New findings on Codazzi tensors in homogeneous spaces.
problem Characterizing Codazzi tensor fields in reductive homogeneous spaces.
method Extending results from Lie groups to reductive homogeneous spaces, analyzing the curvature of canonical connections.
result Invariant Codazzi tensor fields on naturally reductive homogeneous spaces are parallel.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
We prove several Liouville-type non-existence theorems for higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavio…
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
We extend a classical result by Derdzinski and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms) as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"), typical of …
Study Codazzi tensors in space-times, linking to Cotton gravity.
problem Understanding Codazzi tensors and their role in space-times.
method Analyzing geometric properties and proving conditions for Codazzi tensors.
result Codazzi tensors restrict space-times, influencing energy-momentum tensors in Cotton gravity.
Decomposes submanifolds with special tensors into simpler parts.
problem Understanding the structure of submanifolds with special tensors.
method Established a decomposition theorem for submanifolds with nonnegative sectional curvature and a Codazzi tensor with parallel mean curvature.
result Submanifolds with these tensors are locally isometric to a direct product of irreducible factors.
New divergence identity for scalar curvature helps prove rigidity of tensors.
problem Proving rigidity of Codazzi tensors under curvature and invariant conditions.
method Derived a divergence identity for a vector field and applied it to tensor rigidity.
result New proof of Tang-Yan theorem on constant eigenvalues for tensors.
In this paper we deal with the following problem: Find all Riemannian metrics on a manifold that can be realized isometrically as immersed hypersurfaces in the Euclidean space. We study this problem for a wide class of metrics on hypersurfaces arising from Codazzi tensors.
Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …
The paper studies twisted almost Hermitian structures on the 6-sphere.
problem Non-integrability of certain automorphisms on the 6-sphere.
method Analyzes ψ-twisted almost Hermitian structures and identifies g-Codazzi maps. result Proves non-integrability of g-Codazzi maps on the 6-sphere. We prove a lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold depending on the scalar curvature as well as a chosen Codazzi tensor. The inequality generalizes the classical estimate from [2].
This paper addresses a gap in the classifcation of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the the eigenspaces is n−1, then the metric is a warped product where t…
The paper examines properties of W-curvature tensor in relativistic space-times.
problem Investigating the properties and implications of the W-curvature tensor in relativistic space-times. method Analyzing the semi-symmetry and divergence properties of the energy-momentum tensor in relation to the W-curvature tensor. result Space-times with specific properties of the W-curvature tensor are classified as Einstein or Codazzi type. Characterizes pseudo B-symmetric spacetimes and their implications in f(R) gravity.
problem Characterizing pseudo B-symmetric spacetimes and their properties.
method Analyzes Codazzi type of B-tensor and applies f(R) gravity model.
result Pseudo B-symmetric spacetimes with Codazzi type B-tensor are conformally flat and Robertson-Walker spacetimes.
We discuss a gap in Besse's book, recently pointed out by Merton, which concerns the classification of Riemannian manifolds admitting a Codazzi tensors with exactly two distinct eigenvalues. For such manifolds, we prove a structure theorem, without adding extra hypotheses and then we conclude with some application of t…
The paper characterizes metallic pseudo-Riemannian manifolds using conjugate connections and tensor structures.
problem Characterizing metallic pseudo-Riemannian manifolds.
method Using conjugate connections and tensor structures, the paper derives new characterizations and conditions for these manifolds.
result A necessary and sufficient condition for a non-integrable metallic pseudo-Riemannian manifold to be a quasi metallic pseudo-Riemannian manifold is derived.
The paper explores connections and curvature tensors on specific geometric manifolds.
problem Investigating properties of connections and curvature tensors on almost anti-Hermitian manifolds.
method Introduced three types of conjugate connections and proved a Klein group result.
result Derived a necessary and sufficient condition for an anti-Kähler structure.
Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on S×R is the tangent bundle of the Teichmüller space of S, if S is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relat…
Characterizes symmetric Killing tensors on specific Lie groups.
problem Understanding Killing tensors on specific Lie groups.
method Completely characterized left-invariant symmetric Killing tensors on almost abelian Lie groups.
result All such tensors are decomposable into polynomial expressions of Killing vector fields and metric.
Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.
problem Classifying hypersurfaces in the Sol4_0 geometry.
method Analyzing hypersurfaces with Codazzi tensors and parallel second fundamental forms.
result Full classification of hypersurfaces in Sol4_0, including parallel and totally umbilical types.
We study left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.
In this paper we examine the structure of Riemannian manifolds with a special kind of Codazzi tensors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy hypersurfaces for any weakly irreducible holonomy representation with parallel spinors, i.e. with a holonomy group which is a semi…
We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric pro…
Equations link metrics with tensors, revealing curvature constraints.
problem Understanding curvature properties of geometric structures.
method Formal analogies to Einstein-Maxwell equations, studying Codazzi and conformal Killing equations.
result Constraints on scalar curvature of metrics in solutions.
Research explores the space-like embeddings in pseudo-hyperbolic space, finding geometric frames and actions.
problem Flat maximal space-like embeddings in pseudo-hyperbolic space.
method Description of Codazzi tensors, introduction of pseudo-Kähler metrics, Hamiltonian actions, moment maps, and geometric frames.
result Existence of two Hamiltonian actions with moment maps and geometric global Darboux frame.
Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.
problem Calculating curvatures in holomorphic fibrations with degenerate Hermitian forms.
method Theory of Chern connections and curvature forms for degenerate Hermitian forms on holomorphic vector bundles.
result Positive holomorphic sectional curvature in Grassmannian bundles if the base does.
Study on 3D trans-Sasakian manifolds with η-Einstein solitons.
problem Characterizing 3D trans-Sasakian manifolds with η-Einstein solitons.
method Analyzing properties of Codazzi type and cyclic parallel Ricci tensors on 3D trans-Sasakian manifolds.
result Examples and properties of 3D trans-Sasakian manifolds with η-Einstein solitons.
Introduces a new geometric structure for statistical manifolds with degenerate metrics.
problem Degenerate metrics in statistical manifolds affect geometric structures and applications.
method Introduces quasi-Codazzi structure for degenerate metrics and coherent tangent bundles.
result Generalizes geometric structures and relations for statistical models with degenerate metrics.
The paper explores a duality between conformally flat metrics and hyperbolic geometry.
problem Locally conformally flat metrics and their relationship to hyperbolic geometry.
method Analyzes the Gauss-Codazzi equations and their duals in hyperbolic space.
result Identifies a unique solution for B^ when g^ is locally conformally flat. Study on 3D Lie groups finds all generalized Einstein metrics.
problem Classifying generalized Einstein metrics on 3D Lie groups.
method Developed theory of left-invariant generalized pseudo-Riemannian metrics, computed Ricci tensor, determined all metrics.
result Determined all generalized Einstein metrics on three-dimensional Lie groups.
Study on generalized quasi-Einstein structures in contact geometry.
problem Characterizing and understanding generalized quasi-Einstein structures in contact geometry.
method Investigation of properties, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifolds.
result Normal metric contact pair manifolds with generalized quasi-constant curvature are generalized quasi-Einstein manifolds.
The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric…
The study classifies 4D manifolds with specific curvature properties.
problem Classifying 4D Riemannian manifolds with zero divergence curvature.
method Analyzes the Codazzi equation and curvature tensor properties.
result For non-classical 4D manifolds, Ricci tensor has four distinct eigenvalues.
In the presented paper left-invariant pseudo-Riemannian metrics on four-dimensional Lie groups with zero Schouten-Weyl tensor are investigated. The complete classification of these metric Lie groups is obtained in terms of the structure constants of corresponding Lie algebras.
The paper extends Weyl's theorem to equiaffine hypersurfaces.
problem Understanding equiaffine hypersurfaces and their properties.
method Developing a quasi-Codazzi structure and projectively flat dual connection.
result Equiaffine hypersurfaces are characterized by a quasi-Codazzi structure.
On Spinc manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spinc manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and po…
Study odd generalized Einstein metrics on 3D Lie groups.
problem Classify odd generalized Einstein metrics on 3D Lie groups.
method Left-invariant generalized connections, divergence operators, and Ricci tensors.
result Describe all odd generalized Einstein metrics on all 3D Lie groups.
We call a metric m-quasi-Einstein if RicXm (a modification of the m-Bakry-Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and…
We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform Lp-bounded solution sequence for p>2, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated comp…
A n-dimensional Lie group G equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on G. Relatively to this affine structure we show that the left invariant Poisson tensor π+ corresponding to $\om^+$ is po…
Let G be a Lie group of even dimension and let (g,J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure $(g…
We prove the Reilly formula for a class of elliptic divergence differential operator LAu=div(A∇u), where A is a (1,1)-Codazzi tensor field. Then we get some estimates for the first positive eigenvalue of the operator.
Paper proves conditions for 3D submanifolds to embed in 4D space.
problem Conditions for 3D Riemannian submanifolds to embed in R4. method Used symbolic method from classical invariant theory.
result Two known intrinsic conditions are sufficient for embedding.
The curvature of Gauss maps for flat submanifolds is studied in space forms.
problem Understanding the curvature of Gauss maps for flat submanifolds in space forms.
method Analyzing the Codazzi symmetry and using the Weingarten operators to derive the Riemann curvature tensor.
result The Riemann curvature tensor of the Gauss image is determined by the curvature and Weingarten operators of the original submanifold.
The paper classifies metrics on Heisenberg group's cotangent bundle.
problem Investigating moduli spaces of left invariant metrics on cotangent bundles of Heisenberg group.
method Algebraic approach combined with geometrical tools like classification of hyperbolic plane conics.
result Detailed classification of various types of metrics and their properties.
The Bonnet theorem is proven for statistical manifolds.
problem Locally embeddable statistical manifolds in flat spaces.
method Using statistical embedding and the Gauss--Codazzi--Ricci equations.
result Statistical manifolds with specific tensor properties are locally embeddable to flat statistical manifolds.
Study of η-Ricci solitons on Kenmotsu 3-manifolds.
problem Exploring η-Ricci solitons on Kenmotsu 3-manifolds. method Examined various types of η-Ricci solitons on Kenmotsu 3-manifolds, including those with specific curvature conditions. result Existence of proper η-Ricci solitons on Kenmotsu 3-manifolds.