Sasakian & contact geometry

Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections $(\nabla ,{\nabla }^{\ast })$ 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 $\nabla$ and ${\nabla }^{\ast }$, together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products $\mathbb{R}{\times }_f{N}_2$ and $N_1{\times }_f\mathbb{R}$ for (pseudo) Riemannian manifolds $N_1$ and $N_2$, where the soliton equation constrains the warping function; and (ii) statistical submanifolds of Kenmotsu statistical manifolds, where the position of the Reeb field $\xi$ relative to the submanifold and the presence of concircular or concurrent vector fields determine the Einstein, $\eta$-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 ${\nabla }^{\theta }$ 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.

We fix a spacelike two-dimensional left-invariant distribution and its associated almost contact structure on a three-dimensional Lorentzian Lie group, and investigate the star-Ricci contraction of the curvature of the Bott connection. We compute the complete tensor for seven standard bracket families and determine all real parameter solutions of the general algebraic equation and its self-adjoint subclass. Using the symmetric part of the tensor, we also define a Bott affine star-Ricci soliton equation and classify all left-invariant vector field solutions. Every affine solution is steady, while two non-unimodular families admit branches with a nonzero symmetrized tensor. The proofs provide complete component equations, treat all degenerate parameter branches, and establish sufficiency by explicit substitution. We also distinguish almost contact structures from contact metric structures.

We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip $S_a=\{(x,y)\in {\mathbb{R}}^2:0<y<a\}$ endowed with the metric $g_{S_a}=\frac{{\pi }^2}{a^2{\sin }^2(\pi y/a)}(d{x}^2+d{y}^2)$, and on its unit tangent bundle $T^1{S}_a$ endowed with the Sasaki metric $g^S$. On the strip, we show that the Gauss curvature is constant equal to $-1$, that the Lie algebra of Killing fields is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, that every Ricci soliton is trivial (Killing potential, $\lambda =-1$), and that a $2$-Killing field which is conformal, or which is the potential of a hyperbolic Ricci soliton with $\lambda \ne 0$, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of $(T^1{S}_a,g^S)$ is four-dimensional and isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus \mathbb{R}$, that $g^S$ is not Einstein but is $\eta$-Einstein with constant coefficients, ${\mathrm{Ric}}^S=-\frac{3}{2}{g}^S+2\eta \otimes \eta$, and that $(T^1{S}_a,g^S)$ admits no Ricci soliton. For hyperbolic Ricci solitons we prove non-existence for Killing, left-invariant, and $\theta$-invariant potentials with conformal projection, and we reduce the general $\theta$-invariant case to a system of equations on the strip; the classification for an arbitrary potential is left open. Finally, we show that the $\eta$-Ricci solitons of $g^S$ have a Killing potential and $\lambda =\frac{3}{2}$, $\mu =-2$, and that the $\eta$-hyperbolic Ricci solitons obtained in the same classes of potentials are the trivial ones, with a Killing potential, $\mu =-\frac{3}{2}$ and $\nu =2$.

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 $(\kappa ,\mu )$-nullity condition related to curvature. For $\kappa <1$ we find conditions under which a w.c.m. manifold admits a bi-Legendrian structure, and for $\kappa =\mu =0$ establish splitting and classification results.

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 $\eta$-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 $\text{Tilde}\tau$-semi-symmetric, quasi-conformal-semi-symmetric and pseudo-projective-semi-symmetric.

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 $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-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 $S^1$.

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 $\alpha$-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.

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 $\ast -$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.

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.

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.

This paper studies a non-trivial gradient Kähler-Ricci soliton, of complex dimension $n$, with an isometry group of dimension at least $n^2-1$. 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 $2^2$. 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.

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, $SU(3)$, $G_2$, and $\mathrm{Spin}(7)$ geometry. We also show that the transverse geometry is always conformally co-closed, with the conformal factor given by the associated soliton potential.

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 $\alpha ,\beta ,\mu$ and $\rho$. In this paper, we show that when $\alpha =0$, 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.

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 $W\in (-\infty ,{\bar{W}}_0)\cup \{0\}\cup (W_0,+\infty )$, where ${\bar{W}}_0<0$ and $W_0>0$ 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 $W\ge 0$, wherea it is a geodesic of nonvanishing curvature when $W$ is a negative constant. Specially, in the first Heisenberg group $\mathbb{M}(0)$, 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 $\mathbb{M}(0)$ and provide a new proof of the fact that the total curvature of $\gamma \subset \mathbb{M}(0)$ with strictly positive curvature equals $2\pi$.

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 $\eta$-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.

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.

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.

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 $\mathcal{S}$-manifolds (w.a.$\mathcal{S}$-manifolds) focusing on the $f$-$(\kappa ,\mu )$-nullity condition and its special case $R_{X,Y}\xi =0$. We establish several results that generalize known rigidity theorems for almost $\mathcal{S}$-manifolds. First, using the partial Ricci flow, we obtain dynamical characterizations of $\mathcal{S}$-manifolds: starting from a w.a.$\mathcal{S}$-structure satisfying the curvature condition of $\mathcal{S}$-manifolds or the $f$-$(1,\mu )$-nullity condition, the flow evolves the structure exponentially fast toward an $\mathcal{S}$-structure. This extends results of Cappelletti Montano and Di Terlizzi to the weak metric setting. Next, we identify conditions under which a w.a.$\mathcal{S}$-manifold admits a bi-Legendrian structure with totally geodesic foliations. Finally, for w.a.$\mathcal{S}$-manifolds with $\kappa =\mu =0$, we prove a splitting theorem in which one factor is flat, generalizing classical results for almost $\mathcal{S}$-geometry. These findings have consequences for the theory of Sasakian and $\mathcal{S}$- manifolds, the geometry of bi-Legendrian structures, and the behavior of weak metric contact manifolds under curvature constraints.

We study the properties of LP-Sasakian manifolds endowed with generalized 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.

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 $\beta f$-Kenmotsu manifold defined as a generalization of K. Kenmotsu's concept. In this paper, the concept of the $\ast$-Ricci tensor of S. Tashibana is adapted to weak metric $f$-manifolds, the interaction of $\ast$-$\eta$-Ricci soliton with the weak $\beta f$-Kenmotsu structure is studied and new characteristics of $\eta$-Einstein metrics are obtained.