Solitons on Statistical Manifolds and Their Submanifolds

Wed, 28 Oct 2026 00:00:00 GMT

Aliya Naaz Siddiqui, Bang-Yen Chen — Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections (∇,∇^(*)) together with a Riemannian metric. In this chapter we survey the theory of statistical solitons, that is, vector fields and constants that solve the Ricci-soliton equation simultaneously for ∇ and ∇^(*), together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products R×_f N₂ and N₁×_fR for (pseudo) Riemannian manifolds N₁ and N₂, where the soliton equation constrains the warping function; and (ii) statistical submanifolds of Kenmotsu statistical manifolds, where the position of the Reeb field ξ relative to the submanifold and the presence of concircular or concurrent vector fields determine the Einstein, η-Einstein or quasi-Einstein character of the submanifold. We further review the results of Blaga and Chen on gradient solitons and statistical structures, in particular the connections ∇^θ defined by a 1-form, volume formulas and bounds for the Ricci tensor. We also describe Chen-type inequalities for statistical warped product submanifolds, which give the extrinsic context in which such solitons live. Key steps of the proofs, remarks on the sharpness of hypotheses, worked examples, and open problems are included.

η- Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip

Thu, 08 Oct 2026 00:00:00 GMT

Amadou Sy, Ameth Ndiaye, Ghodratallah Fasihi-Ramandi — We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip Sₐ=\(x,y)∈ℝ²:0

Characterization of Sasakian Manifolds by Curvature Conditions

Mon, 05 Oct 2026 00:00:00 GMT

Sourav Nayak, Vladimir Rovenski, Dhriti Sundar Patra — Sasakian manifolds, a subclass of contact metric manifolds, play a central role in modern differential geometry and mathematical physics, where they arise naturally in geometric mechanics, CR geometry, and supersymmetric field theories. The weak contact metric (w.c.m.) structure generalizes the contact metric structure and provides a broader framework for studying contact geometry and its applications. A fundamental problem is to understand how curvature conditions constrain the underlying contact metric structure and, in particular, how they distinguish Sasakian geometry from its weaker variants. Using the partial Ricci flow, we characterize Sasakian structure (among w.c.m. manifolds) under the (κ,μ)-nullity condition related to curvature. For κ<1 we find conditions under which a w.c.m. manifold admits a bi-Legendrian structure, and for κ=μ=0 establish splitting and classification results.

K-contact manifolds admitting some geometric solitons with Semi-Symmetric Non-Metric Connection

Mon, 14 Sep 2026 00:00:00 GMT

Bidhan Mondal, Nirabhra Basu, Arindam Bhattacharyya — In this paper, we introduce some type vector fields with respect to a semi-symmetric non-metric (SSNM) connection. We investigate several geometric properties of a K-contact manifold equipped with an SSNM connection and provide a concrete example to justify the relation between the scalar curvature of the SSNM connection and Levi-Civita connection that we have obtained in this paper. Furthermore, we have found the nature of Riemann solitons, conformal Ricci solitons and conformal η-Ricci-Yamabe solitons on K-contact manifolds admitting a SSNM connection.\\ Finally, we determine the necessary and sufficient conditions for such a manifold to be Tildeτ-semi-symmetric, quasi-conformal-semi-symmetric and pseudo-projective-semi-symmetric.

Sasaki with torsion manifolds and string backgrounds

Sun, 09 Aug 2026 00:00:00 GMT

Beatrice Brienza, Anna Fino, Udhav Fowdar, Gueo Grantcharov — Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a ∇-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact ∇-Einstein manifold in dimension 5 and 7. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with S¹.

Non-compact Ricci Solitons of Finite Volume with Potential Field of Constant Length

Fri, 24 Jul 2026 00:00:00 GMT

Hemangi Madhusudan Shah, Sharief Deshmukh, Mohammad Aqib — We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we characterise 3-dimensional Ricci solitons of constant length and of finite volume on almost α-cosymplectic manifolds. In fact, we show that these spaces are indeed compact quotients of Euclidean spaces. We also study complete, non-compact and finite volume contact metric manifolds in dimension 3 admitting a Reeb vector field as a Ricci soliton. We show that it is a compact spherical space form. We include some examples to illustrate the necessity of the finite-volume and constant-length assumptions.

Hyperbolic solitons on trans-Sasakian space forms and its submanifolds

Mon, 22 Jun 2026 00:00:00 GMT

Bidhan Mondal, Sibsankar Panda, Nirabhra Basu — Kong and Liu introduced the concept of hyperbolic Ricci flow in 2007 and used it to study the wave character of metrics. After that, many mathematicians have used this new geometric flow to study the evolution of manifolds and their structures. Ricci solitons and hyperbolic Ricci solitons are self-similar solitons of the Ricci flow and hyperbolic Ricci flow respectively. In this paper, we introduce the concept of hyperbolic *-Ricci solitons and hyperbolic Ricci-Yamabe solitons on a trans-Sasakian space forms and characterized the nature of some hyperbolic solitons. Additionally, we deduce the Ricci tensors of submanifolds of trans-Sasakian space forms and conformal trans-Sasakian space form and found the nature of solitons on submanifolds. Finally, we have included an example which will justify our result.

On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian 5-manifold

Wed, 20 May 2026 00:00:00 GMT

Shu-Cheng Chang, Yingbo Han, Chien Lin, Chin-Tung Wu — In this paper, we first confirm the Hamilton-Tian conjecture for the Sasaki-Ricci flow in a compact transverse Fano quasi-regular Sasakian 5-manifold with klt foliation singularities. Secondly, we derive the compactness theorem of Sasaki-Ricci solitons on transverse Fano quasi-regular Sasakian 5-manifolds. Then,by the second Sasakian structure theorem, we confirm the Hamilton-Tian conjecture for a compact transverse Fano Sasakian 5-manifold. With its applications, we show that the gradient Sasaki-Ricci soliton orbifold metric on a compact Sasakian 5-manifold is Sasaki-Einstein if M is transverse K-stable.

On weighted extremal Kähler metrics

Sat, 09 May 2026 00:00:00 GMT

Akito Futaki — The notion of weighted extremal Kähler metrics extends the classical notion of Calabi's extremal Kähler metrics, but includes many well-studied objects in Kähler geometry such as Kähler-Ricci solitons and Sasaki-Einstein metrics. In this paper, after explaining how this notion grew out, we will try to survey recent works concerning the YTD conjecture on weighted extremal Kähler metrics.

Kähler-Ricci solitons with almost maximal symmetry

Sat, 02 May 2026 00:00:00 GMT

Ha Tuan Dung, Catherine Searle, Hung Tran — This paper studies a non-trivial gradient Kähler-Ricci soliton, of complex dimension n, with an isometry group of dimension at least n²−1. We show that the isometry group acts by cohomogeneity one and, consequently, admits a special ansatz involving a Sasakian model. In complex dimension two, we can actually say more: namely, that every such soliton has maximal symmetry; that is, the isometry group is exactly of dimension 2². In addition, we prove that, if the isometry group acts by cohomogeneity one on a non-trivial gradient Ricci soliton (not necessarily Kähler), the potential function is invariant by the action.

The canonical symmetry reduction of string backgrounds

Tue, 25 Nov 2025 00:00:00 GMT

Aaron Kennon, Jeffrey Streets — String backgrounds, defined here as metric connections with skew-symmetric torsion and reduced holonomy, yield generalized Ricci solitons relative to the Lee vector field. By a variational argument using the string action, they are also gradient generalized Ricci solitons relative to a potential function. These two observations combine to yield a canonical symmetry, and in this work we derive fundamental features of the transverse geometry, and rigidity phenomena. We prove in a unified conceptual fashion that the transverse geometry satisfies the string generalized Ricci soliton equations (a simplified Hull-Strominger system) in many settings including almost Hermitian, almost contact, SU(3), G₂, and Spin(7) geometry. We also show that the transverse geometry is always conformally co-closed, with the conformal factor given by the associated soliton potential.

Complete gradient Einstein-type Sasakian manifolds with α=0

Tue, 14 Oct 2025 00:00:00 GMT

Shun Maeta — Catino, Mastrolia, Monticelli, and Rigoli have launched an ambitious program to study known geometric solitons from a unified perspective, which they term Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe solitons, and all of their generalizations simultaneously. Einstein-type manifolds are characterized by four constants α, β, μ and ρ. In this paper, we show that when α= 0, complete gradient Einstein-type Sasakian manifolds are trivial or isometric to the unit sphere. As a consequence, many geometric solitons on Sasakian manifolds turn out to be trivial or isometric to the unit sphere.

Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

Sun, 28 Sep 2025 00:00:00 GMT

Jingshi Cui, Peibiao Zhao — In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold M for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature W of M satisfies W ∈ (-∞, W₀ )∪ \ 0\ ∪ (W₀, +∞), where W₀ <0 and W₀ >0 are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when W ≥ 0, wherea it is a geodesic of nonvanishing curvature when W is a negative constant. Specially, in the first Heisenberg group M(0), we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in M(0) and provide a new proof of the fact that the total curvature of γ⊂ M(0) with strictly positive curvature equals 2π.

Curvature tensors and hyperbolic solitons on Lorentzian trans-Sasakian space form

Mon, 22 Sep 2025 00:00:00 GMT

Bidhan Mondal, Nirabhra Basu, Arindam Bhattacharyya — Lorantzian trans-Sasakian space form is a special type of space form in which the nature of even and odd dimensional space form both exist. Various curvature tensors with respect to Levi-Civita connection on the space form are derived in this paper. We have shown that if an odd-dimensional Lorentzian trans-Sasakian space form admits a hyperbolic Ricci soliton and hyperbolic conformal Ricci soliton then they will be η-Einstein. We also obtained the conditions for the solitons to be expanding, steady or shrinking. Finally, an example has been constructed which justifies the results obtained.

Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor

Mon, 01 Sep 2025 00:00:00 GMT

Shu-Cheng Chang, Hongbing Qiu — We establish integral curvature estimates for complete gradient shrinking Sasaki-Ricci solitons. As an application, we show that any such soliton with harmonic Weyl tensor must be a finite quotient of a sphere. This result can be regarded as the Sasaki analogue of the work of Munteanu and Sesum [15] on Ricci solitons.

Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

Tue, 19 Aug 2025 00:00:00 GMT

Shu-Cheng Chang, Yingbo Han, Chin-Tung Wu — In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking Kähler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.

Characterizations of weak almost {𝓢-manifolds with curvature properties

Tue, 12 Aug 2025 00:00:00 GMT

Sourav Nayak, Dhriti Sundar Patra, Vladimir Rovenski — Weak metric structures, introduced by Rovenski and Wolak in 2022, extend Yano's f-structure and almost contact metric structure. In this paper, we investigate curvature phenomena of weak almost cal S-manifolds (w.a.cal S-manifolds) focusing on the f-(κ,μ)-nullity condition and its special case R_(X,Y) ξ=0. We establish several results that generalize known rigidity theorems for almost cal S-manifolds. First, using the partial Ricci flow, we obtain dynamical characterizations of cal S-manifolds: starting from a w.a.cal S-structure satisfying the curvature condition of cal S-manifolds or the f-(1,μ)-nullity condition, the flow evolves the structure exponentially fast toward an cal S-structure. This extends results of Cappelletti Montano and Di Terlizzi to the weak metric setting. Next, we identify conditions under which a w.a.cal S-manifold admits a bi-Legendrian structure with totally geodesic foliations. Finally, for w.a.cal S-manifolds with κ=μ=0, we prove a splitting theorem in which one factor is flat, generalizing classical results for almost cal S-geometry. These findings have consequences for the theory of Sasakian and cal S- manifolds, the geometry of bi-Legendrian structures, and the behavior of weak metric contact manifolds under curvature constraints.

Generalized η-Ricci solitons on LP-Sasakian manifolds admitting the general connection

Mon, 14 Jul 2025 00:00:00 GMT

Murat Altunbaş, Ayşe Karanlık Akpınar — We study the properties of LP-Sasakian manifolds endowed with generalized % η-Ricci solitons associated to the general connection. Finally, the existence of such solitons on a 4-dimensional LP-Sasakian manifold is proved by constructing a non-trivial example.

∗-η-Ricci solitons on weak Kenmotsu f-manifolds

Tue, 10 Jun 2025 00:00:00 GMT

Vladimir Rovenski — Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations and generalized A. Weinstein conjectures. Weak metric f-structures, introduced by the author and R. Wolak as a generalization of Hermitian structure, as well as f-structure allow for a fresh perspective on the classical theory. An important case of such manifolds, which is locally a twisted product, is a weak βf-Kenmotsu manifold defined as a generalization of K. Kenmotsu's concept. In this paper, the concept of the ∗-Ricci tensor of S. Tashibana is adapted to weak metric f-manifolds, the interaction of ∗-η-Ricci soliton with the weak βf-Kenmotsu structure is studied and new characteristics of η-Einstein metrics are obtained.

∗-η-Ricci solitons and Einstein metrics on a weak β-Kenmotsu manifold

Sun, 23 Mar 2025 00:00:00 GMT

Vladimir Rovenski — Weak almost contact metric manifolds (i.e., the complex structure is replaced by a nonsingular skew-symmetric tensor), defined by the author and R. Wolak, allow a new look at the classical theory and find novel applications. An important case of these manifolds, which is locally a twisted product, is a weak β-Kenmotsu manifold defined by the author and D.S. Patra. In the paper, the concept of the ∗-Ricci tensor is adapted to weak almost contact manifolds, the interaction of the ∗-η-Ricci soliton with the weak β-Kenmotsu structure (with β=const) is studied and new characteristics of Einstein metrics are obtained.

Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

Sat, 22 Feb 2025 00:00:00 GMT

Shu-Cheng Chang, Fengjiang Li, Chien Lin, Hongbing Qiu — In this paper, we study several properties of Sasaki-Ricci solitons as singularity models of the Sasaki-Ricci flow. First, we establish several fundamental equations for Sasaki-Ricci solitons, which enable us to derive potential estimates and prove the positivity of the scalar curvature. Then we present two criteria for the transverse rigidity of Sasaki-Ricci solitons. As essential applications, we prove that any low-dimensional Sasaki-Ricci soliton with constant scalar curvature must be Sasaki-Einstein, and that any Sasaki-Ricci soliton with harmonic Weyl tensor is a finite quotient of the sphere.

η-Ricci solitons and η-Einstein metrics on weak β-Kenmotsu f-manifolds

Wed, 18 Dec 2024 00:00:00 GMT

Vladimir Rovenski — Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by V. Rovenski and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow a fresh look at the classical theory. In this paper, we study a new f-structure of this kind, called the weak β-Kenmotsu f-structure, as a generalization of K. Kenmotsu's concept. We prove that a weak β-Kenmotsu f-manifold is locally a twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with β=const and equipped with an η-Ricci soliton structure whose potential vector field satisfies certain conditions are η-Einstein manifolds of constant scalar curvature.

η-Ricci Solitons on Kenmotsu 3-Manifolds

Fri, 22 Nov 2024 00:00:00 GMT

K. De, U. C. De — In the present paper we study η-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider η-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study φ-Ricci symmetric η-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition R.R=Q(S,R) is considered. Finally, an example is constructed to prove the existence of a proper η-Ricci soliton on a Kenmotsu 3-manifold.

The weighted Hermite–Einstein equation

Mon, 12 Aug 2024 00:00:00 GMT

Michael Hallam, Abdellah Lahdili — We introduce a new weighted version of the Hermite–Einstein equation, along with notions of weighted slope (semi/poly)stability, and prove that a vector bundle admits a weighted Hermite–Einstein metric if and only if it is weighted slope polystable. The new equation encompasses several well-known examples of canonical Hermitian metrics on vector bundles, including the usual Hermite–Einstein metrics, Kähler–Ricci solitons, and transversally Hermite–Einstein metrics on certain Sasaki manifolds. We prove that the equation arises naturally as a moment map, that solutions to the equation are unique up to scaling, and demonstrate a weighted Kobayashi–Lübke inequality satisfied by vector bundles admitting a weighted Hermite–Einstein metric. As an application of our techniques, we extend a bound of Tian on the Ricci curvature to a bound on a modified Ricci curvature, related to the existence of Kähler–Ricci solitons. Along the way, we introduce a new weighted vortex equation, as well as a weighted analogue of Gieseker stability. A key technical point is the application of a new extension of Inoue's equivariant intersection numbers to arbitrary weight functions on the moment polytope of a Kähler manifold with Hamiltonian torus action.

On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

Mon, 24 Jun 2024 00:00:00 GMT

Shu-Cheng Chang, Fengjiang Li, Chien Lin, Chin-Tung Wu — In this paper, we derive the uniform L^4-bound of the transverse conic Ricci curvature along the conic Sasaki-Ricci flow on a compact transverse log Fano Sasakian manifold M of dimension five and the space of leaves of the characteristic foliation is not well-formed. Then we first show that any solution of the conic Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold conic Sasaki-Ricci soliton on M_infinite which is a S^1-orbibundle over the unique singular conic Keahler-Ricci soliton on a log del Pezzo orbifold surface. As a consequence, there exists a Keahler-Ricci soliton orbifold metric on its leave space which is a log del Pezzo orbifold surface. Second, we show that the conic Sasaki-Ricci soliton is the conic Sasaki-Einstein if M is transverse log K-polystable. In summary, we have the existence theorems of orbifold Sasaki-Ricci solitons and Sasaki-Einstein metrics on a compact quasi-regular Sasakian manifold of dimension five.

The Smale Conjecture and Minimal Legendrian Graph in S²× S³

Tue, 26 Dec 2023 00:00:00 GMT

Shu-Cheng Chang, Chin-Tung Wu, Liuyang Zhang — In this article, we recapture the Smale conjecture on a Sasakian 3-sphere via the Legendrian mean curvature flow. More precisely, we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian 3-spheres to an isometry via the Legendrian mean curvature flow on the Legendrian graph in S²× S³. By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in S²× S³. Finally, we will address the rigidity theorem of 2-dimensional Legendrian self-shrinkers in ℝ⁵. We are able to reconstruct the Harvey-Lawson special Lagrangian cone in ℂ³ from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.

Kähler Solitons, Contact Structures, and Isoparametric Functions

Tue, 17 Oct 2023 00:00:00 GMT

Hung Tran — All known examples of simply-connected gradient Kähler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function f and the scalar curvature SS. In this article, we consider the case that f and SS are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.

Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds

Fri, 30 Jun 2023 00:00:00 GMT

Roberto Mossa, Giovanni Placini — We discuss local Sasakian immersion of Sasaki-Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by a large class of fiber products of homogeneous Sasakian manifolds are, in fact, η-Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a Kähler-Ricci soliton on ℂⁿ which admits no local holomorphic isometry into products of homogeneous bounded domains with flat Kähler manifolds and generalized flag manifolds.

Einstein-type metrics and generalized Ricci solitons on weak f-K-contact manifolds

Mon, 12 Jun 2023 00:00:00 GMT

Vladimir Rovenski — A weak metric f-structure (f,Q,ξᵢ,ηⁱ,g) (i=1,…,s), generalizes the metric f-structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak f-K-contact structure, which is a weak f-contact structure, whose characteristic vector fields are Killing. We show that ker f of a weak f-contact manifold defines a g-foliation with an abelian Lie algebra. Then we characterize weak f-K-contact manifolds among all weak metric f-manifolds by the property known for f-K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak f-K-contact manifold. We show that for s>1, an Einstein weak f-K-contact manifold is Ricci flat, then find sufficient conditions for a weak f-K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We prove positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak f-K-contact structure to an f-K-contact structure. We define an η-Ricci soliton and η-Einstein structures on a weak metric f-manifold (which for s=1, give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak f-K-contact manifold with an η-Ricci soliton structure of constant scalar curvature to be η-Einstein.

Neckpinch — Sasakian & contact geometry