Modified Ricci Almost Solitons

• Modified Ricci almost solitons are generalized Einstein-type structures defined by the standard Ricci soliton equation enhanced with an additional quadratic 1-form term.
• They unify classical Ricci solitons and n-quasi-Einstein metrics, emerging naturally from warped-product constructions and the modified Ricci-harmonic flow.
• This framework applies to diverse geometric settings—including almost contact and B-metric scenarios—offering insights into rigidity, spectral obstructions, and local structure.

Searching arXiv for papers on modified Ricci almost solitons and closely related generalized soliton structures. arXiv search query: "modified Ricci almost soliton quasi-Einstein Ricci almost soliton". Modified Ricci almost solitons form a family of generalized Einstein-type structures in which the Ricci tensor is balanced not only by a Lie-derivative or Hessian term, as in Ricci soliton theory, but also by an additional quadratic term built from a $1$-form. In the formulation introduced as a distinct class of metrics, a modified Ricci almost soliton is a Riemannian manifold $(M^m,g)$ with a vector field $X$ and a $1$-form $\omega$ such that

$$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$

where $\lambda$ is a smooth function in the almost case and $\theta>0$ is a real parameter; in the gradient case one writes $X=\nabla\eta$ and $\omega=d\xi$, so that

$(M^m,g)$0

This framework contains Ricci solitons when $(M^m,g)$1, is closely related to $(M^m,g)$2-quasi-Einstein metrics, and arises naturally from warped-product constructions and from a modified Ricci-harmonic flow (Filho, 29 Sep 2025).

1. Definitions and terminological scope

The expression “modified Ricci almost soliton” does not occur with a single universal normalization across the literature represented here. In the narrow sense just described, the modification is the extra $(M^m,g)$3 term. In contact and almost contact settings, however, closely related structures are often written with $(M^m,g)$4 or with both $(M^m,g)$5 and $(M^m,g)$6 terms. The basic forms used in the cited works are as follows (Filho, 29 Sep 2025, Blaga, 2017, Manev, 2020).

Variant	Defining equation	Context
Modified Ricci almost soliton	$(M^m,g)$7	Riemannian Einstein-type metric
Almost $(M^m,g)$8-Ricci soliton	$(M^m,g)$9	$X$0 and Kenmotsu geometry
Ricci-like / almost Ricci-like soliton	$X$1	Almost contact B-metric geometry

In all of these variants, the adjective “almost” means that the coefficient corresponding to the soliton constant is allowed to be a smooth function rather than a constant. That convention is inherited from almost Ricci soliton theory. What changes from one subliterature to another is the tensorial form of the “modification.” On $X$2-manifolds, the term “modified Ricci almost solitons” is explicitly identified with almost $X$3-Ricci solitons, namely with the addition of the $X$4 term (Blaga, 2017). On Sasaki-like almost contact B-metric manifolds, Ricci-like solitons are presented as a further generalization, since the associated B-metric $X$5 also enters the equation (Manev, 2020).

A recurrent source of confusion is therefore terminological rather than geometric. The common core is the replacement of the classical Ricci soliton equation by a nonconstant soliton coefficient together with an additional structure-adapted symmetric $X$6-tensor.

2. Relation to quasi-Einstein metrics, warped products, and flows

The modified Ricci soliton class was introduced as a unifying Einstein-type class containing both Ricci solitons and $X$7-quasi-Einstein metrics. The underlying motivation is the structure of Ricci solitons realized as warped products: on the base, one obtains an equation of the form

$X$8

which can be rewritten in the style of a modified Ricci soliton with an additional quadratic term in a differential. Likewise, the standard $X$9-quasi-Einstein equation

$1$0

fits this framework after a suitable identification of variables (Filho, 29 Sep 2025).

The same work places modified Ricci almost solitons inside a flow-theoretic picture. A modified Ricci soliton appears as part of a special solution of the modified Ricci-harmonic flow

$1$1

where $1$2. The corresponding modified Ricci-harmonic solitons satisfy the modified Ricci soliton equation together with

$1$3

This formulation shifts the subject away from a purely static generalization of Ricci solitons and toward a self-similar-solution theory for coupled geometric flows.

Concrete examples reinforce the breadth of the class. Berger spheres provide nontrivial modified Ricci soliton structures, and an explicit modified Ricci-harmonic soliton is constructed on a warped product of the form $1$4 (Filho, 29 Sep 2025). A plausible implication is that the modified theory is best understood not as a minor perturbation of Ricci soliton theory, but as a bridge between solitons, Ricci-Hessian equations, and warped-product Einstein geometry.

3. Compact rigidity and spectral obstructions

In the compact Riemannian setting with constant scalar curvature, modified Ricci almost solitons exhibit a strong rigidity comparable to classical Lichnerowicz-Obata phenomena. If $1$5 is a compact modified Ricci almost soliton with constant scalar curvature $1$6 and if $1$7 is an eigenfunction of the Laplacian with eigenvalue $1$8, then

$1$9

with equality if and only if $\omega$0 is isometric to the standard sphere $\omega$1 of radius

$\omega$2

In the equality case, the structure is necessarily gradient up to rescaling and a constant (Filho, 29 Sep 2025).

The same source gives the explicit sphere data. For $\omega$3, any gradient modified Ricci almost soliton with conformal $\omega$4 has

$\omega$5

$\omega$6

$\omega$7

where $\omega$8 is a height function. Thus the rigid case is not only classified up to isometry; its potentials are explicitly described.

An integral identity underlies this rigidity: $\omega$9 Under constant scalar curvature, this collapses toward the vanishing of the traceless Ricci tensor in the rigid case. There is also an obstruction statement: if $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$0, then $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$1 and $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$2 is Killing, so the manifold is Einstein and the modified soliton is trivial. Complementing rigidity, a Myers-type compactness criterion is obtained under the inequality

$$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$3

together with boundedness assumptions on $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$4 and a positive lower bound along geodesics (Filho, 29 Sep 2025).

4. Four-dimensional and pseudo-Riemannian local structure

Although much of the modified Ricci almost soliton literature is Riemannian, several decisive structural results come from four-dimensional pseudo-Riemannian almost soliton theory. For four-dimensional half conformally flat proper gradient Ricci almost solitons, the geometry splits according to the causal character of $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$5. If $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$6, then $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$7 is locally isometric to a warped product $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$8, with $$\mathrm{Ric}+\frac{1}{2}\mathcal{L}_X g=\lambda g+\theta\,\omega\otimes\omega,$$9 of constant sectional curvature, and the metric is in fact locally conformally flat. If $\lambda$0, then the soliton is locally realized on the cotangent bundle $\lambda$1 of an affine surface $\lambda$2 endowed with a modified Riemannian extension; these examples are self-dual, not locally conformally flat, and are steady traceless $\lambda$3-Einstein solitons (Brozos-Vázquez et al., 2016).

The null-gradient case is especially important because it shows that half conformal flatness alone does not force local conformal flatness. The local metric is encoded by affine-surface data, the potential has the form $\lambda$4, and the soliton function satisfies $\lambda$5 in the construction. This sharply separates the non-null rigid regime from the null flexible regime.

A related dichotomy appears for four-dimensional half-conformally flat gradient $\lambda$6-almost Ricci solitons in Lorentzian and neutral signature. If $\lambda$7 is a non-zero constant, then $\lambda$8 is locally a warped product $\lambda$9, where $\theta>0$0 is Einstein and in fact of constant sectional curvature. If $\theta>0$1, then $\theta>0$2 is locally a Walker manifold. In neutral signature, explicit models have local metric

$\theta>0$3

The same work relates gradient almost Ricci solitons on standard static spacetimes to Ricci-Hessian class type equations on the spatial fiber (Güler, 2020).

These results matter for modified Ricci almost solitons because they identify the geometric mechanisms through which an “almost” soliton equation can either collapse to a warped-product classification or open a genuinely కొత్త local geometry. The decisive distinction is not merely curvature pinching, but the isotropic versus non-isotropic behavior of the potential gradient.

5. Contact, almost contact, and Kenmotsu realizations

In Lorentzian concircular geometry, an almost $\theta>0$4-Ricci soliton is a quadruple $\theta>0$5 satisfying

$\theta>0$6

and in the gradient case this becomes

$\theta>0$7

On $\theta>0$8-manifolds, sharp lower and upper bounds are obtained for $\theta>0$9 in the gradient case, and a Bochner-type formula is established: $X=\nabla\eta$0 These formulas are used to deduce rigidity consequences and to control the scalar curvature under curvature conditions such as $X=\nabla\eta$1 and $X=\nabla\eta$2 (Blaga, 2017).

On Kenmotsu manifolds, almost $X=\nabla\eta$3-Ricci solitons are strongly constrained. If a Kenmotsu metric is an $X=\nabla\eta$4-Ricci soliton and is either $X=\nabla\eta$5-Einstein, or has potential vector field an infinitesimal contact transformation, or has potential vector field collinear to the Reeb vector field, then the metric is Einstein with scalar curvature

$X=\nabla\eta$6

For a gradient almost $X=\nabla\eta$7-Ricci soliton, the condition $X=\nabla\eta$8 again forces the manifold to be Einstein with the same scalar curvature. In dimension three, the Einstein case corresponds to constant sectional curvature $X=\nabla\eta$9 (Patra et al., 2020).

A related but distinct generalization is the Ricci $\omega=d\xi$0-soliton, motivated by the Ricci-Bourguignon flow

$\omega=d\xi$1

On a $\omega=d\xi$2-dimensional almost Kenmotsu $\omega=d\xi$3-Einstein manifold, if the metric admits a Ricci $\omega=d\xi$4-soliton, then the manifold is Kenmotsu of constant sectional curvature $\omega=d\xi$5 and the soliton is expanding with $\omega=d\xi$6 (Azami et al., 2019). The same paper explicitly notes that the presence of the $\omega=d\xi$7 parameter does not introduce new geometries in that setting.

D-homothetically deformed Kenmotsu manifolds provide another structured setting. There one studies almost Ricci and almost Riemann solitons with gradient, solenoidal, or Reeb potential vector fields, obtains explicit formulas for the deformed Ricci and scalar curvatures, and derives lower bounds for the Ricci curvature of the initial Kenmotsu manifold when the deformed manifold admits a gradient almost Riemann or almost Ricci soliton (Blaga, 2020).

6. B-metric extensions, adjacent modifications, and recurrent rigidity patterns

On almost contact B-metric manifolds, the soliton equation is enlarged further to the Ricci-like form

$\omega=d\xi$8

or, in the gradient almost case,

$\omega=d\xi$9

For Sasaki-like almost contact B-metric manifolds with potential Reeb vector field, existence of a Ricci-like soliton is equivalent to the Einstein-like condition

$(M^m,g)$00

with $(M^m,g)$01 and $(M^m,g)$02 (Manev, 2019). When the potential is arbitrary, the Ricci tensor is forced to be purely vertical,

$(M^m,g)$03

and both scalar curvatures $(M^m,g)$04 are constant and equal to $(M^m,g)$05; in the gradient almost Ricci-like case the soliton coefficients are necessarily constant (Manev, 2020).

If the potential is vertical, $(M^m,g)$06, then $(M^m,g)$07 must be constant and one has

$(M^m,g)$08

Thus the manifold is $(M^m,g)$09-Einstein, and a broad list of additional curvature properties—Ricci cyclic parallel, Codazzi type, Ricci $(M^m,g)$10-symmetric, almost pseudo Ricci symmetric, special weakly Ricci symmetric—force the Einstein condition (Manev, 2020). Explicit Lie-group models in dimensions $(M^m,g)$11 and $(M^m,g)$12 realize these conclusions.

An adjacent modification is obtained by replacing the Ricci tensor with the $(M^m,g)$13-Ricci tensor. On Kenmotsu manifolds, if the metric is a $(M^m,g)$14-Ricci soliton, then the soliton constant is necessarily $(M^m,g)$15; in dimension $(M^m,g)$16, the manifold has constant sectional curvature $(M^m,g)$17. For gradient almost $(M^m,g)$18-Ricci solitons, either the manifold is Einstein or the potential vector field is collinear with the characteristic vector field $(M^m,g)$19 on an open set (Venkatesh et al., 2019).

Across these variants, a stable pattern emerges. The addition of $(M^m,g)$20, $(M^m,g)$21, $(M^m,g)$22, or $(M^m,g)$23-Ricci terms enlarges the formal class of equations, but many natural geometric settings remain rigid: compact constant-scalar-curvature metrics tend toward spheres, Kenmotsu and almost Kenmotsu structures tend toward Einstein or constant-curvature models, and Sasaki-like B-metric manifolds force vertical Ricci tensors or constant coefficients. The principal exceptions occur in null or isotropic regimes, especially in four-dimensional pseudo-Riemannian geometry, where Walker structures and modified Riemannian extensions produce genuinely nontrivial local models.