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.
Classifies hypersurfaces with constant curvature in product spaces.
problem Classifying hypersurfaces with constant curvature in product spaces.
method Analyzing hypersurfaces in R^k x S^{n-k+1} and R^k x H^{n-k+1} for 2 <= k <= n-1.
result Complete description of hypersurfaces with constant curvature in product spaces of space forms.
Survey on geodesics on tetrahedra in curved spaces.
problem Understanding geodesics on tetrahedra in curved spaces.
method Analyzing geodesics on regular tetrahedra in spaces of constant curvature.
result Results on the behavior of simple closed geodesics.
Geodesics in Randers spaces of constant curvature are classified.
Paper shows rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
problem Understanding rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
method Pointwise hypersurface invariant analysis for minimal hypersurfaces in spaces of constant curvature.
result Rotationally symmetric minimal hypersurfaces in 5D spaces are rigid.
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.
2-stein submanifolds in space forms have constant curvature if normal connection is flat or codimension is 2.
problem Characterizing submanifolds with constant curvature in space forms.
method Analyzing submanifolds with flat normal connection or codimension 2.
result 2-stein submanifolds have constant curvature under specified conditions.
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
Curves with constant curvature are flexible and can be deformed.
problem Understanding the flexibility of curves with constant curvature.
method Proving the parametric C1-dense relative h-principle for curves of constant curvature. result Two knots of constant curvature are isotopic and homotopic if their self-linking numbers are equal.
Study of curves in dual space with constant curvature and torsion.
problem Classifying curves in dual space with specific geometric properties.
method Defined curvature and torsion for curves in dual space, classified curves with constant properties, and proved existence theorems.
result Established fundamental theorem of existence for dual curves with prescribed curvature and torsion.
Local rigidity proved for convex hypersurfaces in spaces of constant curvature.
problem Proving rigidity of convex hypersurfaces in spaces of constant curvature.
method Analyzing isometric convex hypersurfaces in spaces of constant curvature of dimension n≥4. result Two convex isometric hypersurfaces are congruent locally around their corresponding under strict convexity isometries.
Constructs orthogonal coordinates in curved spaces.
problem Separating variables in curved spaces.
method Explicit construction of orthogonal coordinates and transformations.
result Explicit formulas for Killing tensors and Stäckel matrices.
Method constructs orthogonal curvilinear coordinates in constant curvature spaces.
problem Creating orthogonal coordinates in spaces of constant curvature.
method Modification of Krichever's method for Euclidean space, applied to constant curvature spaces.
result Examples of orthogonal coordinate systems on the sphere and hyperbolic plane constructed.
Some examples of three-dimensional metrics of constant curvature defined by solutions of nonlinear integrable differential equations and their generalizations are constructed. The properties of Riemann extensions of the metrics of constant curvature are studied. The connection with the theory of normal Riemann spaces a…
We study isometric immersions of surfaces of constant curvature into the homogeneous spaces H2xR and S2xR. In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c>0 into H2xR and a unique one into S2xR when c>1, up to isometries of the ambient space. Mor…
Unified proof of end-point estimates for Radon transform on curved spaces.
problem Proving end-point estimates for the totally-geodesic Radon transform on spaces of constant curvature.
method Unified geometric approach to prove end-point estimates for Radon transform on spaces of constant curvature.
result Unified formula for the k-plane transform of radial functions on spaces of constant curvature. In this paper, we generalize Magnanini-Sakaguchi's result [MS3] from Euclidean space to spaces of constant curvature. More precisely, we show that if a conductor satisfying the exterior geodesic sphere condition in the space of constant curvature has initial temperature 0 and its boundary is kept at temperature 1 (at a…
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.
We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.
The classification of Finsler spaces of constant curvature is an interesting and important topic of research in differential geometry. In this paper we obtain necessary and sufficient conditions for generalized Kropina space to be of constant flag curvature.
The paper derives height estimates for surfaces with constant curvature in warped product spaces.
problem Estimating heights of surfaces with constant curvature in warped product spaces.
method Use of conformal parameters and geometric applications to derive height estimates.
result Derives height estimates for surfaces with positive extrinsic or mean curvature in RimesfR2. The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.
The aim of the present paper is to give two \emph{intrinsic} generalizations of Akbar-Zadeh's theorem on a Finsler space of constant curvature. Some consequences, of these generalizations, are drown.
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their curvature quotients.
The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.
problem Investigating the Funk-Finsler metric in spaces of constant curvature.
method Explicitly computed S-curvature, Riemann curvature, Ricci curvature, and flag curvature. result The S-curvature and flag curvature of the Funk-Finsler metric in hyperbolic, spherical, and Euclidean spaces are bounded. In this paper, we continue studying the 6-dimensional pseudo-Riemannian space V^6(g_{ij}) with signature [++--], which admits projective motions, i. e. continuous transformation groups preserving geodesics. In particular, we determine a necessary and sufficient condition that the 6-dimensional rigid h-spaces have const…
Paper proves rigidity of convex hypersurfaces in various spaces.
problem Proving the uniqueness of convex hypersurfaces in multidimensional spaces.
method Generalizing Senkin's theorem to higher dimensions and constant curvature spaces.
result Rigidity of convex hypersurfaces in En+1, n≥3. This paper solves part of a problem by constructing surfaces with specific curvature and singularities.
problem Solving an open problem by Gálvez, Hauswirth, and Mira regarding constant curvature metrics with conical singularities.
method Established a geometric correspondence between metrics and isometric immersions into Euclidean 3-space, constructing a family of surfaces.
result Explicitly constructed a family of surfaces with constant curvature one and two conical singularities.
The two-body problem with a central interaction on simply connected constant curvature spaces of an arbitrary dimension is considered. The explicit expression for the quantum two-body Hamiltonian via a radial differential operator and generators of the isometry group is found. We construct a self-adjoint extension of t…
We relate the total curvature and the isoperimetric deficit of a curve γ in a two-dimensional space of constant curvature with the area enclosed by the evolute of γ. We provide also a Gauss-Bonnet theorem for a special class of evolutes.
Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…
The aim of the present paper is to provide an intrinsic investigation of two special Finsler spaces whose defining properties are related to Berwald connection, namely, Finsler space of scalar curvature and of constant curvature. Some characterizations of a Finsler space of scalar curvature are proved. Necessary and su…
We generalize a theorem by J. Choe on capillary surfaces for arbitrary 3-dimensional spaces of constant curvature. The main tools in this paper are an extension of a theorem of H. Hopf due to S.-S. Chern and two index lemmas by J. Choe.
The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (in…
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.
Compact spacelike hypersurface with constant curvature and boundary angles must be part of a hyperboloid.
problem Characterizing compact spacelike hypersurfaces in Minkowski space with constant curvature and boundary conditions.
method Using an auxiliary function and an associated integral equality, the authors prove the rigidity of the hypersurface.
result Compact spacelike hypersurfaces with constant curvature and boundary angles are rigid, being parts of hyperboloids unless entirely in the boundary hyperplane.
Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
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 on polyharmonic curves on spheres and space forms.
problem Classifying polyharmonic curves of constant curvature.
method Analyzing curves on spheres and space forms, deriving explicit families.
result New insights into higher order variational problems.
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. Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces with constant curvature in hyperbolic space.
method Deriving curvature estimates to prove existence for all curvature values.
result Existence of smooth hypersurfaces for all possible curvature values.
Study of Moncrief lines' behavior in curved space-times.
problem Understanding the asymptotic behavior of Moncrief lines in curved space-times.
method Analysis of geodesic laminations and convergence to Thurston boundary.
result Moncrief lines converge to a unique point in the Thurston boundary.
The paper provides an intrinsic proof of a theorem about Landsberg spaces.
problem Proving Numata's theorem on Landsberg spaces of scalar curvature.
method Intrinsic point of view and coordinate-free proof using Finsler geometry.
result All Landsberg spaces of dimension n≥3 of non-zero scalar curvature are Riemannian spaces of constant curvature. 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.
Classifies self-similar curve shortening flows in hyperbolic 2-space.
problem Classifying self-similar curve shortening flows in hyperbolic 2-space.
method Analyzes and classifies solutions in hyperbolic 2-space.
result Completes the classification of self-similar curve shortening flows in constant curvature model spaces in 2-dimensions.
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 periodic polyhedra found in curved spaces.
problem Existence of periodic polyhedra in curved spaces.
method Using Archimedean solids and transformations, constructing polyhedra with specific properties.
result Existence of compact polyhedral surfaces in spaceforms.
Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.
problem Investigating curvature measures in spherical, hyperbolic, and de Sitter spaces.
method Establishing a unifying framework for curvature measures in real-analytic spaces of constant curvature.
result Floating bodies and duality in non-Euclidean spaces are connected to curvature measures in Euclidean space.