇ f (∆ f ) + 2ρ(1 − nρ)g(∇R, ∇ f )
(2.21)
− (n(2ρ + 1) − 2)g(∇λ, ∇ f ).
Remark 2.5
(2.22)
and
The analogs of (2.15) and (2.16) for a RB soliton (M n , g, X, λ, ρ) are
(1 − nρ)
∆∣X∣2 = (1 − nρ)∣∇X∣2 + (nρ − 1) Ric(X, X) + nρ∇ X div X
2
+ 2ρ(1 − nρ)g(∇R, X)
(1 − nρ)
(∆ − ∇ X )∣X∣2 = (1 − nρ)∣∇X∣2 + λ(nρ − 1)∣X∣2 + ρ(nρ − 1)R∣X∣2
2
+ nρ∇ X div X + 2ρ(1 − nρ)g(∇R, X).
(2.23)
The proofs are special cases of the proof of Lemma 2.4 with ∇λ = 0.
3 Proofs of the Results
We start this section by proving the following lemma that will be used in the proofs
of Theorems 1.1 and 1.2.
Lemma 3.1 Let (M n , g, X, λ, ρ), n ≥ 3, be a RB almost soliton. If X is a nontrivial
conformal vector field with L X g = 2ψg, then R and λ − ψ are constant.
Proof
The soliton equation is
(3.1)
1
R i j + (L X g) i j = λg i j + ρRg i j ,
2
where λ ∶ M → R is a function. If X is a nontrivial conformal vector field, then we
have
(3.2)
L X g = 2ψg,
for some function ψ ∶ M → R, ψ ≠ 0. So (3.1) becomes
(3.3)
R i j = (λ − ψ + ρR)g i j .
Taking the divergence of (3.3), we get
∇ i R i j = ∇ i (λ − ψ + ρR)g i j ,
which implies
(3.4)
1
( − ρ)∇ j R = ∇ j (λ − ψ).
2
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
600
S. Dwivedi
On the other hand, tracing (3.3) and taking the covariant derivative, we get
(3.5)
(1 − nρ)∇ j R = n∇ j (λ − ρ).
So from (3.4) and (3.5), we get
1
(1 − nρ)∇ j R = n( − ρ)∇ j R.
2
Thus, if M is connected, then R is constant, and hence, λ − ψ is constant.
(3.6)
∎
Remark 3.2 If (M n , g, X, λ, ρ), n ≥ 3, is a RB soliton and X is a conformal vector
field with L X g = 2ψg for some function ψ ∶ M → R, then the proof of Lemma 3.1
shows that R and ψ are constant as in this case ∇λ = 0.
We prove Theorem 1.1, which we restate here.
Theorem 3.3 Let (M n , g, X, λ, ρ), n ≥ 3, be a RB soliton and suppose that the vector
field X is a conformal vector field.
If M is compact, then X is a Killing vector field, and hence (M n , g, X, λ, ρ) is a
trivial RB soliton.
(ii) There is no nontrivial, complete noncompact RB soliton (M n , g, X, λ, ρ) with
conformal vector field X.
(i)
Proof
Suppose X is a conformal vector field with potential ψ ∶ M → R, i.e.,
L X g = 2ψg,
(3.7)
then from Remark 3.2, we know that R and ψ are constant.
Taking the trace of (3.7), we get
2 div X = 2nψ,
which upon integration over compact M gives
(3.8)
0 = ∫ 2 div Xdv = 2n Vol(M)ψ,
M
which implies that ψ = 0. So X is a Killing vector field, and hence (M n , X, g, λ, ρ) is
a trivial RB soliton.
If M is noncompact and a gradient RB soliton with X = ∇ f , then X being conformal implies
∇ i ∇ j f = ψg i j
and by Remark 3.2, ψ is constant. If ψ = 0, then X is a Killing vector field and M is
a trivial RB soliton. If ψ ≠ 0, then from [Tas65, Theorem 2], we conclude that M n is
isometric to the Euclidean space.
∎
Next we prove Corollary 1.1.
Proof Since M n is compact, we know from Theorem 1.1 that X is a Killing vector
field, and hence we have Ric = (λ + ρR)g. So we can apply a classical theorem due to
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
Some results on Ricci-Bourguignon solitons and almost solitons
601
Lichnerowicz [Lic58], which states that if Ric ≥ k, where k > 0 is a constant, then the
n
k. So we get
first eigenvalue of the Laplacian λ 1 satisfies λ 1 ≥ n−1
λ 1 ≥ (λ + ρR)
n
.
n−1
Moreover, for the equality case, we can apply Obata’s theorem [Oba62], to conclude
that equality occurs in the above inequality if and only if M n is isometric to a
(λ+ρR)
Euclidean sphere of constant curvature n−1 .
∎
We now prove Theorem 1.2, which we restate here.
Theorem 3.4 Let (M n , g, X, λ, ρ), n ≥ 3, be a compact RB almost soliton. If X is a
nontrivial conformal vector field, then M n is isometric to a Euclidean sphere.
Proof Suppose X is a nontrivial conformal vector field with potential function ψ ∶
M → R, i.e.,
L X g = 2ψg,
with ψ ≠ 0. Since (M n , g, X, λ, ρ) is a compact RB almost soliton with n ≥ 3, Lemma
3.1 tells us that R and λ − ψ are constant. So from Lemma 2.3 in [Yan70, p. 52], we
conclude that R ≠ 0 or else ψ would be 0. Taking the Lie derivative of (3.3), we get
L X Ric = L X (λ − ψ + ρR)g,
and since (λ − ψ), ρ and R are all constant, so we get
L X Ric = 2(λ − ψ + ρR)ψg.
(3.9)
Now we can apply [Yan70, Theorem 4.2, p. 54], to conclude that M is isometric to
a Euclidean sphere.
∎
We proceed to the proof of Theorem 1.5.
Proof
We see from (2.15) of Lemma 2.4 that
(1 − nρ)
∆∣X∣2 = (1 − nρ)∣∇X∣2 + (nρ − 1) Ric(X, X) + nρ∇ X div X
2
+ 2ρ(1 − nρ)g(∇R, X) − (n(2ρ + 1) − 2)g(∇λ, X).
Integrating above over compact M, we get
0 = ∫ [(1 − nρ)∣∇X∣2 + (nρ − 1) Ric(X, X) + nρ∇ X div X
M
+ 2ρ(1 − nρ)g(∇R, X) − (n(2ρ + 1) − 2)g(∇λ, X)]dv.
(3.10)
Since ρ ≠ n1 , we get
nρ
∇ X div X − 2ρg(∇R, X)
nρ − 1
(n(2ρ + 1) − 2)
−
g(∇λ, X)]dv,
nρ − 1
2
∫ ∣∇X∣ dv = ∫ [Ric(X, X) +
M
(3.11)
M
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
602
S. Dwivedi
so if (1.6) holds, then ∣∇X∣2 = 0, and hence X is a Killing vector field. Thus,
(M n , g, X, λ, ρ) is trivial.
∎
The proof of Corollary 1.2 is a special case of the proof of Theorem 1.5, where we
use (2.22) of Remark 2.5.
Next, we prove Theorem 1.7, which we restate here.
Theorem 3.5
Let (M n , g, ∇ f , λ, ρ) be a compact gradient RB almost soliton. Then
2
∫ ∣∇ f −
(3.12)
M
R 2
(n − 2)
∫ ∣ Ric − g∣ dv =
∫ g(∇R, ∇ f )dv.
n
2n
M
M
(3.13)
Proof
(n − 2)
∆f 2
g∣ dv =
∫ g(∇R, ∇ f )dv,
n
2n
M
To prove (3.12), we take the divergence of (2.4) of Proposition 2.1 to get
(1 − 2ρ(n − 1))∆R + ∆∣∇ f ∣2 − 2(n − 1)∆λ = 2ρg(∇R, ∇ f ) + 2g(∇λ, ∇ f )
(3.14)
+ (2ρR + 2λ)∆ f .
By commuting the covariant derivatives, we have
∇ i ∇ i (g(∇ j f , ∇ j f )) = 2∇ i (g(∇ i ∇ j f , ∇ j f ))
= 2g(∇ i ∇ i ∇ j f , ∇ j f ) + 2∣∇2 f ∣2
= 2g(∇ j ∇ i ∇ i f − R i ji l ∇ l f , ∇ j f ) + 2∣∇2 f ∣2
= 2g(∇(∆ f ), ∇ f ) + 2 Ric(∇ f , ∇ f ) + 2∣∇2 f ∣2 ,
so (3.14) becomes
(1 − 2ρ(n − 1))∆R + 2g(∇(∆ f ), ∇ f ) + 2 Ric(∇ f , ∇ f ) + 2∣∇2 f ∣2 − 2(n − 1)∆λ
(3.15)
= 2ρg(∇R, ∇ f ) + 2g(∇λ, ∇ f ) + (2ρR + 2λ)∆ f .
From (2.1) of Proposition 2.1, we know that ∆ f = nλ + (nρ − 1)R, which on differentiation and using (2.5) becomes
0 = ∇ i ∆ f + (1 − nρ)∇ i R − n∇ i λ
= (1 − nρ)∇ i R + ∇ j ∇ i ∇ j f − R i l ∇ l f − n∇ i λ
= (1 − nρ)∇ i R + ∇ j (−R i j + λg i j + ρRg i j ) − R i l ∇ l f − n∇ i λ
1
= ( − ρ(n − 1))∇ i R − R i l ∇ l f + (1 − n)∇ i λ,
2
and hence
(3.16)
2 Ric(∇ f , ∇ f ) = (1 − 2ρ(n − 1))g(∇R, ∇ f ) + 2(1 − n)g(∇λ, ∇ f ).
So, using (3.16) and ∆ f = nλ + (nρ − 1)R, the left-hand side of (3.15) becomes
(1 − 2ρ(n − 1))∆R + 2∣∇2 f ∣2 − 2(n − 1)∆λ + 2g(∇λ, ∇ f ) + (2ρ − 1)g(∇R, ∇ f ),
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
Some results on Ricci-Bourguignon solitons and almost solitons
603
and hence (3.15) becomes
(3.17) (1 − 2ρ(n − 1))∆R + 2∣∇2 f ∣2 − 2(n − 1)∆λ = g(∇R, ∇ f ) + (2ρR + 2λ)∆ f .
Since ∣∇2 f −
∆f
n
g∣2 = ∣∇2 f ∣2 −
(1 − 2ρ(n − 1))∆R + 2∣∇2 f −
(∆ f )2
,
n
(3.17) becomes
(∆ f )2
∆f 2
g∣ = g(∇R, ∇ f ) + (2ρR + 2λ)∆ f − 2
n
n
+ 2(n − 1)∆λ
= g(∇R, ∇ f ) + (2ρR + 2λ)∆ f
(∆ f )
(nλ + (nρ − 1)R) + 2(n − 1)∆λ
n
2
= g(∇R, ∇ f ) + R∆ f + 2(n − 1)∆λ.
n
−2
(3.18)
Integrating (3.18) over compact M, we obtain
2
∫ 2∣∇ f −
M
(3.19)
∆f 2
2
g∣ dv = ∫ [g(∇R, ∇ f ) + R∆ f ]dv
n
n
M
(n − 2)
=
∫ g(∇R, ∇ f )dv,
n
M
where we have used integration by parts in the first equality to get the second equality.
This proves (3.12).
To prove (3.13), note that
(3.20)
R
R
Ric − g = −∇2 f + λg + ρRg − g
n
n
R
2
= −∇ f + (λ + ρR − )g
n
∆
f
= −∇2 f +
g,
n
and then (3.13) follows from (3.12).
∎
Remark 3.6 Since a gradient RB soliton is a special case of a gradient RB almost
soliton, the proof of Theorem 1.7, with ∇λ = 0, shows that the same integral formulas
(3.12) and (3.13) hold for a compact gradient RB solitons as well.
Finally, using Theorem 1.7, we prove Corollary 1.3.
Proof Observe that any of the assertions of Corollary 1.3 enable us to conclude that
the right-hand side of (3.13) is less than or equal to zero, and hence Ric = Rn g. So, from
(2.5), we see that
∇ i ∇ j f = (λ + R(ρ −
1
))g,
n
and hence ∇ f is a nontrivial conformal vector field, so from Theorem 1.2, we get that
M n is isometric to a Euclidean sphere.
∎
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press
604
S. Dwivedi
Acknowledgement The author would like to thank his advisor Spiro Karigiannis for
his constant encouragement and advice. The author is also grateful to the anonymous
referee for various useful remarks and suggestions that have improved the quality of
the paper.
References
[ABR11]
C. Aquino, A. Barros, and E. Ribeiro, Jr., Some applications of the Hodge–de Rham
decomposition to Ricci solitons. Results Math. 60(2011), nos. 1–4, 245–254.
https://doi.org/10.1007/s00025-011-0166-1
[Aub70]
[BR12]
T. Aubin, Métriques riemanniennes et courbure. J. Differ. Geom. 4(1970), 383–424.
A. Barros and E. Ribeiro, Jr., Some characterizations for compact almost Ricci solitons. Proc.
Am. Math. Soc. 140(2012), no. 3, 1033–1040.
https://doi.org/10.1090/S0002-9939-2011-11029-3
J. P. Bourguignon, Ricci curvature and Einstein metrics. In: Global differential geometry and
global analysis (Berlin, 1979), Lecture Notes in Mathematics, 838, Springer, Berlin,
Germany, New York, NY, 1981, pp. 42–63.
[CCD+17] G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, and L. Mazzieri, The Ricci-Bourguignon
flow. Pac. J. Math. 287(2017), no. 2, 337–370. https://doi.org/10.2140/pjm.2017.287.337
[CM16]
G. Catino and L. Mazzieri, Gradient Einstein solitons. Nonlinear Anal. 132(2016), 66–94.
[Bou81]
https://doi.org/10.1016/j.na.2015.10.021
[CMM15] G. Catino, L. Mazzieri, and S. Mongodi, Rigidity of gradient Einstein shrinkers. Commun.
Contemp. Math. 17(2015), no. 6, 1550046, 18 pp.
https://doi.org/10.1142/S0219199715500467
[Lic58]
A. Lichnerowicz, Géométrie des groupes de transformations. Travaux et Recherches
Mathématiques, III, Dunod, Paris, 1958.
[Oba62]
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere. J.
Math. Soc. Jpn. 14(1962), 333–340. https://doi.org/10.2969/jmsj/0140333
[PW09]
P. Petersen and W. Wylie, Rigidity of gradient Ricci solitons. Pac. J. Math. 241(2009), no. 2,
329–345. https://doi.org/10.2140/pjm.2009.241.329
[PRRS11] S. Pigola, M. Rigoli, M. Rimoldi, and A. G. Setti, Ricci almost solitons. Ann. Sc. Norm. Super.
Pisa Cl. Sci. 10(2011), no. 4, 757–799.
[Tas65]
Y. Tashiro, Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc.
117(1965), 251–275. https://doi.org/10.2307/1994206
[Yan70]
K. Yano, Integral formulas in Riemannian geometry. Pure and Applied Mathematics, 1,
Marcel Dekker, Inc., New York, NY, 1970.
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
e-mail: s2dwived@uwaterloo.ca
https://doi.org/10.4153/S0008439520000673 Published online by Cambridge University Press