Study on singularities of discrete Weingarten surfaces in Riemannian and Lorentzian spaceforms.
problem Analysis of singularities in discrete Weingarten surfaces.
method Defined and analyzed singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in 3D Riemannian and Lorentzian spaceforms.
result Discussed singularities of discrete surfaces with non-zero constant Gaussian curvature and parallel surfaces of discrete minimal and maximal surfaces.
Study classifies semi-discrete linear Weingarten surfaces with Weierstrass-type representations and analyzes their singularities.
problem Characterizing semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities.
method Established properties, classified, and analyzed the singularities of semi-discrete linear Weingarten surfaces in Riemannian and Lorentzian spaceforms.
result Defined and analyzed the singularities of semi-discrete linear Weingarten surfaces, including those with non-zero constant Gaussian curvature, parallel surfaces of minimal and maximal surfaces, and constant mean curvature 1 surfaces in de Sitter 3-space.
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
Solves curvature equations using parabolic flows in various spaces.
problem Finding strictly convex, spacelike solutions to curvature equations.
method Employing curvature flows without global terms in Riemannian and Lorentzian spaces.
result Solves a broad class of elliptic prescribed curvature equations.
The question whether a Riemannian manifold is geodesically connected can be studied from geometrical as well as variational methods, and accurate results can be obtained by using the associated distance and related properties of the positive-definiteness. It is natural to state this problem in manifolds with (possibly …
In this paper we numerically construct CMC deformations of the Lawson minimal surfaces ξg,1 using a spectral curve and a DPW approach to CMC surfaces in spaceforms.
Wind Riemannian structures generalize Randers metrics and are classified for constant flag curvature.
problem Classifying wind Riemannian structures of constant flag curvature.
method Using the natural data for Zermelo navigation problem, constructing WRS from a Riemannian metric and a vector field (wind).
result Local and global classification of wind Riemannian structures of constant flag curvature.
The paper constructs Dirac-harmonic maps using spinors.
problem Unclear conditions for Dirac-harmonic maps in previous work.
method Uses harmonic and twistor spinors to construct Dirac-harmonic maps.
result Shows conditions for existence of solutions under special assumptions.
In Euclidean and Hyperbolic space, and the hemisphere in Sn, geodesic balls maximize the gap λ2−λ1 of Dirichlet eigenvalues, amoung domains with fixed λ1. We prove an upper bound on λ2−λ1 for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms.
Closed curves in 3D and 4D spheres are dense in Sobolev norms.
problem Characterizing closed curves in 3D and 4D spheres.
method Density in Sobolev W2,2-norm. result Spaces of closed finite gap curves are dense in the spaces of closed curves.
We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works…
Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.
problem Classifying conformally flat hypersurfaces and characterizing rotational hypersurfaces.
method Derives a sharp inequality relating eigenvalues of trace-free matrices and applies it to hypersurfaces.
result New proof of the classification of conformally flat hypersurfaces and construction of a functional for rotational hypersurfaces.
In n-dimensional Euclidean space E^n, harmonic curvatures of a non-degenerate curve defined by Özdamar and Hacisalihoğlu [4]. In this paper, We define a new type of curves called LC helix when the angle between tangent of this curve and LC parallel vector field in space form is constant. Furthermore, several characteri…
Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Emery's curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distribut…
Stability results for geometric equations in warped product spaces.
problem Geometric partial differential equations in warped product spaces.
method Stability theorem development for level sets of functions.
result Quantitative stability theorems for Serrin's problem and Alexandroff's theorem.
Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
problem Density of closed finite gap curves in Sobolev spaces.
method Proof of density in Sobolev spaces for curves in hyperbolic and Euclidean 3-spaces.
result Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…
Cosmologists are taking a renewed interest in multiconnected spherical 3-manifolds (spherical spaceforms) as possible models for the physical universe. To understand the formation of large scale structures in such a universe, cosmologists express physical quantities, such as density fluctuations in the primordial plasm…
Defines metrics for Lorentzian spaces and explores maximal developments.
problem Defining metrics and maximal developments in Lorentzian spaces.
method Defining Lorentzian spaces and using functorial properties.
result Explicit non-spacetime example of maximal globally hyperbolic Lorentzian space.
Survey of connected holonomy groups in Lorentzian manifolds.
problem Classifying connected holonomy groups of Lorentzian manifolds.
method Simplified construction of Lorentzian metrics and applications to specific manifolds.
result Obtained a simplification in constructing Lorentzian metrics with all possible connected holonomy groups.
Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
problem Global optimality of extremal trajectories in a series of Lorentzian structures.
method Analysis of a one-parametric series of left-invariant Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
result Properties of the Lorentzian structures deform to those of the sub-Lorentzian structure in a limit case.
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.
It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …
Study gluing of Lorentzian length spaces and their causal ladder properties.
problem Compatibility of Lorentzian amalgamation with length space properties.
method Conditions for gluing Lorentzian length spaces and criteria for causal ladder preservation.
result Gluing of Lorentzian length spaces yields again a Lorentzian length space under certain conditions.
Defines doubling conditions for Lorentzian spaces linked to curvature bounds.
problem Relating doubling conditions to curvature bounds in Lorentzian geometry.
method Defines doubling conditions using chronological diamonds and proves implications with timelike curvature bounds.
result Relates doubling to curvature bounds in Lorentzian geometry.
Research explores Lorentzian distances on a specific geometric plane.
problem Investigating Lorentzian structures on a 2D geometric plane.
method Analyzes sectional curvature, attainable sets, and Lorentzian length maximizers.
result Describes distance properties and spheres in the context of Lorentzian geometry.
The study examines conditions for high-dimensional Lorentzian cobordisms with specific structure groups.
problem Conditions for the existence of Lorentzian and weak Lorentzian cobordisms with $\Spin(1, n)_0$ structure group.
method Analysis of necessary and sufficient conditions, computation of cobordism groups, restrictions on gravitational kinks.
result Restrictions on the gravitational kink numbers of $\Spin(1, n)_0$-weak Lorentzian cobordisms.
The study proves a transverse diameter theorem for Lorentzian foliations.
problem Understanding the geometry of foliations in Lorentzian spacetimes.
method Developed a novel causality structure on leaf spaces via transverse Lorentzian geometry.
result Derived a transverse diameter theorem for Lorentzian foliations and orbifolds.
Finds a limit on Lorentzian manifolds.
problem Limiting the number of Lorentzian manifolds.
method Functor from Lorentzian to Riemannian manifolds.
result Finiteness result on Lorentzian manifolds.
Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.
The paper classifies 3D Lorentzian Ein(2) Lie groups.
problem Classifying 3D Lorentzian Ein(2) Lie groups. method Complete classification through mathematical analysis.
result Three-dimensional Lorentzian Ein(2) Lie groups have been completely classified. Study submersions from Lorentzian Sasakian manifolds, finding conditions and decompositions.
problem Characterizing submersions from Lorentzian Sasakian manifolds.
method Analyzing semi-Riemannian submersions from Lorentzian (para)Sasakian manifolds.
result Decomposition theorems for anti-invariant submersions.
Study finds all 4D homogeneous semi-symmetric Lorentzian manifolds.
problem Characterizing semi-symmetric Lorentzian manifolds.
method Used algebraic curvature tensors to determine all 4D manifolds.
result Found all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.
Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.
problem Characterizing singularities on timelike minimal surfaces.
method Constructing timelike minimal surfaces as Lorentzian harmonic maps and analyzing their singularities.
result Criteria for cuspidal edges, swallowtails, and cuspidal cross caps are provided.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
Defines new metrics for Lorentzian spaces and their convergence.
problem Defining metrics for Lorentzian spaces and their convergence.
method Abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, defining bounded Lorentzian-metric spaces, and proving stability under GH limits.
result GH limits of Lorentzian-metric spaces are isometric and homeomorphic.
Compact group found for special Lorentzian manifolds.
problem Understanding the conformal group of specific Lorentzian manifolds.
method Analyzing real analytic Lorentzian manifolds with finite fundamental group.
result The conformal group is compact for these manifolds.
Study on Lorentzian spaces with curvature bounds, proving comparison theorems.
problem Understanding curvature bounds in Lorentzian spaces.
method Introduced normalized angle for Lorentzian pre-length spaces, proving comparison theorems.
result Established local Lorentzian Toponogov theorem and Alexandrov convexity property.
Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.
problem Identifying Lorentzian manifolds embeddable in Minkowski spacetime.
method Characterization and proof of embeddability conditions.
result Lorentzian manifolds embeddable in Minkowski spacetime coincide with globally hyperbolic spacetimes.
Diagonalizes metrics of 3D Lorentzian manifolds.
problem Diagonalizing metrics of 3D Lorentzian manifolds.
method Applying the technique of moving frames.
result Every smooth Lorentzian 3-manifold admits an atlas with a diagonal metric.
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We in…
If an m+2-manifold M is locally modeled on $\RR^{m+2}$ with coordinate changes lying in the subgroup $G=\RR^{m+2}\rtimes ({\rO}(m+1,1)\times \RR^+)$ of the affine group ${\rA}(m+2)$, then M is said to be a \emph{Lorentzian similarity manifold}. A Lorentzian similarity manifold is also a conformally flat Lorentzia…
Proves existence of longest paths in sub-Lorentzian problems.
problem Existence of longest paths in sub-Lorentzian problems.
method Generalizes classical theorem for Lorentzian manifolds to sub-Lorentzian problems.
result Longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups.
Study classifies Riemann solitons on specific 3D Lorentzian groups.
problem Classifying Riemann solitons on three-dimensional Lorentzian Lie groups.
method Complete classification through analysis of left-invariant structures.
result Comprehensive classification of Riemann solitons on these groups.
Three functors link Lorentzian geometry concepts.
problem Finiteness results, singularity theorems, boundary constructions.
method Review of three functors from Lorentzian categories.
result Novel functor from ordered measure spaces to Lorentzian pre-length spaces.
The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentz…
Holonomy of Weyl connections in Lorentzian space classified.
problem Classifying holonomy algebras of Weyl connections in Lorentzian signature.
method Classification through construction of examples of Weyl connections with all possible holonomy algebras.
result Examples of Weyl connections with all possible holonomy algebras constructed.