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Irina Tsyganok

Publications and source records attributed to Irina Tsyganok.

At least 19 recordsLinked to original sources
arXiv2026

The Harmonic Variational Principle for the Einstein-Hilbert Functional

Let (M,g) be a compact n-dimensional Riemannian manifold, n>2. We introduce a restricted variational principle for the Einstein-Hilbert functional by requiring the admissible metric variations to satisfy the harmonic gauge condition. We derive the corresponding Euler-Lagrange equation and show that a metric is critical with respect to all volume-preserving harmonic variations if and only if its Einstein tensor differs from a multiple of the metric by an element of the image of the adjoint Bianchi operator. We prove that every harmonic critical metric determines a compact Ricci soliton whose soliton constant is given by the normalized Einstein-Hilbert functional. By Perelman's theorem, every such metric is in fact the metric of a compact gradient Ricci soliton. Conversely, every compact gradient Ricci soliton satisfies the restricted Euler-Lagrange equation. Thus, a compact Riemannian metric is harmonic critical if and only if it is the metric of a compact gradient Ricci soliton. We further show that the gauge one-form differs from the negative differential of a soliton potential by a Killing one-form. In particular, if the Ricci tensor is negative definite, then the gauge one-form vanishes and the metric is Einstein. Moreover, every non-Einstein harmonic critical metric is necessarily shrinking.

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arXiv2026

A Restricted Chen-Nagano Variational Principle for the Einstein-Hilbert Functional

This paper introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional on compact Riemannian manifolds. Instead of considering arbitrary symmetric variations of the metric, we restrict the variational problem to an infinite-dimensional subspace determined by the Chen-Nagano gauge constraint. We derive the corresponding restricted Euler-Lagrange equations and obtain a novel structural characterization of critical metrics. The resulting criticality condition is expressed by the equation $E_g = B_g^{*}(θ) + c\,g$ which may be regarded as a restricted counterpart to the classical Einstein equation. Furthermore, we demonstrate that this variational framework naturally leads to generalized Ricci almost soliton structures and, in the gradient case, to gradient Ricci almost solitons. Several global rigidity consequences of this restricted principle are also established.

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arXiv2026

An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds

We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected Riemannian manifold whose Ricci curvature has global minimum $\operatorname{Ric}_{\min}$, and that the target manifold $(\overline{M},\overline{g})$ has non-positive sectional curvature along $f(M)$. We prove that \[ E_2(f) \ge \operatorname{Ric}_{\min}\, E_1(f). \] We further analyze the equality case and obtain rigidity results: equality holds if and only if $f$ is totally geodesic and of constant rank. Applications to maps into Hadamard manifolds are also presented. To the best of our knowledge, this is the first geometric inequality directly relating the Dirichlet energy and the bienergy of smooth maps. This result establishes a direct connection between the Ricci curvature of the domain and higher-order variational energies.

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arXiv2026

Extensions and Applications of Stein-Weiss Operators to the Study of Traceless Symmetric Tensors

First-order differential operators arising from the representation-theoretic decomposition of the covariant derivative play a central role in Riemannian geometry. In this paper, we study Stein-Weiss $O(n)$-gradients acting on covariant symmetric trace-free tensors of arbitrary rank $p \ge 2$. By analyzing the decomposition of $T^*M \otimes S_0^p(M)$ into its $O(n)$-irreducible components, we explicitly describe the corresponding generalized gradients and compute Weitzenbock formulas for their adjoint compositions. These results extend Bouguignon four-dimensional formulas for $p = 2$ and generalize previous work of other authors to higher-rank symmetric tensors. The formulas obtained provide a unified framework for understanding second-order Stein-Weiss operators and yield tools applicable to deformation complexes, curvature estimates, and stability problems in geometric analysis. The article continues the authors' earlier investigations of Stein-Weiss operators on natural tensor bundles.

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arXiv2025

Back to harmonic mappings of compact Riemannian manifolds

In this paper, we address several interconnected problems in the theory of harmonic maps between Riemannian manifolds. First, we present necessary background and establish one of the main results of the paper: a criterion characterizing when a smooth submersion or diffeomorphism between Riemannian manifolds is harmonic. This result provides a useful analytic condition for verifying the harmonicity of geometric mappings. Second, we investigate the L2-orthogonal decomposition of the pullback metric associated with a harmonic map. We analyze the structure of this decomposition and discuss its geometric implications, particularly in the context of the energy density and trace conditions. Finally, we study harmonic symmetric bilinear forms and harmonic Riemannian metrics. Special attention is given to their role in the theory of harmonic identity maps. We derive new results that link these notions and demonstrate how they contribute to the broader understanding of harmonicity in geometric analysis.

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arXiv2025

Geometric Interpretations and Applications of the Berger-Ebin and York $L^2$-Orthogonal Decompositions

The Berger-Ebin and York $L^2$-orthogonal decompositions of the vector space of symmetric bilinear differential two-forms are fundamental tools in global Riemannian geometry. In this paper, we investigate the structure of Ricci tensors on compact Riemannian manifolds, with a particular focus on compact Ricci almost solitons, utilizing both the Berger-Ebin and York $L^2$-orthogonal decompositions. In addition, we explore applications of the York $L^2$-orthogonal decomposition to the theory of submanifolds and to the study of harmonic maps between Riemannian manifolds.

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arXiv2025

Some numerical characteristic inequalities of compact Riemannian manifolds

In this paper, we consider numerical characteristics of the connected compact Riemannian manifold (M, g) such as the supremum and infimum of the scalar curvature s, Ricci curvature Ric and sectional curvature sec, as well as their applications. Below are two examples of proven results. The first statement: If (M, g) be a connected, compact Riemannian manifold of even dimension n > 3 whose Ricci and sectional curvatures satisfy the strict inequality n Inf (sec) > Sup (Ric), then M is diffeomorphic to the Euclidean n-dimensional sphere of some radius r or the real projective n-dimensional space. The second statement: There is no harmonic immersion of an n-dimensional connected, complete Riemannian manifold (M, g) into the Euclidean n-sphere of radius r if there exists inf(Ric) such that Inf (Ric) > n/2r^2.

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arXiv2023

Back to almost Ricci solitons

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.

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arXiv2022

Geometry in the large on Hadamard manifolds

In this paper, we prove several Liouville-type theorems on the non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds and, in particular, on Riemannian symmetric spaces of non-compact type. These theorems supplement the well-known vanishing theorems for the above tensors, obtained using the Bochner technique for compact Riemannian manifolds. In turn, the proofs of our theorems will use well-known Liouville-type theorems on the non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, which we have partially modified for the case of Hadamard manifolds and, in particular, Riemannian symmetric spaces of noncompact type.

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arXiv2019

On the Betti and Tachibana numbers of compact Einstein manifolds

Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein manifold $(M,g)$ with Einstein constant $α>0$ is a homo-logical sphere when the minimum of its sectional curvatures $> α/(n+ 2)$; in particular, $(M,g)$ is a spherical space form when the minimum of its sectional curvatures $> α/ n$. Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold $(M,g)$ with $α< 0$.

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arXiv2019

The Bourguignon Laplacian and harmonic symmetric bilinear forms

The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In this case, there are the exterior differential and codifferential defined on the vector space of these differential one-forms. Then a symmetric bilinear form is said to be harmonic if it is closed and coclosed as a one-form with values in the cotangent bundle of a Riemannian manifold. In the present paper we prove that the kernel of the little known Bourguignon Laplacian is a finite-dimensional vector space of harmonic symmetric bilinear forms on a compact Riemannian manifold. We also prove that every harmonic symmetric bilinear form on a compact Riemannian manifold with non-negative sectional curvature is invariant under parallel translations. In addition, we investigate the spectral properties of the little studied Bourguignon Laplacian.

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arXiv2019

From harmonic mappings to Ricci flow due to the Bochner technique

The present paper is devoted to the study a global aspect of the geometry of harmonic mappings and, in particular, infinitesimal harmonic transformations, and represents the application of our results to the theory of Ricci solutions and the Ricci flow. These results will be obtained using the methods of Geometric analysis and, in particular, due to theorems of Yau, Li and Schoen on the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions.

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arXiv2019

Geometry in the large of the kernel of Lichnerowicz Laplacians and its applications

There are very few general theorems on the kernel of the well-known Lichnerowicz Laplacian. In the present article we consider the geometry of the kernel of this operator restricted to covariant (not necessarily symmetric or skew-symmetric) tensors. Our approach is based on the analytical method, due to Bochner, of proving vanishing theorems for the null space of Laplace operator. In particular, we pay special attention to the kernel of the Lichnerowicz Laplacian on Riemannian symmetric spaces of compact and noncompact types. In conclusion, we give some applications to the theories of infinitesimal Einstein deformations and the stability of Einstein manifolds.

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arXiv2018

From harmonic mappings to Ricci solitons

The paper is devoted to the study of the global geometries of harmonic mappings and infinitesimal harmonic transformations and presents their applications to the theory of Ricci solitons.

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