Conformal Ricci–Bourguignon Solitons

• Conformal Ricci–Bourguignon solitons are defined by a modified RB flow where a conformal vector field deforms the metric, reducing the equation to an Einstein-type condition.
• They appear in diverse contexts such as compact manifolds, sequential warped products, and Lorentzian spacetimes, illustrating rigidity and sphere theorems.
• The research reconciles multiple normalization conventions, highlighting the interplay of conformal fields, gradient potentials, and deformation parameters in establishing rigidity.

Searching arXiv for the cited papers to ground the article in the current literature. I’m checking arXiv records relevant to conformal Ricci–Bourguignon solitons and closely related Ricci–Bourguignon soliton literature. 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 (Catino et al., 2015, Dwivedi, 2018, Kaya et al., 2023, Rehman et al., 13 Aug 2025).

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 $\lambda$ (Catino et al., 2015).

A conformal vector field $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$0 is defined by

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$1

and, in the gradient setting, a conformal associated vector field means

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$2

Allowing $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$3 to vary produces an RB almost soliton,

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$4

with gradient form

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$5

This extension is central in compact rigidity results because conformality then yields

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$6

rather than the constant-$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$7 reduction alone (Dwivedi, 2018).

Several later papers use different normalizations. On sequential warped products, the soliton equation is written

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$8

with gradient form

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$9

That paper also explicitly notes a convention change: its Ricci soliton definition uses $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$0, whereas its Ricci–Bourguignon soliton uses $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$1. In the conformal case, the authors assume $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$2 has factor $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$3, with $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$4 constant, and then use the reduced balance

$$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$5

in their component calculations (Kaya et al., 2023).

Other variants enlarge the framework further. The $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$6-almost Ricci–Bourguignon equation is

$$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$7

or, in the gradient case,

$$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$8

Under conformality, $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$9, it reduces to

$X$0

(Bousso et al., 4 May 2025).

The Vaidya analysis adopts yet another convention, drawn from conformal Ricci soliton and conformal RB soliton literature: $X$1 For $X$2 and $X$3, the paper sets

$X$4

Accordingly, the cited works do not present a single universal normalization for the phrase “conformal Ricci–Bourguignon soliton” (Rehman et al., 13 Aug 2025).

2. Einstein reduction under conformality

In the standard RB soliton convention, substituting

$X$5

into

$X$6

gives

$X$7

Tracing the soliton equation yields

$X$8

Eliminating $X$9 gives the rigidity identity

$\mathcal{L}_X g=2\psi g$0

Thus every conformal Ricci–Bourguignon soliton is Einstein in this normalization, and for $\mathcal{L}_X g=2\psi g$1 the scalar curvature is constant. Consequently, $\mathcal{L}_X g=2\psi g$2 is also constant by the traced relation (Catino et al., 2015).

This reduction remains the basic mechanism in later compact results. In the almost-soliton setting,

$\mathcal{L}_X g=2\psi g$3

while in the gradient conformal case the pure-trace condition is

$\mathcal{L}_X g=2\psi g$4

Tracing then gives

$\mathcal{L}_X g=2\psi g$5

and hence

$\mathcal{L}_X g=2\psi g$6

The same paper develops identities for $\mathcal{L}_X g=2\psi g$7, 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 (Dwivedi, 2018).

On compact manifolds the conformal reduction is even stronger. Since $\mathcal{L}_X g=2\psi g$8, integrating over a compact manifold gives $\mathcal{L}_X g=2\psi g$9, so the conformal field is actually Killing. In the standard soliton case this implies that a compact conformal RB soliton is trivial, with

$(M^n,g,X,\lambda)$0

The noncompact statement in the same source is also rigid: there is no nontrivial complete noncompact RB soliton with conformal vector field $(M^n,g,X,\lambda)$1. In the gradient case, $(M^n,g,X,\lambda)$2 with $(M^n,g,X,\lambda)$3 constant forces Euclidean geometry by Tashiro’s theorem, contradicting nontriviality unless $(M^n,g,X,\lambda)$4 (Dwivedi, 2018).

The $(M^n,g,X,\lambda)$5-almost theory preserves the same pattern. Once $(M^n,g,X,\lambda)$6, the equation

$(M^n,g,X,\lambda)$7

again makes the Ricci tensor pointwise proportional to the metric: $(M^n,g,X,\lambda)$8 A plausible implication is that conformality remains a mechanism for forcing Einstein structure even when the deformation term is weighted by a nonconstant factor $(M^n,g,X,\lambda)$9 (Bousso et al., 4 May 2025).

3. Compact rigidity, integral identities, and sphere theorems

The compact theory of conformal RB almost solitons is dominated by sphere rigidity. If $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$0, $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$1, is a compact RB almost soliton and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$2 is a nontrivial conformal vector field, then $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$3 is isometric to a Euclidean sphere. The proof runs through the Einstein-type reduction

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$4

constancy of $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$5 and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$6, and a Yano characterization of constant sectional curvature. Compactness and nontriviality of the conformal factor force the round sphere (Dwivedi, 2018).

In the compact gradient setting the decisive inputs are the integral identities

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$7

and

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$8

If $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$9 is constant, the right-hand side vanishes and the metric is Einstein. Plugging this back into the almost-soliton equation gives

$$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$0

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: $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$1 is constant; $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$2; or $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$3 is homogeneous (Dwivedi, 2018).

The compact $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$4-almost extension produces weighted analogues of these formulas. A key identity is

$$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$5

When $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$6 is conformal, the Kazdan–Warner identity on compact manifolds,

$$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$7

annihilates the right-hand side and forces the traceless Ricci tensor to vanish. The resulting sphere theorem states that a compact $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$8-almost Ricci–Bourguignon soliton with $$\operatorname{Ric}-\rho\,R\,g+\nabla^2 f=\lambda g.$$9 is isometric to a standard sphere if either

$$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$0

or $$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$1 is a nontrivial conformal vector field. Constant scalar curvature also yields Einstein structure and then sphere rigidity by the same mechanism (Bousso et al., 4 May 2025).

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 $$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$2-almost frameworks, and even there the global outcome is usually round-sphere rigidity rather than a broader moduli space (Dwivedi, 2018, Bousso et al., 4 May 2025).

4. Sequential warped products and spacetime specializations

A detailed conformal analysis is available for sequential warped products

$$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$3

where $$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$4 and $$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$5. The Ricci tensor decomposes by factors: $$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$6

$$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$7

$$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$8

with mixed components zero, and

$$\operatorname{div}X+(1-n\rho)R=n\lambda, \qquad \Delta f+(1-n\rho)R=n\lambda.$$9

If $\lambda$0, then $\lambda$1 splits into factor terms plus the warping contributions $\lambda$2 and $\lambda$3 (Kaya et al., 2023).

In the conformal case, that paper assumes $\lambda$4 is conformal with factor $\lambda$5, with $\lambda$6 constant, and uses

$\lambda$7

Under the hypotheses

$\lambda$8

the factor equations become

$\lambda$9

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$00

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$01

Hence $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$02, $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$03, and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$04 are Einstein under the stated hypotheses, with Einstein factors given by the bracketed coefficients. If $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$05 and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$06 are constant, then $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$07 and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$08, so the total space reduces to a scaled product Einstein space (Kaya et al., 2023).

The converse direction is also explicit. If $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$09 are Einstein with factors $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$10 and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$11, $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$12, then

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$13

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$14

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$15

Thus each component $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$16 is conformal on its factor whenever the right-hand side is constant (Kaya et al., 2023).

The same framework yields spacetime specializations. For the sequential standard static model

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$17

with $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$18 conformal and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$19, $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$20, the Einstein factors of $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$21 and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$22 are

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$23

For the sequential generalized Robertson–Walker model

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$24

with $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$25 conformal and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$26, the spatial fibers are Einstein with

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$27

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 (Kaya et al., 2023).

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

On Vaidya spacetime, the conformal RB equation is treated in the form

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$28

because in the paper’s conventions the scalar curvature is

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$29

For the line element

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$30

the Ricci tensor satisfies

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$31

Writing

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$32

the component equation reduces to a linear PDE system, whose complete solution is

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$33

The decisive constraint is

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$34

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, $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$35 is required, and the potential is

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$36

The paper further classifies the soliton by the sign of $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$37,

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$38

and justifies this by linear stability heuristics rather than a spectral analysis (Rehman et al., 13 Aug 2025).

A low-dimensional counterpart appears in the Cigar Ricci–Bourguignon almost soliton. The metric is

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$39

with potential

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$40

It satisfies the steady almost-gradient equation

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$41

In polar form,

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$42

and after the change $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$43,

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$44

Its Gaussian curvature is

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$45

A rigidity theorem then states: if $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$46 is a complete steady almost gradient RB soliton with $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$47, $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$48 has a zero, and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$49, then $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$50 is isometric, up to homothety, to Hamilton’s Cigar Ricci–Bourguignon soliton (Aqib et al., 27 May 2026).

The same work relates conformal fields to the larger class of mixed Killing fields, defined by

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$51

Every conformal field is mixed Killing, with mixed Killing factor

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$52

on $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$53 when $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$54. On the cigar RB geometry, the complete conformal algebra is $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$55-dimensional, spanned by

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$56

while the full space of complete mixed Killing fields has dimension $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$57. Angular mixed Killing fields are necessarily rotational Killing, and radial mixed Killing fields

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$58

are mixed Killing if and only if

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$59

with conformality exactly when $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$60 (Aqib et al., 27 May 2026).

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

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$61

and conformal Ricci pseudosymmetry is the condition

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$62

on the set where $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$63. In that setting, an RB soliton is written

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$64

The main “conformal” consequence is an algebraic restriction on the mixed curvature component $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$65, governed by whether $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$66 is an eigenvector of the symmetric tensor $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$67 with the distinguished eigenvalue

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$68

If that eigenvalue condition holds, then $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$69; otherwise,

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$70

for the explicit $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$71 given in the theorem (Yan et al., 14 Mar 2025).

Within the same framework, generator geometry yields strong RB-soliton rigidity. If the integral curves of $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$72 are geodesic, then $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$73, so the manifold reduces to a pseudo generalized quasi-Einstein manifold. If $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$74 is torse-forming,

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$75

then again $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$76, and $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$77 is an eigenvector of $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$78 with eigenvalue $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$79. In the conharmonically flat case,

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$80

and the RB soliton is steady if and only if $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$81 (Yan et al., 14 Mar 2025).

That paper also records an immediate inference for genuinely conformal potentials, although it does not develop it as a main theorem: if one assumes

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$82

then the equation

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$83

gives

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$84

forcing $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$85. 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 (Yan et al., 14 Mar 2025).

Parameter issues must also be separated carefully. For the flow itself, short-time existence and strict parabolicity are established under

$$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$86

This condition is essential in the RB flow analysis and its maximum-principle arguments. By contrast, several conformal soliton papers state results for general $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$87, with no sign restriction on $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$88 or $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$89, and often do not use the shrinking/steady/expanding terminology in theorems. In particular, the sequential warped-product analysis imposes no sign restriction on $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$90 or $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$91, whereas the Vaidya paper classifies by $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$92, not by $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$93, because $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$94 includes a fixed shift by the conformal pressure term (Catino et al., 2015, Kaya et al., 2023, Rehman et al., 13 Aug 2025).

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 $$\operatorname{Ric}-\rho\,R\,g+\tfrac12\mathcal{L}_X g=\lambda g,$$95 limit; and in the steady positive-curvature surface setting they collapse, up to homothety, to the cigar model (Dwivedi, 2018, Kaya et al., 2023, Rehman et al., 13 Aug 2025, Aqib et al., 27 May 2026).