The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.
problem Characterizing almost Ricci solitons on Finsler measure spaces.
method Introducing and investigating gradient almost Ricci solitons, proving conditions for existence.
result Conditions for the existence of gradient almost Ricci solitons on Finsler measure spaces.
The study classifies h-almost Ricci-Yamabe solitons in various paracontact manifolds.
problem Classifying h-almost Ricci-Yamabe solitons in paracontact geometry.
method Characterization and classification of para-Kenmotsu, para-Sasakian, and para-cosymplectic manifolds.
result Characterizations and classifications of various paracontact manifolds.
The paper investigates geometrical aspects of static spacetime with almost gradient Ricci solitons.
problem Geometrical properties of static spacetime with almost gradient Ricci solitons.
method Analyzing conditions and properties of static spacetime with almost gradient Ricci solitons.
result Conditions and properties of static spacetime with almost gradient Ricci solitons are determined.
Study on Ricci-Bourguignon solitons and almost solitons, generalizing previous results.
problem Analyzing solitons and almost solitons in the context of Ricci-Bourguignon flow.
method Generalizing results for Ricci solitons and introducing Ricci-Bourguignon almost solitons, proving results and deriving integral formulas.
result Compact gradient Ricci-Bourguignon almost solitons with constant scalar curvature or conformal associated vector field are isometric to Euclidean spheres.
Study k-almost Ricci solitons on contact metric manifolds.
problem Characterize k-almost Ricci solitons on contact metric manifolds.
method Prove isometric properties and extend results for k-almost gradient Ricci solitons and k-almost Ricci solitons.
result Compact K-contact metric manifolds that are k-almost gradient Ricci solitons are isometric to a unit sphere.
The paper identifies invariant solutions for gradient Ricci almost solitons.
problem Understanding invariant solutions for gradient Ricci almost solitons.
method Analyzing conformal metrics invariant under translation and rotation.
result Identified all conformal metrics invariant under translation and rotation as gradient Ricci almost solitons.
Paper studies equations for a type of soliton with applications.
problem Analyzing a specific type of soliton structure.
method Analytic techniques to establish elliptic and integral equations.
result Established basic structural equations for almost Ricci-harmonic solitons.
Study on Ricci-like solitons and gradient solitons on specific manifolds.
problem Characterizing solitons on Sasaki-like almost contact B-metric manifolds.
method Introduced and studied Ricci-like solitons with arbitrary potential and gradient solitons. Proved properties of the Ricci tensor and soliton coefficients.
result Gradient almost Ricci-like solitons have constant soliton coefficients.
Study on non-gradient Ricci almost solitons in warped products.
problem Understanding non-gradient Ricci almost solitons.
method Construction method and explicit example in warped products.
result Rigidity result for Gaussian soliton.
New volume comparison theorem for gradient Ricci almost solitons.
problem Volume comparison and rigidity of gradient Ricci almost solitons.
method Established a new volume comparison theorem with Bakry-Emery Ricci curvature.
result New volume rigidity result for gradient Ricci almost solitons.
The paper studies ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds.
problem Investigating properties of ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds. method Analyzing the metric properties and vector fields of Kenmotsu manifolds to derive conditions for ∗-Ricci solitons and gradient almost ∗-Ricci solitons. result Characterizations of ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds, including conditions for constant curvature and Einstein metrics. The paper explores ∗-conformal η-Ricci solitons in Kenmotsu manifolds.
problem Characterizing ∗-conformal η-Ricci solitons in Kenmotsu manifolds. method Analyzing the properties of Kenmotsu metrics and manifolds under ∗-conformal η-Ricci solitons. result Kenmotsu metrics as ∗-conformal η-Ricci solitons are Einstein if the soliton vector field is contact. Paper finds gradient estimates for warped product gradient almost Ricci solitons.
problem Nonexistence of certain gradient almost Ricci solitons.
method Modified Li-Yau technique to estimate warping function gradients.
result Nonexistence theorem for specific gradient almost Ricci solitons.
The paper studies integral formulas for a specific type of soliton.
problem Integral formulas for compact gradient h-almost Ricci-Bourguignon solitons.
method Investigation of integral formulas and proving properties of solitons.
result Compact, non-trivial h-almost Ricci-Bourguignon solitons are isometric to a Euclidean sphere under certain conditions.
The study examines almost Ricci-Yamabe solitons on almost Kenmotsu manifolds and their properties.
problem Characterizing almost Ricci-Yamabe solitons on almost Kenmotsu manifolds.
method Analyzing the conditions for almost Ricci-Yamabe solitons to be η-Einstein and proving local isometry for certain manifolds.
result Properties of almost Ricci-Yamabe solitons on (2n+1)-dimensional (κ,μ)′-AKMs. Study classifies gradient almost Ricci solitons in Lorentzian and neutral signatures.
problem Classifying gradient almost Ricci solitons in different signatures.
method Proved local isometry and constructed examples.
result Found that gradient almost Ricci solitons are locally isometric to specific types of manifolds.
Study on almost η-Ricci solitons in ε-para Sasakian manifolds.
problem Characterizing almost η-Ricci solitons in specific geometric structures.
method Analysis of curvature conditions and Codazzi condition to estimate Ricci curvature and scalar curvature.
result Gradient η-Ricci solitons' expansion or shrinking properties based on spacelike or timelike conditions.
Characterizes potential function of almost conformal Ricci solitons on Sasakian manifolds.
problem Characterizing potential functions of almost conformal Ricci solitons on Sasakian manifolds.
method Characterization through the potential function f and non-dynamical scalar field p. result Established a sufficient condition for an almost conformal Ricci soliton to be an almost conformal gradient Ricci soliton.
Study on solitons in deformed Kenmotsu manifolds with specific vector fields.
problem Analyzing geometric solitons in deformed Kenmotsu manifolds.
method Examined almost Riemann and Ricci solitons in a D-homothetically deformed Kenmotsu manifold with specific vector fields. result Explicitly obtained Ricci and scalar curvatures for some cases, provided a lower bound for Ricci curvature.
We introduce the concept {\it h-almost Ricci soliton} which extends naturally the {\it almost Ricci soliton} by Pigola-Rigoli-Rimoldi-Setti and show that a compact nontrivial h-almost Ricci soliton of dimension no less than three with h having defined signal and constant scalar curvature is isometric to a standar…
Study bounds Ricci curvature in (LCS)n-manifolds with almost η-Ricci solitons.
problem Bounding Ricci curvature in (LCS)n-manifolds with specific solitons. method Provided lower and upper bounds for Ricci curvature, derived a Bochner-type formula.
result Derived bounds and consequences for (LCS)n-manifolds with almost η-Ricci solitons. Conditions for statistical structures on manifolds derived from solitons.
problem Characterizing statistical structures on manifolds from soliton equations.
method Analyzing gradient solitons on statistical manifolds to derive conditions for statistical structures.
result Established necessary and sufficient conditions for statistical structures under various soliton types.
We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an…
Study shows Kähler gradient Ricci solitons have limited symmetry groups.
problem Understanding the symmetry groups of Kähler gradient Ricci solitons.
method Connection to almost contact metric structure.
result Group of isometries is at most n^2, with equality characterized.
Gradient almost para-Ricci-like solitons have constant coefficients and scalar curvatures.
problem Characterizing gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian Π-manifolds. method Proving constant coefficients and scalar curvatures through analysis of soliton properties.
result Constant coefficients and scalar curvatures for gradient almost para-Ricci-like solitons.
The paper characterizes Einstein metrics in Kenmotsu manifolds using specific soliton types.
problem Characterizing Einstein metrics in Kenmotsu manifolds using specific soliton types.
method Proving properties of Kenmotsu metrics as η-Ricci solitons and gradient η-Ricci solitons. result Kenmotsu metrics as η-Ricci solitons are Einstein if certain conditions are met. Study on para-Sasakian metrics and their solitons.
problem Characterizing para-Sasakian metrics with conformal η-Ricci solitons.
method Analyzing the properties of para-Sasakian metrics under conformal η-Ricci solitons.
result Para-Sasakian metrics admitting conformal η-Ricci solitons are η-Einstein.
The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a ste…
The aim of this note is to prove that any compact non-trivial almost Ricci soliton (Mn,g,X,λ) with constant scalar curvature is isometric to a Euclidean sphere Sn. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreo…
On an n-dimensional complete manifold M, consider an h-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and dh/du>0, then the manifold M is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. More…
The study classifies contact metric manifolds based on Ricci-Yamabe solitons.
problem Classifying contact metric manifolds based on Ricci-Yamabe solitons.
method Analyzing specific types of solitons in contact metric manifolds.
result Contact metric manifolds are classified based on the properties of Ricci-Yamabe solitons.
In this paper we introduce the notion of generalized quasi--Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi--Einstein manifolds. We prove that a complete generalized quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a war…
The paper characterizes specific almost Kenmotsu manifolds with Ricci-Yamabe solitons.
problem Characterizing almost Kenmotsu manifolds with Ricci-Yamabe solitons.
method Analyzing curvature properties and potential vector fields.
result Locally isometric structures of specific almost Kenmotsu manifolds.
The paper studies special solitons in Riemannian manifolds and finds conditions for compactness.
problem Characterizing conditions for compactness in Riemannian manifolds with specific soliton structures.
method Analyzes gradient ρ-Einstein solitons and almost η-Ricci solitons, proving conditions for compactness and non-negative scalar curvature.
result A sufficient condition for a Riemannian manifold to be compact is derived, involving gradient ρ-Einstein solitons and almost η-Ricci solitons.
Study classifies gradient almost Ricci solitons with harmonic Weyl tensor.
problem Characterizing the local structure of gradient almost Ricci solitons with harmonic Weyl tensor.
method Local representation as multiply warped products, analysis of eigenvalues, and classification based on Weyl tensor properties.
result Classification of gradient almost Ricci solitons with harmonic Weyl tensor, extending previous results.
Study on solitons in specific geometric manifolds, proving manifold properties and presenting examples.
problem Exploring solitons in almost coKähler and asymptotically harmonic manifolds.
method Analyzing η-Einstein and gradient η-Einstein solitons in various geometric frameworks. result Proves manifold properties and presents examples validating results.
Study on almost Riemann solitons with gradient or torse-forming vector fields.
problem Characterizing almost Riemann solitons with specific vector fields.
method Using Bochner formula and properties of gradient and torse-forming vector fields.
result Explicit expressions for the soliton function λ under gradient and torse-forming conditions. Characterizes Ricci almost solitons on semi-Riemannian warped products.
problem Characterizing Ricci almost solitons on semi-Riemannian warped products.
method Characterization using potential function dependence and fiber properties.
result Fiber is necessarily an Einstein manifold; base and warped product are also Einstein manifolds under certain conditions.
Gradient Ricci almost solitons were introduced by Pigola, Rigoli, Rimoldi and Setti. They are defined as solitons except that the metric coefficient is required to be a smooth function rather than a constant. It is shown that any almost soliton is conformal to another almost soliton having a soliton function which is m…
Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.
problem Deriving critical metrics for the Einstein-Hilbert functional on compact Riemannian manifolds.
method Restricts the variational problem to an infinite-dimensional subspace.
result Derives a novel structural characterization of critical metrics.
This paper studies Kähler-Ricci solitons on Heisenberg groups and related metrics.
problem Investigating Kähler-Ricci solitons on Heisenberg groups and related metrics.
method Developed an ansatz for Kähler metrics, specialized to frame-dependent PDEs for gradient Kähler-Ricci solitons, and examined curvature properties and asymptotics.
result Found complete expanding gradient Kähler-Ricci solitons under the action of the (2m-1)-dimensional Heisenberg group.
We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geome…
Study on mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.
problem Characterizing mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.
method Analyzing the condition LVLVg=fLVg and using rigidity phenomena. result Dimension of complete mixed Killing fields is 5 and a basis is explicitly determined.
Study characterizes Einstein manifolds in almost Ricci solitons.
problem Characterizing Einstein manifolds in almost Ricci solitons.
method Geometric dynamics and geometric analysis methods.
result Characterizes Einstein manifolds in the class of complete almost Ricci solitons.
Study new Einstein-like metrics and their properties.
problem Characterize a new class of quasi-Einstein metrics.
method Investigate modified Ricci solitons and their relationships.
result Prove rigidity of standard spheres under specific conditions.
The study examines Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes.
problem Investigating Ricci solitons and curvature inheritance in Robinson-Trautman spacetimes.
method Analyzing the existence of Ricci solitons and curvature inheritance properties on Robinson-Trautman spacetimes.
result Robinson-Trautman spacetimes admit various types of Ricci solitons and curvature inheritance.
New examples of solitons found as warped products.
problem Constructing new soliton examples.
method Warped products and explicit descriptions using elementary functions.
result Complete examples of Ricci almost solitons and Ricci-Bourguignon solitons.
Paper examines properties of specific solutions to Yamabe flow.
problem Characterizing quotient almost Yamabe solitons.
method Investigates quotient almost Yamabe solitons and presents conditions for their rigidity.
result Sufficient conditions for quotient almost Yamabe solitons to be trivial or isometric with a sphere.