Vladimir Rovenski — Neckpinch

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.

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.

∗-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.

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.

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.

Back to almost Ricci solitons

Fri, 07 Apr 2023 00:00:00 GMT

Vladimir Rovenski, Sergey Stepanov, Irina Tsyganok — In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.

Generalized Ricci solitons and Einstein metrics on weak K-contact manifolds

Mon, 09 Jan 2023 00:00:00 GMT

Vladimir Rovenski — We study metric structures on a smooth manifold (introduced in our recent works and called a weak contact metric structure and a weak K-structure) which generalize the metric contact and K-contact structures, and allow a new look at the classical theory. First, we characterize weak K-contact manifolds among all weak contact metric manifolds by the property well known for K-contact manifolds, and find when a Riemannian manifold endowed with a unit Killing vector field forms a weak K-contact structure. Second, we find sufficient conditions for a weak K-contact manifold with parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.