Researchers develop Q-curvature for convex hypersurfaces using ambient metrics.
problem Calculating Q-curvature for convex hypersurfaces in projective manifolds.
method Using the ambient metric, they construct GJMS operators and relate Q-curvature to the logarithmic coefficient in volume expansion.
result Derived first and second variation formulas for strictly convex domains.
Study volume expansion on convex domains using Blaschke metric.
problem Volume expansion of Blaschke metric on strictly convex domains.
method Expressed logarithmic coefficient L as integrals of affine invariants over the boundary and formulated intrinsic geometry as conformal Codazzi structure.
result L is a global conformal invariant of the boundary.
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. Symmetry reduction solves Gauss-Codazzi equations for curved surfaces.
problem Solving Gauss-Codazzi equations for surfaces with curvature.
method Symmetry reduction of the equations to a form solvable by the sixth Painlevé function.
result Known solutions of the equations can be recovered through this method.
The study proves non-existence theorems for Codazzi tensors on Riemannian manifolds.
problem Proving non-existence theorems for Codazzi tensors on Riemannian manifolds.
method Using theorems connecting manifold geometry and subharmonic functions.
result Several Liouville-type non-existence theorems for Codazzi tensors.
In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We obtain Gauss-Codazzi-Ricci equations and a conformal Bonnet theorem characterizi…
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.
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.
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 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. New approach classifies conformal Killing vector fields for FLRW space-time.
problem Classifying conformal Killing vector fields for FLRW space-time.
method Introduced new perspective on conformal Killing vector fields for FLRW space-time, considering three cases for the conformal factor.
result Nine conformal vector fields on FLRW, six of which are Killing and the rest non-Killing.
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…
We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We…
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 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.
In [dLMu05], DeLellis and Müller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface Σ⊆R3 with area normalized to H2(Σ)=4π, it was shown that ∥AΣ−id∥L2(Σ)≤C∥AΣ0∥L2(Σ) , and building on…
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.
E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from sol…
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 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.
Study classifies 4D gradient Ricci solitons with harmonic Weyl curvature.
problem Characterizing 4D gradient Ricci solitons with harmonic Weyl curvature.
method Motivated by Cao-Chen's and Deditzinski's works, the method involves Codazzi tensors and local isometries.
result Proves soliton metrics are locally isometric to four types of metrics.
Investigates curvature properties of Robinson-Trautman metric.
problem Examines curvature characteristics of Robinson-Trautman metric.
method Analyzes various pseudosymmetric structures and properties of the metric.
result The metric exhibits multiple curvature properties including Roter type, 2-quasi-Einstein, and Riemann compatible.
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.
In this paper, we study the structure of the singular set for a C1 smooth surface in the 3-dimensional Heisenberg group H1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation …
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…
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…
The study proves a theorem for surfaces using Codazzi operators and investigates parallel mean curvature surfaces.
problem Understanding surfaces with parallel mean curvature in product spaces.
method Intrinsic Klotz-Osserman theorem and Simons' formula.
result The existence of surfaces with parallel mean curvature in product spaces with non-positive Gaussian curvature.
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.
The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.
problem Classifying hypersurfaces in Nil^4.
method Using Lie group structure and Codazzi conditions.
result Characterization and classification of minimal hypersurfaces in Nil^4.
The study defines and analyzes nearly Kähler and Kähler-Codazzi manifolds.
problem Defining and analyzing nearly Kähler and Kähler-Codazzi manifolds.
method Definition and comparison of nearly Kähler and Kähler-Codazzi manifolds in various geometries.
result Nearly Kähler type manifolds exist only in Hermitian and para-Hermitian contexts, while Kähler-Codazzi type manifolds reduce to Kähler type manifolds in all four geometries.
Study of contravariant pseudo-Hessian manifolds and their Poisson structures.
problem Understanding properties of contravariant pseudo-Hessian manifolds.
method Investigation of flat connections and symmetric bivector fields satisfying a contravariant Codazzi equation.
result Association of a Poisson tensor to contravariant pseudo-Hessian manifolds.
Study lift metrics and connections on tangent bundles of Riemannian manifolds.
problem Investigate geometric properties of tangent bundles and their lifts.
method Analyze lift metrics and connections on TM of (M,g), and study statistical and Codazzi couples. result Prove a result on 1-Stein and Osserman structures on TM. The paper proves weak continuity of Cartan structural system on semi-Riemannian manifolds with lower regularity.
problem Weak continuity of the Cartan structural system on semi-Riemannian manifolds with lower regularity.
method Formulated and proved a geometric compensated compactness theorem, deduced Lp weak continuity of the Cartan structural system. result Weak continuity of the Cartan structural system and Gauss-Codazzi-Ricci system on semi-Riemannian manifolds with lower regularity.
The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the…
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.
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…
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.
Integrability of the (2+1)-dimensional Gauss-Codazzi-Mainardi equation is considered. It is shown that this equation is the particular cases of the Yang-Mills-Higgs-Bogomolny and self-dual Yang-Mills equations.
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.
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.
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].
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…
We prove a global smooth isometric immersion for negatively curved surfaces with finite total curvature.
problem Finding a sufficient condition for a complete negatively curved surface to be isometrically embedded in R^3.
method Developed new techniques to overcome slow decay and oscillations of Gauss curvature, reformulating the Gauss-Codazzi equations as a symmetric hyperbolic system.
result Proved the global existence of a smooth solution to the Gauss-Codazzi system, achieving a global smooth isometric immersion of the surface into R^3.
The paper develops a comprehensive theory of submanifolds in conformal geometries.
problem Understanding submanifolds in conformal geometries of arbitrary dimension.
method Using conformal tractor calculus, the paper provides a new framework for studying submanifolds.
result The theory includes a new notion of distinguished submanifolds and characterizes them in various dimensions.
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.
In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give essentially sharp generalizations of some classical theorems of surface theory that mainly depen…
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.
Unified approach to global weak rigidity of Riemannian manifolds and their immersions.
problem Global weak rigidity of Gauss-Codazzi-Ricci equations and isometric immersions of Riemannian manifolds.
method Unified intrinsic approach, div-curl structure, compensated compactness theorem, global intrinsic div-curl lemma.
result Established global weak rigidity of GCR equations and isometric immersions of Riemannian manifolds with lower regularity.