Modified Ricci Solitons in Geometry

• Modified Ricci solitons are Einstein-type metrics on complete Riemannian manifolds that incorporate an extra tensor term, generalizing Ricci solitons, n-quasi-Einstein metrics, and almost solitons.
• They arise naturally from warped product constructions and conformal reductions, providing deep insights into flow dynamics and rigidity phenomena through modified Ricci-harmonic flows.
• The theory offers explicit models, eigenvalue bounds, and rigidity results, while extending the framework to include flux-coupled and higher-order curvature modifications.

Modified Ricci solitons are a class of Einstein type metrics defined on a complete Riemannian manifold $(M^m,g)$ endowed with a vector field $X$ and a $1$-form $\omega$ by

$$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$

for real $\lambda$ and $\theta>0$. In the gradient case, $X=\nabla\eta$ and $\omega=d\xi$, so the equation becomes

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$

This framework contains Ricci solitons and $X$0-quasi-Einstein metrics, is closely related to warped product constructions, and arises as the self-similar sector of a modified Ricci-harmonic flow (Filho, 29 Sep 2025).

1. Defining equation and basic variants

The defining modified Ricci soliton equation is

$X$1

Its gradient form is

$X$2

If $X$3 is allowed to be a nonconstant function, the structure is called a modified Ricci almost soliton (Filho, 29 Sep 2025).

The same source places modified Ricci solitons in direct relation with two established classes. Ricci solitons are recovered by setting $X$4 and $X$5, so that

$X$6

The $X$7-quasi-Einstein equation is

$X$8

and is presented there as a special case of the modified Ricci soliton equation. In parallel, the theory of Ricci almost solitons replaces the constant soliton parameter in the classical gradient Ricci soliton equation by a smooth function: $X$9 That construction generalizes gradient Ricci solitons and introduces the shrinking, steady, expanding, and indefinite cases according to the sign behavior of $1$0 (Pigola et al., 2010).

Structure	Equation	Relation stated in the literature
Modified Ricci soliton	$1$1	General class
Gradient modified Ricci soliton	$1$2	Gradient case
Ricci soliton	$1$3	Obtained when $1$4, $1$5
$1$6-quasi-Einstein metric	$1$7	Presented as a special case
Ricci almost soliton	$1$8	Variable soliton function

A recurrent point of terminology is that “modified,” “almost,” and “generalized” do not denote the same alteration. The data indicate that modified Ricci solitons add a tensor term $1$9, whereas almost solitons replace the constant coefficient by a function. This suggests terminological caution when comparing results across subliteratures.

2. Warped products, quasi-Einstein metrics, and conformal reductions

A principal structural motivation for modified Ricci solitons is their relation to warped products. For a Ricci soliton realized as a warped product $\omega$0, the metric on the base $\omega$1 satisfies

$\omega$2

Letting $\omega$3, one has

$\omega$4

and therefore

$\omega$5

In this way, the base metric of a warped product Ricci soliton becomes a modified Ricci soliton with $\omega$6 and $\omega$7 (Filho, 29 Sep 2025).

The same warped-product viewpoint interacts with the quasi-soliton formalism. If $\omega$8 is a warped product gradient Ricci soliton, then the base metric $\omega$9 satisfies

$$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$0

with $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$1. Under the functional dependence assumption $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$2, such a base metric is called a special quasi-soliton. For Kähler special quasi-solitons of real dimension at least four, the metric satisfies a Ricci-Hessian equation on an open set, and if that equation is standard then the metric is a Riemannian product; when $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$3, one factor is a Kähler-Einstein manifold of codimension $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$4 (Maschler, 2015).

Conformal reductions furnish an analogous route. If $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$5 is a gradient Ricci soliton conformal to a Kähler metric $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$6, then, in terms of $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$7,

$$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$8

with the key hypothesis $$\operatorname{Ric}+\frac12\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$9. In dimension $\lambda$0, the partial classification obtained under this hypothesis yields four possibilities on an open dense set: $\lambda$1 is a Kähler-Ricci soliton; $\lambda$2 satisfies a Ricci-Hessian equation; $\lambda$3 and $\lambda$4 is Einstein; or $\lambda$5 and $\lambda$6 is a non-Einstein steady gradient Ricci soliton (Maschler, 2015).

These constructions show that modified Ricci solitons are not merely formal perturbations of the Ricci soliton equation. They encode the effective geometry seen on the base of a warped product and, in the conformal-Kähler setting, reduce naturally to Ricci-Hessian type equations.

3. Flow interpretation and explicit models

Modified Ricci solitons appear as special solutions of the modified Ricci-harmonic flow

$\lambda$7

where $\lambda$8 is a family of maps and $\lambda$9 for a fixed $\theta>0$0-form $\theta>0$1 on $\theta>0$2. A modified Ricci-harmonic soliton is a triple $\theta>0$3 satisfying

$\theta>0$4

and the self-similar solutions evolve by

$\theta>0$5

This gives a direct dynamical interpretation of the modified Ricci soliton equation (Filho, 29 Sep 2025).

An explicit example is given by

$\theta>0$6

with $\theta>0$7 endowed with the Euclidean metric. Choosing

$\theta>0$8

one verifies

$\theta>0$9

Hence $X=\nabla\eta$0 realizes a modified Ricci-harmonic soliton and, after suitable scaling and diffeomorphism, a special solution to the modified Ricci-harmonic flow (Filho, 29 Sep 2025).

A related coupled theory is the harmonic-Ricci soliton system,

$X=\nabla\eta$1

with gradient version

$X=\nabla\eta$2

In that setting, the modified curvature quantities are the $X=\nabla\eta$3-Ricci tensor $X=\nabla\eta$4, the $X=\nabla\eta$5-scalar curvature $X=\nabla\eta$6, and the associated $X=\nabla\eta$7-Schouten, $X=\nabla\eta$8-Cotton, $X=\nabla\eta$9-Weyl, and $\omega=d\xi$0-Bach tensors (Anselli, 2020).

4. Rigidity, eigenvalue bounds, and dynamical stability

The compact theory includes a rigidity theorem in the spirit of Lichnerowicz and Obata. If $\omega=d\xi$1, $\omega=d\xi$2, is a compact modified Ricci almost soliton with constant scalar curvature $\omega=d\xi$3, and if $\omega=d\xi$4 is an eigenfunction of the Laplacian with eigenvalue $\omega=d\xi$5,

$\omega=d\xi$6

then

$\omega=d\xi$7

Equality holds if and only if $\omega=d\xi$8 is isometric to the standard sphere $\omega=d\xi$9, where

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$0

and the structure is gradient, up to rescaling and addition of a constant (Filho, 29 Sep 2025).

The broader almost-soliton literature provides analogous rigidity and isolation phenomena. For Ricci almost solitons,

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$1

the weighted Laplacian

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$2

and the Bakry-Émery Ricci tensor

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$3

play a central role. The theory derives differential identities such as

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$4

together with existence, rigidity, a-priori curvature estimates, isolation phenomena, and topological consequences under weighted maximum principles and volume comparison (Pigola et al., 2010).

Dynamical stability is developed for a curvature-normalized Ricci flow

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$5

whose stationary solutions satisfy

$$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$6

Using weighted little Hölder spaces, maximal regularity theory, and Simonett’s theorem, strict linear stability implies dynamical stability for a class of Einstein metrics and for a class of non-Einstein Ricci solitons, with many explicit examples on simply connected solvable Lie groups (Williams et al., 2013). In this sense, rigidity and stability enter the modified-soliton landscape both through elliptic characterizations and through normalized flow dynamics.

5. Flux-coupled and higher-order modifications

One major extension of the Ricci soliton paradigm introduces a closed $$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$7-form $$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$8. Generalized Ricci solitons are pairs $$\operatorname{Ric}+\nabla d\eta=\lambda g+\theta\,d\xi\otimes d\xi.$$9 satisfying

$X$00

The steady, gradient, $X$01-invariant case on $X$02 reduces to the ansatz

$X$03

and yields the ODE system

$X$04

For each $X$05 below the threshold $X$06, the resulting local solution extends to a complete, positively curved, $X$07-invariant, steady generalized Ricci soliton of gradient type, and varying $X$08 gives a one-parameter family of pairwise non-isometric metrics (Podestà et al., 2024).

The Heterotic-Ricci flow introduces both a $X$09-form $X$10 and a non-negative parameter $X$11: $X$12 When $X$13, it reduces to the generalized Ricci flow; when $X$14 and $X$15, it reduces to a constrained version of the RG-2 flow. The corresponding Heterotic soliton equations involve $X$16, $X$17, the curvature square term $X$18, and the Bianchi identity. In dimension three, compact strong Heterotic solitons are classified, up to finite cover, as hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a left-invariant metric; moreover, all Einstein three-dimensional Heterotic solitons have constant dilaton, and Einstein Heterotic solitons with constant dilaton are rigid (Moroianu et al., 2023).

These theories extend the modified-soliton idea beyond an added $X$19 term. They incorporate torsion, flux, and higher-curvature corrections while preserving the central soliton principle: stationary or self-similar behavior for a geometric flow after allowing diffeomorphisms and, in some settings, scaling.

6. Special geometries, Lorentzian models, and curvature-modified classifications

On Sasaki-like almost contact B-metric manifolds, a Ricci-like soliton with arbitrary potential vector field $X$20 is defined by

$X$21

For a Sasaki-like manifold of dimension $X$22, any such soliton with constants $X$23 must satisfy

$X$24

and its Ricci tensor is

$X$25

The scalar curvatures relative to $X$26 and $X$27 are both constant and equal to $X$28. In the gradient almost Ricci-like case,

$X$29

the soliton coefficients are proved to be constant, and explicit $X$30- and $X$31-dimensional Lie group examples are exhibited (Manev, 2020).

A Lorentzian counterpart appears in the study of conformal Ricci-Bourguignon solitons on Vaidya spacetime. There the soliton equation is

$X$32

and the Vaidya metric in Eddington-Finkelstein coordinates is

$X$33

With $X$34 and $X$35, the admissible vector field coefficients reduce to

$X$36

where $X$37. In the gradient case, when $X$38,

$X$39

The decisive rigidity result is that the PDE system is compatible only when the mass function vanishes, $X$40, so such solitons exist only in flat Minkowski spacetime; no genuinely radiating Vaidya spacetime admits them (Rehman et al., 13 Aug 2025).

A different curvature modification appears in four-dimensional gradient shrinking Ricci solitons through the modified curvature tensor

$X$41

and the associated modified sectional curvature $X$42. In dimension four, if

$X$43

then the soliton is locally Kähler. For compact solitons with lower bounds on $X$44 and $X$45, the gap theorems stated in the source force the manifold to be isometric to $X$46 or $X$47, and a Hitchin-Thorpe type inequality is obtained under positive modified sectional curvature (Cao et al., 25 Sep 2025).

Across these examples, the common theme is that a “modified” Ricci soliton equation typically acquires extra tensorial data—$X$48-forms, fluxes, torsion, conformal terms, auxiliary metrics, or curvature corrections—and that the resulting flexibility is often sharply constrained by rigidity, classification, or stability theorems.