---
title: Conformal Ricci–Bourguignon Solitons
url: https://www.emergentmind.com/topics/conformal-ricci-bourguignon-soliton
type: topic
---

# Conformal Ricci–Bourguignon Solitons

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A conformal Ricci–Bourguignon soliton is a Ricci–Bourguignon soliton for which the potential vector field is conformal, so the self-similarity of the Ricci–Bourguignon flow is generated by a conformal deformation rather than merely by a Killing field or a gradient potential. In the standard Ricci–Bourguignon normalization, the flow is
\[
\partial_t g=-2\bigl(\operatorname{Ric}-\rho\,R\,g\bigr),
\]
and a soliton satisfies
\[
\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,
\]
or, in the gradient case,
\[
\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.
\]
If \(X\) is conformal, \(\mathcal{L}_X g=2\psi g\), the equation collapses to an Einstein-type condition. Recent work develops this theme in compact rigidity theory, sequential warped products, Lorentzian models such as Vaidya spacetime, and steady almost-gradient surface geometries, while also showing that the defining convention is not uniform across the literature [1507.00324, 1809.11103, 2303.01257, 2508.09985].

## 1. Definitions, normalizations, and competing conventions

In the standard convention used for the Ricci–Bourguignon flow, an RB soliton \((M^n,g,X,\lambda)\) satisfies
\[
\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,
\]
with gradient form
\[
\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.
\]
The trace gives
\[
\operatorname{div}X+(1-n\rho)R=n\lambda,
\qquad
\Delta f+(1-n\rho)R=n\lambda.
\]
The usual shrinking, steady, and expanding nomenclature is determined by the sign of \(\lambda\) [1507.00324].

A conformal vector field \(X\) is defined by
\[
\mathcal{L}_X g=2\psi g,
\]
and, in the gradient setting, a conformal associated vector field means
\[
\nabla^2 f=\psi g.
\]
Allowing \(\lambda\) to vary produces an RB almost soliton,
\[
\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda(x)g,
\]
with gradient form
\[
\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda(x)g.
\]
This extension is central in compact rigidity results because conformality then yields
\[
\operatorname{Ric}-\rho\,R\,g=(\lambda(x)-\psi)g
\]
rather than the constant-\(\lambda\) reduction alone [1809.11103].

Several later papers use different normalizations. On sequential warped products, the soliton equation is written
\[
\operatorname{Ric}+\mathcal{L}_X g=\lambda g+\rho R g,
\]
with gradient form
\[
\operatorname{Ric}+\operatorname{Hess}u=\lambda g+\rho R g.
\]
That paper also explicitly notes a convention change: its Ricci soliton definition uses \(+2\mathcal{L}_Xg\), whereas its Ricci–Bourguignon soliton uses \(+\mathcal{L}_Xg\). In the conformal case, the authors assume \(X\) has factor \(2\alpha\), with \(\alpha\) constant, and then use the reduced balance
\[
\operatorname{Ric}(Y,Z)=(\lambda+\rho R-\alpha)g(Y,Z)
\]
in their component calculations [2303.01257].

Other variants enlarge the framework further. The \(h\)-almost Ricci–Bourguignon equation is
\[
\operatorname{Ric}+\frac{h}{2}\mathcal{L}_\xi g=(\lambda+\rho R)g,
\]
or, in the gradient case,
\[
\operatorname{Ric}+h\,\nabla^2 f=(\lambda+\rho R)g.
\]
Under conformality, \(\mathcal{L}_\xi g=2\psi g\), it reduces to
\[
\operatorname{Ric}=(\lambda+\rho R-h\psi)g
\]
[2505.02193].

The Vaidya analysis adopts yet another convention, drawn from conformal Ricci soliton and conformal RB soliton literature:
\[
\mathcal{L}_X g+2S=\Bigl(2\beta-\Bigl(p+\tfrac{2}{n}\Bigr)\Bigr)g+2\alpha Rg.
\]
For \(n=4\) and \(R=0\), the paper sets
\[
\kappa:=2\beta-\Bigl(p+\tfrac12\Bigr),
\qquad
\mathcal{L}_X g+2S=\kappa g.
\]
Accordingly, the cited works do not present a single universal normalization for the phrase “conformal Ricci–Bourguignon soliton” [2508.09985].

## 2. Einstein reduction under conformality

In the standard RB soliton convention, substituting
\[
\mathcal{L}_X g=2\psi g
\]
into
\[
\operatorname{Ric}-\rho R g+\tfrac12\mathcal{L}_X g=\lambda g
\]
gives
\[
\operatorname{Ric}=(\lambda-\psi+\rho R)g.
\]
Tracing the soliton equation yields
\[
n\psi+(1-n\rho)R=n\lambda,
\qquad
\psi=\lambda-\frac{1-n\rho}{n}R.
\]
Eliminating \(\psi\) gives the rigidity identity
\[
\operatorname{Ric}=\frac{R}{n}g.
\]
Thus every conformal Ricci–Bourguignon soliton is Einstein in this normalization, and for \(n\ge 3\) the scalar curvature is constant. Consequently, \(\psi\) is also constant by the traced relation [1507.00324].

This reduction remains the basic mechanism in later compact results. In the almost-soliton setting,
\[
\operatorname{Ric}-\rho R g=(\lambda(x)-\psi)g,
\]
while in the gradient conformal case the pure-trace condition is
\[
\nabla^2 f=\psi g.
\]
Tracing then gives
\[
(1-n\rho)R+n\psi=n\lambda
\]
and hence
\[
(1-n\rho)R=n(\lambda-\psi).
\]
The same paper develops identities for \(\nabla R\), the commutator of Ricci derivatives, and a Bochner-type formula for RB almost solitons, all of which are used to convert this Einstein-type reduction into global rigidity on compact manifolds [1809.11103].

On compact manifolds the conformal reduction is even stronger. Since \(\operatorname{div}X=n\psi\), integrating over a compact manifold gives \(\psi=0\), so the conformal field is actually Killing. In the standard soliton case this implies that a compact conformal RB soliton is trivial, with
\[
\lambda=\frac{R}{n}-\rho R.
\]
The noncompact statement in the same source is also rigid: there is no nontrivial complete noncompact RB soliton with conformal vector field \(X\). In the gradient case, \(\nabla^2 f=\psi g\) with \(\psi\) constant forces Euclidean geometry by Tashiro’s theorem, contradicting nontriviality unless \(\psi=0\) [1809.11103].

The \(h\)-almost theory preserves the same pattern. Once \(\mathcal{L}_\xi g=2\psi g\), the equation
\[
\operatorname{Ric}+\frac{h}{2}\mathcal{L}_\xi g=(\lambda+\rho R)g
\]
again makes the Ricci tensor pointwise proportional to the metric:
\[
\operatorname{Ric}=(\lambda+\rho R-h\psi)g.
\]
A plausible implication is that conformality remains a mechanism for forcing Einstein structure even when the deformation term is weighted by a nonconstant factor \(h\) [2505.02193].

## 3. Compact rigidity, integral identities, and sphere theorems

The compact theory of conformal RB almost solitons is dominated by sphere rigidity. If \((M^n,g,X,\lambda,\rho)\), \(n\ge 3\), is a compact RB almost soliton and \(X\) is a nontrivial conformal vector field, then \((M^n,g)\) is isometric to a Euclidean sphere. The proof runs through the Einstein-type reduction
\[
\operatorname{Ric}=(\lambda-\psi+\rho R)g,
\]
constancy of \(R\) and \(\lambda-\psi\), and a Yano characterization of constant sectional curvature. Compactness and nontriviality of the conformal factor force the round sphere [1809.11103].

In the compact gradient setting the decisive inputs are the integral identities
\[
\int_M \Big|\nabla^2 f-\frac{\Delta f}{n}g\Big|^2\,d\mu
=
\frac{n-2}{2n}\int_M \langle \nabla R,\nabla f\rangle\,d\mu,
\]
and
\[
\int_M \Big|\operatorname{Ric}-\frac{R}{n}g\Big|^2\,d\mu
=
\frac{n-2}{2n}\int_M \langle \nabla R,\nabla f\rangle\,d\mu.
\]
If \(R\) is constant, the right-hand side vanishes and the metric is Einstein. Plugging this back into the almost-soliton equation gives
\[
\nabla^2 f=\bigl(\lambda+\rho R-\tfrac{R}{n}\bigr)g,
\]
so the associated gradient vector field is conformal. The same paper derives the corollary that a nontrivial compact gradient RB almost soliton is isometric to a Euclidean sphere if any of the following hold: \(R\) is constant; \(\int_M\langle \nabla R,\nabla f\rangle\,d\mu\le 0\); or \((M^n,g)\) is homogeneous [1809.11103].

The compact \(h\)-almost extension produces weighted analogues of these formulas. A key identity is
\[
\int_M \frac{1}{n}\,h\,\Bigl|\operatorname{Ric}-\frac{R}{n}g\Bigr|^2\,dV_g
=
\frac{n-2}{2n}\int_M \langle \nabla R,\xi\rangle\,dV_g.
\]
When \(\xi\) is conformal, the Kazdan–Warner identity on compact manifolds,
\[
\int_M \langle \nabla R,\xi\rangle\,dV_g=0,
\]
annihilates the right-hand side and forces the traceless Ricci tensor to vanish. The resulting sphere theorem states that a compact \(h\)-almost Ricci–Bourguignon soliton with \(n>3\) is isometric to a standard sphere if either
\[
\int_M \langle \nabla^2 h,\operatorname{Ric}\rangle\,dV_g=0
\]
or \(\nabla h\) is a nontrivial conformal vector field. Constant scalar curvature also yields Einstein structure and then sphere rigidity by the same mechanism [2505.02193].

A recurring misconception is that conformality necessarily enlarges the compact RB soliton class. In the standard compact soliton setting it does the opposite: conformality collapses the equation to Einstein and then to Killing triviality. New nontrivial compact examples appear only after moving to almost-soliton or \(h\)-almost frameworks, and even there the global outcome is usually round-sphere rigidity rather than a broader moduli space [1809.11103, 2505.02193].

## 4. Sequential warped products and spacetime specializations

A detailed conformal analysis is available for sequential warped products
\[
M=(M_1\times_f M_2)\times_h M_3,
\qquad
g=(g_1\oplus f^2 g_2)\oplus h^2 g_3,
\]
where \(f:M_1\to \mathbb{R}_+\) and \(h:M_1\times M_2\to \mathbb{R}_+\). The Ricci tensor decomposes by factors:
\[
\operatorname{Ric}(X_1,Y_1)=\operatorname{Ric}_1(X_1,Y_1)-\frac{n_2}{f}\operatorname{Hess}f(X_1,Y_1)-\frac{n_3}{h}\operatorname{Hess}h(X_1,Y_1),
\]
\[
\operatorname{Ric}(X_2,Y_2)=\operatorname{Ric}_2(X_2,Y_2)-f^\sharp g_2(X_2,Y_2)-\frac{n_3}{h}\operatorname{Hess}h(X_2,Y_2),
\]
\[
\operatorname{Ric}(X_3,Y_3)=\operatorname{Ric}_3(X_3,Y_3)-h^\sharp g_3(X_3,Y_3),
\]
with mixed components zero, and
\[
f^\sharp=f\Delta f+(n_2-1)|\nabla f|^2,
\qquad
h^\sharp=h\Delta h+(n_3-1)|\operatorname{grad}h|^2.
\]
If \(X=X_1+X_2+X_3\), then \(\mathcal{L}_X g\) splits into factor terms plus the warping contributions \(2fX_1(f)g_2\) and \(2h(X_1+X_2)(h)g_3\) [2303.01257].

In the conformal case, that paper assumes \(X\) is conformal with factor \(2\alpha\), with \(\alpha\) constant, and uses
\[
\operatorname{Ric}(Y,Z)=(\lambda+\rho R-\alpha)g(Y,Z).
\]
Under the hypotheses
\[
\operatorname{Hess}f=\sigma g,
\qquad
\operatorname{Hess}h=\psi g,
\]
the factor equations become
\[
\operatorname{Ric}_1(Y_1,Z_1)=\bigl[\lambda+\rho R-\alpha+n_2\sigma+n_3\psi\bigr]g_1(Y_1,Z_1),
\]
\[
\operatorname{Ric}_2(Y_2,Z_2)=\bigl[\lambda f+\rho Rf-\alpha f+(n_3\psi)f+f^\sharp\bigr]g_2(Y_2,Z_2),
\]
\[
\operatorname{Ric}_3(Y_3,Z_3)=\bigl[\lambda h+\rho Rh-\alpha h+h^\sharp\bigr]g_3(Y_3,Z_3).
\]
Hence \(M_1\), \(M_2\), and \(M_3\) are Einstein under the stated hypotheses, with Einstein factors given by the bracketed coefficients. If \(f\) and \(h\) are constant, then \(\sigma=\psi=0\) and \(f^\sharp=h^\sharp=0\), so the total space reduces to a scaled product Einstein space [2303.01257].

The converse direction is also explicit. If \((M_i,g_i)\) are Einstein with factors \(\mu_i\) and \(\operatorname{Hess}f=\sigma g\), \(\operatorname{Hess}h=\psi g\), then
\[
L_{X_1}g_1(Y_1,Z_1)=2[\lambda+\rho R-\mu_1+n_2\sigma+n_3\psi]g_1(Y_1,Z_1),
\]
\[
L_{X_2}g_2(Y_2,Z_2)=2[(\lambda+\rho R)f-\mu_2+f^\sharp+(n_3\psi)f-fX_1(f)]g_2(Y_2,Z_2),
\]
\[
L_{X_3}g_3(Y_3,Z_3)=2[(\lambda+\rho R)h-\mu_3+h^\sharp-h(X_1+X_2)(h)]g_3(Y_3,Z_3).
\]
Thus each component \(X_i\) is conformal on its factor whenever the right-hand side is constant [2303.01257].

The same framework yields spacetime specializations. For the sequential standard static model
\[
M=(M_1\times_f M_2)\times_h I,
\qquad
g=(g_1\oplus f^2g_2)\oplus h^2(-dt^2),
\]
with \(X=X_1+X_2+w\partial_t\) conformal and \(\operatorname{Hess}f=\sigma g\), \(\operatorname{Hess}h=\psi g\), the Einstein factors of \(M_1\) and \(M_2\) are
\[
\mu_1=-\frac{\Delta h}{h}+n_2\sigma+\psi,
\qquad
\mu_2=-\frac{\Delta h}{h}f+f^\sharp+\psi f.
\]
For the sequential generalized Robertson–Walker model
\[
M=(I\times_f M_2)\times_h M_3,
\qquad
g=(-dt^2\oplus f^2 g_2)\oplus h^2 g_3,
\]
with \(X=w\partial_t+X_2+X_3\) conformal and \(\operatorname{Hess}h=\psi g\), the spatial fibers are Einstein with
\[
\mu_2=\Bigl[-\Bigl(\frac{n_2\dot f}{f}\Bigr)-\Bigl(\frac{n_3\dot h}{h}\Bigr)\Bigr]f+f^\diamond+n_3\psi,
\qquad
\mu_3=\Bigl[-\Bigl(\frac{n_2\dot f}{f}\Bigr)-\Bigl(\frac{n_3\dot h}{h}\Bigr)\Bigr]h+h^\sharp.
\]
These formulas show that, on sequential warped products, conformal RB solitons are controlled by a precise factorwise Einstein decomposition rather than by an undifferentiated global equation [2303.01257].

## 5. Lorentzian Vaidya geometry and the two-dimensional cigar model

On Vaidya spacetime, the conformal RB equation is treated in the form
\[
\mathcal{L}_X g+2S=\kappa g,
\qquad
\kappa=2\beta-\Bigl(p+\tfrac12\Bigr),
\]
because in the paper’s conventions the scalar curvature is
\[
R=0.
\]
For the line element
\[
ds^2=\Bigl(\frac{2m(u)}{r}-1\Bigr)du^2-2\,dr\,du+r^2d\theta^2+r^2\sin^2\theta\,d\phi^2,
\]
the Ricci tensor satisfies
\[
S_{uu}=\frac{2\dot m(u)}{r^2},
\qquad
S_{\mu\nu}=0 \text{ otherwise.}
\]
Writing
\[
X=A\partial_u+B\partial_r+C\partial_\theta+D\partial_\phi,
\]
the component equation reduces to a linear PDE system, whose complete solution is
\[
A=\frac{\kappa u}{2}+\Psi,\qquad B=\frac{\kappa r}{2},\qquad C=0,\qquad D=\psi_3.
\]
The decisive constraint is
\[
m(u)\equiv 0,
\]
which is necessary and sufficient for the soliton to exist. Hence the geometry must reduce to flat Minkowski spacetime in null coordinates. In the gradient case, \(\psi_3=0\) is required, and the potential is
\[
f(u,r)=-\frac{\kappa u}{2}\Bigl(r-\frac{u}{2}\Bigr)-\Psi(r+u)+\Psi_2.
\]
The paper further classifies the soliton by the sign of \(\beta\),
\[
\beta>0:\text{ expanding},\qquad
\beta=0:\text{ steady},\qquad
\beta<0:\text{ shrinking},
\]
and justifies this by linear stability heuristics rather than a spectral analysis [2508.09985].

A low-dimensional counterpart appears in the Cigar Ricci–Bourguignon almost soliton. The metric is
\[
g_{\mathrm{Cigar\text{-}RB}}(t)=\frac{dx^2+dy^2}{E(t)+x^2+y^2},
\qquad
E(t)=e^{4(1-2\rho)t},
\]
with potential
\[
f(x,y,t)=-(1-2\rho)\log\bigl(E(t)+x^2+y^2\bigr),
\qquad
\xi=\nabla f=-2(1-2\rho)\Bigl(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}\Bigr).
\]
It satisfies the steady almost-gradient equation
\[
\operatorname{Ric}+\nabla^2 f-\rho Rg=0.
\]
In polar form,
\[
g=\frac{dr^2+r^2d\theta^2}{E(t)+r^2},
\]
and after the change \(r=\sqrt{E}\sinh s\),
\[
g=ds^2+\psi(s)^2d\theta^2,
\qquad
\psi(s)=\tanh s.
\]
Its Gaussian curvature is
\[
K=\frac{2E}{E+r^2}>0.
\]
A rigidity theorem then states: if \((\Sigma^2,g,f)\) is a complete steady almost gradient RB soliton with \(\rho\neq \tfrac12\), \(\nabla f\) has a zero, and \(K>0\), then \((\Sigma^2,g)\) is isometric, up to homothety, to Hamilton’s Cigar Ricci–Bourguignon soliton [2605.28970].

The same work relates conformal fields to the larger class of mixed Killing fields, defined by
\[
L_VL_Vg=f\,L_Vg.
\]
Every conformal field is mixed Killing, with mixed Killing factor
\[
f=\frac{V(\lambda)}{\lambda}+2\lambda
\]
on \(\{\lambda\neq 0\}\) when \(L_Vg=2\lambda g\). On the cigar RB geometry, the complete conformal algebra is \(4\)-dimensional, spanned by
\[
\partial_x,\quad \partial_y,\quad -y\partial_x+x\partial_y,\quad x\partial_x+y\partial_y,
\]
while the full space of complete mixed Killing fields has dimension \(5\). Angular mixed Killing fields are necessarily rotational Killing, and radial mixed Killing fields
\[
V=w(s)\partial_s
\]
are mixed Killing if and only if
\[
w(s)^2=A\,\tanh^2 s+B,
\]
with conformality exactly when \(B=0\) [2605.28970].

## 6. Terminological scope, adjacent frameworks, and parameter issues

The term “conformal” does not always refer to a conformal potential field. In the mixed super quasi-Einstein literature, “conformal” means conformal curvature tensor and conformal Ricci pseudosymmetry. The Ricci tensor is decomposed as
\[
\operatorname{Ric}(X,Y)=\Psi_1 g(X,Y)+\Psi_2A(X)A(Y)+\Psi_3B(X)B(Y)
+\Psi_4[A(X)B(Y)+B(X)A(Y)]+\Psi_5D(X,Y),
\]
and conformal Ricci pseudosymmetry is the condition
\[
(C(X,Y)\cdot \operatorname{Ric})(Z,W)=F_{\operatorname{Ric}}\,Q(g,\operatorname{Ric})(Z,W;X,Y)
\]
on the set where \(\operatorname{Ric}\neq \tfrac{r}{n}g\). In that setting, an RB soliton is written
\[
2L_Ug+\operatorname{Ric}=(\lambda+\rho r)g.
\]
The main “conformal” consequence is an algebraic restriction on the mixed curvature component \(R(X,Y,\xi_1,\xi_2)\), governed by whether \(\xi_2\) is an eigenvector of the symmetric tensor \(D\) with the distinguished eigenvalue
\[
m=-\frac{R(\xi_2,\xi_1,\xi_1,\xi_2)}{n-2}+D(\xi_2,\xi_2).
\]
If that eigenvalue condition holds, then \(R(X,Y,\xi_1,\xi_2)=0\); otherwise,
\[
R(X,Y,\xi_1,\xi_2)=E(X)A(Y)-E(Y)A(X)
\]
for the explicit \(E\) given in the theorem [2503.11296].

Within the same framework, generator geometry yields strong RB-soliton rigidity. If the integral curves of \(\xi_1\) are geodesic, then \(\Psi_4=0\), so the manifold reduces to a pseudo generalized quasi-Einstein manifold. If \(\xi_1\) is torse-forming,
\[
\nabla_X\xi_1=fX-A(X)\xi_1,
\]
then again \(\Psi_4=0\), and \(\xi_2\) is an eigenvector of \(D\) with eigenvalue \(D(\xi_2,\xi_2)\). In the conharmonically flat case,
\[
r=0,\qquad n\Psi_1+\Psi_2+\Psi_3=0,
\]
and the RB soliton is steady if and only if \(\operatorname{div}\xi_1=0\) [2503.11296].

That paper also records an immediate inference for genuinely conformal potentials, although it does not develop it as a main theorem: if one assumes
\[
L_Ug=2\phi g,
\]
then the equation
\[
2L_Ug+\operatorname{Ric}=(\lambda+\rho r)g
\]
gives
\[
\operatorname{Ric}=(\lambda+\rho r-4\phi)g,
\]
forcing \(\Psi_2=\Psi_3=\Psi_4=\Psi_5=0\). Thus the manifold is Einstein. This aligns with the Einstein reduction found in the standard RB soliton convention, even though the surrounding notion of “conformal” in that paper is curvature-theoretic rather than vector-field-theoretic [2503.11296].

Parameter issues must also be separated carefully. For the flow itself, short-time existence and strict parabolicity are established under
\[
\rho<\frac{1}{2(n-1)}.
\]
This condition is essential in the RB flow analysis and its maximum-principle arguments. By contrast, several conformal soliton papers state results for general \(\rho\in\mathbb{R}\), with no sign restriction on \(\lambda\) or \(\rho\), and often do not use the shrinking/steady/expanding terminology in theorems. In particular, the sequential warped-product analysis imposes no sign restriction on \(\lambda\) or \(\rho\), whereas the Vaidya paper classifies by \(\beta\), not by \(\kappa\), because \(\kappa=2\beta-(p+\tfrac12)\) includes a fixed shift by the conformal pressure term [1507.00324, 2303.01257, 2508.09985].

Taken together, these results show that conformal Ricci–Bourguignon solitons are less a single rigidly normalized equation than a family of closely related Einstein-reduction mechanisms. In the standard compact theory they are trivial as RB solitons and spherical as RB almost solitons; on sequential warped products they force factorwise Einstein geometry with explicit warping corrections; on Vaidya spacetime they exist only in the flat \(m=0\) limit; and in the steady positive-curvature surface setting they collapse, up to homothety, to the cigar model [1809.11103, 2303.01257, 2508.09985, 2605.28970].

Source: https://www.emergentmind.com/topics/conformal-ricci-bourguignon-soliton