Statistical manifolds with constant curvature are projectively flat and symmetric.
problem Characterizing statistical manifolds with constant curvature.
method Analyzing the curvature and projective flatness properties of statistical manifolds.
result Statistical manifolds with constant curvature are projectively flat and symmetric.
Classifies minimal immersions from S2 into specific flag manifolds.
problem Classifying minimal immersions from S2 into specific flag manifolds. method Classification based on constant curvature and low-dimensional flag manifolds.
result Primitive minimal immersions of constant curvature from S2 into F2,1,1 and F2,2,1 are classified. Compact Einstein manifolds with specific curvature operators are constant curvature spaces.
problem Characterizing compact Einstein manifolds with certain curvature operators.
method Bochner technique and Li's result [10]
result Compact Einstein manifolds with specific curvature operators are constant curvature spaces.
Study on constant curvature immersions of surfaces into flag manifolds.
problem Investigate constant curvature immersions of Riemann surfaces into flag manifolds.
method Investigate pseudoholomorphic maps and invariant metrics on flag manifolds.
result Unitarily equivalent primitive immersions of the two-sphere into full flag manifolds have constant curvature under all invariant metrics.
Investigates immersions in Sn using complex spinors.
problem Generalizing immersions of Riemann surfaces to Spin-manifolds.
method Uses complex spinors and the Dirac equation.
result Investigates submanifolds of SpinC-manifolds of constant curvature.
The paper studies constant curvature holomorphic two-spheres in complex Grassmann manifold.
problem Investigating constant curvature holomorphic two-spheres in complex Grassmann manifold.
method Exploring the theory of functions of one complex variable to determine curvature distribution and construct examples.
result Explicit characterization and construction of non-homogeneous constantly curved holomorphic two-spheres.
This paper solves Hilbert's fourth problem for constant curvature metrics.
problem Classifying metric geometries with shortest straight lines in constant curvature settings.
method Analyzing Finsler manifolds with constant flag curvature, deriving distance formulas, and proving global geometry theorems.
result Complete characterization of global geometry for constant flag curvature metrics.
We generalize here our general procedure for constructing constant curvature maps of 2-spheres into Grassmannian manifolds G(m,n) this time concentrating our attention on maps which are non-holomorphic. We present some expressions describing these solutions in the general case and discuss how to use these results to co…
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
The condition for the curvature of a statistical manifold to admit a kind of standard hypersurface is given. We study the statistical hypersurfces of some types of the statistical manifolds (M,∇,g), which enable (M,∇(α),g),∀α∈R to admit the structure of a constant curvature.
In this paper we prove that the holonomy group of a simply connected locally projectively flat Finsler manifold of constant curvature is a finite dimensional Lie group if and only if it is flat or it is Riemannian.
We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified, respectively, with different integral-differential formulas and semigroup inequaliti…
Classifies weakly Einstein hypersurfaces in spaces of constant curvature.
problem Identifying weakly Einstein hypersurfaces in spaces of constant curvature.
method Complete classification through tensor analysis and geometric properties.
result Hypersurfaces are either products of spaces of constant curvature or rotation hypersurfaces.
A Riemannian manifold is called IP, if the eigenvalues of its skew-symmetric curvature operator are pointwise constant. It was previously shown that for all n\ge 4, except n=7, any IP manifold either has constant curvature, or is a warped product, with some specific function, of a line and a space of constant curvature…
Sharp mapping properties and regularization for X-ray transform on disks of constant curvature.
problem Sharp mapping properties and regularization of X-ray transform.
method Derive functional relations and mapping properties using elliptic differential operators.
result Theoretical possibility of regularized inversions for X-ray transform.
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.
A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study a triharmonic isometric immersion into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-enegy and the L^4-norm of the tension field are fin…
New statistical manifolds derived from identity map biharmonicity.
problem Deriving new statistical manifolds from identity map biharmonicity.
method Statistical biharmonicity of identity maps, semi-equiaffine condition, constant curvature.
result Determined statistical structures of new class of manifolds.
Study on statistical manifolds with product structures and their properties.
problem Investigating statistical manifolds with almost product structures.
method Proving properties of para-Kähler-like statistical manifolds and deriving properties of statistical submersions compatible with almost product structures.
result The statistical structure of a para-Kähler-like statistical manifold of constant curvature is a Hessian structure.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
The Eisenhart problem of finding parallel tensors treated already in the framework of quasi-constant curvature manifolds in \cite{x:j} is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on p…
The study examines properties of f-contact manifolds with constant curvature.
problem Investigating constant curvature in f-contact manifolds. method Analyzing (κ,μ)-nullity condition and f-sectional curvature. result An f-(κ,μ) manifold with constant f-sectional curvature implies specific conditions on μ and κ. The paper studies the holonomy of spherically symmetric Finsler metrics.
problem Investigating the holonomy group of spherically symmetric projective Finsler metrics of constant curvature.
method Analyzing the holonomy group for n-dimensional projective Finsler metrics of constant curvature, focusing on the spherically symmetric case. result For a simply connected manifold, the holonomy group is isomorphic to Diffo(Sn−1), the connected component of the identity of the group of smooth diffeomorphisms on the (n−1)-dimensional sphere. Develops discrete geometry for non-constant curvature surfaces.
problem Modeling surfaces of non-constant curvature, especially with non-constant negative curvature.
method Derived and numerically integrated Lelieuvre formulas for C1,1 hyperbolic surfaces. Proposed iterative and fast marching methods for solving implicit equations and computing geodesic distances. result Explicit construction of immersions is not provided, but equations are described implicitly.
The study characterizes constant curvature manifolds using ruled surfaces.
problem Characterizing manifolds of constant curvature using ruled surfaces.
method Investigating ruled surfaces in 3d Riemannian manifolds, finding stiction curve, distribution parameter, and fundamental forms.
result Identifies necessary and sufficient conditions for extrinsically flat surfaces to be ruled and proves manifold properties.
The study proves that geodesic spherical curves characterize manifolds of constant curvature.
problem Characterizing manifolds of constant curvature using spherical curves.
method Proving the converse of the known linear equation for RM frames, and providing two additional characterizations.
result Geodesic spherical curves on a manifold characterize constant sectional curvature.
In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers…
Study biharmonic curves in warped product manifolds with curvature analysis.
problem Characterize biharmonic curves in warped product manifolds.
method Establish a main theorem, analyze four cases, construct examples.
result Reveal curvature-related characteristics of biharmonic curves.
The paper studies special Finsler spaces with Hp-scalar curvature.
problem Characterizing and investigating Finsler spaces with specific scalar curvatures.
method Intrinsic investigation and various conditions for transformations between Finsler spaces.
result Conditions for transforming Finsler spaces of scalar curvature to those of Hp-scalar curvature. We prove families of uniform (Lr,Ls) resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author \cite{KRS}. In the case of the sphere we take advantage of the fact that the half-wave group of th…
We confirm a conjecture of Hamilton: On compact manifolds the normalized Ricci flow evolves metrics with positive curvature operators to limit metrics with constant curvature.
The study finds obstructions to certain Riemannian metrics using Lorentzian geometry.
problem Finding obstructions to curvature distinguished Riemannian metrics.
method Dual Lorentzian metrics and Penrose's plane wave limit.
result Necessary local conditions for certain Riemannian metrics.
The paper analyses the extrema of p-energy functional on a Finsler space with constant curvature.
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Ad…
Symmetry algebras of Killing vector fields and conformal Killing vectors fields can be extended to Killing-Yano and conformal Killing-Yano superalgebras in constant curvature manifolds. By defining Z-gradations and filtrations of these superalgebras, we show that the second cohomology groups of them are triv…
In this paper, we study rigidity problems for hypersurfaces with constant curvature quotients H2kH2k+1 in the warped product manifolds. Here H2k is the k-th Gauss-Bonnet curvature and H2k+1 arises from the first variation of the total integration of $…
Only the 6-sphere has constant curvature hypersurfaces in nearly Kähler manifolds.
problem Characterizing hypersurfaces with constant curvature in six-dimensional nearly Kähler manifolds.
method Proving hypersurfaces are η-quasi umbilical in specific spaces, then using non-existence results. result Only the 6-sphere has constant curvature hypersurfaces in nearly Kähler manifolds.
The article studies spinor and tensor fields on curved spaces, deriving formulas and spectra.
problem Understanding spinor and tensor fields on curved spaces.
method Weitzenböck-type formulas, explicit factorization of Laplace operator, representation theory.
result Explicit factorization of the Laplace operator and spectra calculation on constant curvature spaces.
Study constructs closed curves with constant curvature on cylinders and tori.
problem Creating closed curves with constant curvature.
method ODEs and symmetry arguments, starting with cylinders, then tori, and finally Frenet-Serret equations.
result Closed constant curvature space curves constructed on cylinders and tori.
The paper characterizes contact metric manifolds with specific solitons.
problem Characterizing contact metric manifolds with ∗-conformal Ricci solitons. method Analyzing properties of (2n+1)-dimensional N(k)-contact metric manifolds. result The manifold is locally isometric to a flat (n+1)-dimensional manifold and an n-dimensional manifold of constant curvature 4. Study of Schwarzian derivative on Finsler manifolds with constant curvature.
problem Characterizing the role of the Schwarzian derivative in Finsler manifolds of constant curvature.
method Developed integrability conditions and rigidity results for Möbius equations on Finsler manifolds.
result Complete Finsler manifolds of positive constant Ricci curvature are homeomorphic to the n-sphere if they admit non-trivial Möbius mappings.
We show a geometric rigidity of isometric actions of non compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.
New space for polarized manifolds with constant curvature metrics.
problem Constructing moduli spaces for polarized manifolds.
method Constructing a moduli space with constant scalar curvature Kähler metrics.
result The moduli space admits a natural Kähler metric.
The abstract finds conditions for creating curves of constant curvature.
problem Finding conditions for closed embedded curves of constant curvature.
method Using Almgren-Pitts min-max method for geodesic curvature.
result Closed embedded curves of any prescribed constant curvature found in S2. The study proves conditions for constant curvature submanifolds in space forms.
problem Understanding the conditions for constant curvature submanifolds in space forms.
method Analyzing isometric immersions and properties of normal bundles.
result Substantial codimension is p=n−1 for specific curvature conditions. Study on hypersurfaces in pseudo-Euclidean space with constant curvature or rotational properties.
problem Characterizing hypersurfaces in pseudo-Euclidean space.
method Defined and studied warped product hypersurfaces with constant sectional curvature or rotational properties.
result Hypersurfaces in pseudo-Euclidean space either have constant curvature or are contained in rotational hypersurfaces.
Paper finds conditions for generalized Kropina spaces to have constant curvature.
problem Classifying Finsler spaces of constant curvature.
method Obtained necessary and sufficient conditions for generalized Kropina spaces to be of constant flag curvature.
result Conditions for generalized Kropina spaces to have constant curvature.
A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics wit…