arXiv subjects

Vladimir Rovenski

Publications and source records attributed to Vladimir Rovenski.

At least 19 recordsLinked to original sources

arXiv2026

Einstein connection of a weak almost contact metric manifold

Advances in modern physics since Einstein have made the nonsymmetric metric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric associated with gravity, and $F\ne0$ is a skew-symmetric tensor associated with electromagnetism, more attractive than ever. Einstein considered a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=G(T(Y,X),Z)$. In this paper, we explicitly present the Einstein connection of $G=g+F$ using a weak almost contact structure $(f,ξ,η)$ with $g(X,fY)=F(X,Y)$ with a natural condition (trivial in the almost contact case). We discuss special Einstein connections, and give an example in terms of the weighted product of almost Hermitian manifold and a real line.

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arXiv2026

Characterizations of weak almost ${\mathcal S}$-manifolds with curvature properties

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.

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arXiv2026

Einstein connection of nonsymmetric pseudo-Riemannian manifold

A.Einstein considered a linear connection $\nabla$ with torsion $T$ on a smooth manifold equipped with a nonsymmetric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric associated with gravity, and $F\ne0$ is a skew-symmetric tensor associated with electromagnetism, such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. In this paper, we explicitly present the Einstein connection of a nonsymmetric pseudo-Riemannian manifold with non-degenerate $F$, satisfying the $f^2$-torsion condition $T(f^2X,Y)=T(X,f^2Y)=f^2 T(X,Y)$, where $g(X,fY)=F(X,Y)$, and show that in the almost Hermitian case, it reduces to the M.Prvanović's (1995) solution. We also explicitly present the Einstein connection of almost contact metric manifolds satisfying the $f^2$-torsion condition, discuss special Einstein connections, and give example in terms of weighted product of almost Hermitian manifolds.

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arXiv2026

On the splitting of weak nearly ${\cal C}$-manifolds

The interest of mathematicians in metric $f$-manifolds, in particular, almost contact metric manifolds, is motivated by the study of the geometry and dynamics of contact foliations, as well as their applications in physics. Weak metric $f$-manifolds, defined by V. Rovenski and R. Wolak (2022), open a new perspective on classical theory of $f$-manifolds and discover new applications. In this paper, we study manifolds of this type, called weak nearly ${\cal C}$-manifolds, which generalize almost ${\cal C}$-manifolds. We find conditions under which a $(2n+s)$-dimensional weak nearly ${\cal C}$-manifold becomes locally a Riemannian product, and characterize $(4+s)$-dimensional weak nearly ${\cal C}$-manifolds. The consequences of these theorems present new results for nearly ${\cal C}$-manifolds.

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arXiv2025

Bochner's technique in Einstein's non-symmetric geometry

A. Einstein considered a manifold with a non-symmetric (0,2)-tensor $G=g+F$, where $g$ is a Riemannian metric and $F\ne0$, and a connection $\nabla$ with torsion $T$ such that $(\nabla_X G)(Y,Z)=-G(T(X,Y),Z)$. Guided by the almost Lie algebroid construction on a vector bundle, we define the basic concepts of Bochner's technique for Einstein's non-symmetric geometry, give a clear example of the Einstein's connection $\nabla$, prove Weitzenböck type decomposition formula and obtain vanishing results about the null space of the Bochner and Hodge type Laplacians.

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arXiv2025

Weak metric structures on generalized Riemannian manifolds

In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the first author and R. Wolak), which generalize almost complex and almost contact metric $f$-contact structures. We consider generalized metric connections (i.e., linear connections preserving $G$) with totally skew-symmetric torsion (0,3)-tensor. For rank$(F)=\dim M$ and non-conformal tensor $A^2$, where $A$ is a skew-symmetric (1,1)-tensor adjoint to $F$, we apply weak almost Hermitian structures to fundamental results (by the second author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold is a weighted product of several nearly Kähler manifolds corresponding to eigen-distributions of $A^2$. For rank$(F)<\dim M$ we apply weak $f$-structures and obtain splitting results for generalized Riemannian manifolds.

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arXiv2025

$\ast$-$η$-Ricci solitons on weak Kenmotsu $f$-manifolds

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 $\ast$-Ricci tensor of S. Tashibana is adapted to weak metric $f$-manifolds, the interaction of $\ast$-$η$-Ricci soliton with the weak $βf$-Kenmotsu structure is studied and new characteristics of $η$-Einstein metrics are obtained.

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arXiv2025

Weak Nearly $\mathcal S$- and Weak Nearly $\mathcal C$- Manifolds

The recent interest of geometers in the $f$-structures of K. Yano is motivated by the study of the dynamics of contact foliations, as well as their applications in theoretical physics. Weak metric $f$-structures on a smooth manifold, recently introduced by the author and R. Wolak, open a new perspective on the theory of classical structures. In the paper, we define structures of this kind, called weak nearly ${\cal S}$- and weak nearly ${\cal C}$- structures, study their geometry, e.g. their relations to Killing vector fields, and characterize weak nearly ${\cal S}$- and weak nearly ${\cal S}$- submanifolds in a weak nearly Kähler manifold.

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arXiv2025

Applications of Weak Metric Structures to Non-Symmetrical Gravitational Theory

Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric part. Such structures naturally arise in spacetime models in theoretical physics, where $F$ can be defined as an almost complex or almost contact metric (a.c.m.) structure. In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the second author and R.~Wolak), which generalize almost complex and a.c.m. structures. We consider linear connections with totally skew-symmetric torsion that satisfy both the Einstein metricity condition and the $A$-torsion condition, where $A$ is a skew-symmetric (1,1)-tensor adjoint to~$F$. In the almost Hermitian case, we prove that the manifold with such a connection is weak nearly K\" ahler, the torsion is completely determined by the exterior derivative of the fundamental 2-form and the Nijenhuis tensor, and the structure tensors are parallel, while in the weak a.c.m. case, the contact distribution is involutive, the Reeb vector field is Levi-Civita parallel, and the structure tensors are also parallel with respect to both connections. For rank$(F)=\dim M$, we apply weak almost Hermitian structures to fundamental results (by the first author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold equipped with an Einstein's connection is a weighted product of several nearly Kähler manifolds. For~rank$(F)<\dim M$ we apply weak almost Hermitian and weak a.c.m. structures and obtain splitting results for generalized Riemannian manifolds equipped with Einstein's connections.

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arXiv2025

$\ast$-$η$-Ricci solitons and Einstein metrics on a weak $β$-Kenmotsu manifold

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 $\ast$-Ricci tensor is adapted to weak almost contact manifolds, the interaction of the $\ast$-$η$-Ricci soliton with the weak $β$-Kenmotsu structure (with $β=const$) is studied and new characteristics of Einstein metrics are obtained.

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arXiv2025

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

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.

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arXiv2025

Geometry of weak metric $f$-manifolds: a survey

A weak $f$-structure on a smooth manifold, introduced by the author and R. Wolak (2022), generalizes K. Yano's (1961) $f$-structure. This generalization allows us to revisit classical theory and discover new applications related to Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. This article reviews the results on weak metric $f$-manifolds, where the complex structure on the contact distribution of a metric $f$-structure is replaced with a nonsingular skew-symmetric tensor, and explores its distinguished classes.

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arXiv2024

Godbillon-Vey type functional for almost contact manifolds

Many contact metric manifolds are critical points of curvature functionals restricted to spaces of associated metrics. The Godbillon-Vey functional has never been considered in a variational context in contact geometry. Recently we extended this functional from foliations to arbitrary plane fields on a 3-dimensional manifold, so, the following question arises: can one use the Godbillon-Vey functional to find optimal almost contact manifolds? In the paper, we introduce a Godbillon-Vey type functional for a 3-dimensional almost contact manifold and find its Euler-Lagrange equations for all variations preserving the Reeb vector field. We construct critical (for our functional) 3-dimensional almost contact manifolds having a double-twisted product structure, these solutions belong to the class $C_{5}\oplus C_{12}$ according to Chinea-Gonzalez classification.

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arXiv2024

Weak quasi contact metric manifolds and new characteristics of K-contact and Sasakian manifolds

Quasi contact metric manifolds (introduced by Y. Tashiro and then studied by several authors) are a natural extension of the contact metric manifolds. Weak almost contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, were defined by the author and R. Wolak. In this paper, we study a weak analogue of quasi contact metric manifolds. Our main results generalize some well known theorems and provide new criterions for K-contact and Sasakian manifolds in terms of conditions on the curvature tensor and other geometric objects associated with the weak quasi-contact metric structure.

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arXiv2024

Weak Almost Contact Structures: a Survey

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of contact manifolds. The paper surveys recent results (concerning geodesic and Killing fields, rigidity and splitting theorems, Ricci-type solitons and Einstein-type metrics, etc.) in this new field of Riemannian geometry.

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arXiv2024

Geometric inequalities for CR-submanifolds

We study two kinds of curvature invariants of Riemannian manifold equip\-ped with a complex distribution $D$ (for example, a CR-submanifold of an almost Hermitian manifold) related to sets of pairwise orthogonal subspaces of the distribution. One kind of invariant is based on the mutual curvature of the subspaces and another is similar to Chen's $δ$-invariants. We compare the mutual curvature invariants with Chen-type invariants and prove geometric inequalities with intermediate mean curvature squared for CR-submanifolds in almost Hermitian spaces. In the case of a set of complex planes, we introduce and study curvature invariants based on the concept of holomorphic bisectional curvature. As applications, we give consequences of the absence of some $D$-minimal CR-submanifolds in almost Hermitian manifolds.

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arXiv2024

On the splitting of weak nearly cosymplectic manifolds

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is approximated by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak (2022), allowed a new look at the theory of contact manifolds. This article studies the curvature and topology of new structures of this type, called the weak nearly cosymplectic structure and weak nearly Kähler structure. We find conditions under which weak nearly cosymplectic manifolds become Riemannian products and characterize 5-dimensional weak nearly cosymplectic manifolds. Our theorems generalize results by H. Endo (2005) and A. Nicola-G. Dileo-I. Yudin (2018) to the context of weak almost contact geometry.

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arXiv2024

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

A weak metric $f$-structure $(f,Q,ξ_i,η^i,g)\ (i=1,\ldots,s)$, 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 $\mathfrak{g}$-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.

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