Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in (2,2) pseudo-hyperbolic space has finite volume. Researchers classify complete maximal submanifolds in pseudo-hyperbolic space.
problem Classifying complete maximal submanifolds in pseudo-hyperbolic space.
method Full classification through mathematical analysis.
result A complete classification of complete maximal p-dimensional spacelike submanifolds in Hp,q. Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.
problem Finding maximal surfaces with given boundary curves in pseudo-hyperbolic spaces.
method Defined and proved the existence of unique solutions using asymptotic Plateau problem and analysis of pseudo-holomorphic curves.
result Existence and uniqueness of maximal surfaces with specified boundary conditions.
Spinorial methods prove inequalities for special hypersurfaces.
problem Proving inequalities for hypersurfaces in pseudo-hyperbolic manifolds.
method Spinorial techniques to prove inequalities.
result Proves a Heintze-Karcher type inequality.
Higgs bundles and pseudo-hyperbolic geometry help study maximal representations.
problem Understanding maximal representations into Hermitian Lie groups.
method Higgs bundle theory and pseudo-hyperbolic geometry.
result Interference between Higgs bundle theory and pseudo-hyperbolic geometry.
We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic sp…
Developed a half-space model for pseudo-hyperbolic space.
problem Modeling pseudo-hyperbolic space for any dimensions.
method Created an isometric embedding of pseudo-hyperbolic space into a half-space.
result Geodesics, totally geodesic submanifolds, horospheres, and isometry group are described in the half-space model.
We classify the semi-Riemannian submersions from a pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are connected and totally geodesic. Also we obtain the classification of the semi-Riemannian submersions from a complex pseudo-hyperbolic space onto a Riemannian manifold under the …
Characterizes polygonal surfaces in pseudo-hyperbolic spaces.
problem Understanding polygonal surfaces in pseudo-hyperbolic spaces.
method Characterizes polygonal surfaces by total curvature finiteness and asymptotic flatness, using comparison of ideal boundaries.
result Polygonal surfaces have parabolic type and polynomial quartic differential.
The study of curvature in spacelike submanifolds in pseudo-hyperbolic spaces.
problem Curvature properties of spacelike submanifolds in pseudo-hyperbolic spaces.
method Analyzing scalar and Ricci curvatures, proving bounds and characterizing submanifolds.
result Nonpositive scalar curvature for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces.
The study characterizes quasiperiodic surfaces in pseudo-hyperbolic spaces with curvature conditions.
problem Characterizing quasiperiodic surfaces in pseudo-hyperbolic spaces.
method Curvature conditions, Gromov hyperbolicity, conformal hyperbolicity.
result Limit curves of quasiperiodic surfaces in the Einstein Universe have canonical quasisymmetric parametrizations.
The paper analyzes Gauss maps of Lorentzian surfaces in anti-de Sitter space.
problem Determining the type numbers of pseudo-hyperbolic Gauss maps of Lorentzian surfaces.
method Investigates the type numbers of pseudo-hyperbolic Gauss maps of oriented Lorentzian surfaces with constant curvatures in anti-de Sitter space.
result Investigates the behavior of type numbers of pseudo-hyperbolic Gauss maps along parallel families of surfaces.
Minimal surfaces linked to Higgs bundles in pseudo-hyperbolic spaces.
problem Proving Labourie's theorem and extending it to new cases.
method Establishing infinitesimal rigidity of minimal surfaces and using it to prove the theorem.
result New proof of Labourie's theorem and extension to Collier's components.
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.
We classify the effective and transitive actions of a Lie group G on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that G is a closed, connected Lie subgroup of SO0(n−r,r+1), the connected component of the indefinite special orthogonal group. Assumin…
We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three and the metrics induced on fibres are negative definite. Also, we obtain the classificat…
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
Study Riemannian geometry of maximal surface group representations in pseudo-hyperbolic space.
problem Characterize the geometry of maximal surface group representations in pseudo-hyperbolic space.
method Introduced a scalar product on the first cohomology group, leading to a Riemannian metric on the smooth locus.
result Found totally geodesic sub-varieties and orbifold structures in the space of representations.
Compactifies a component by studying metric degeneration.
problem Compactify the SO(2,3)-Hitchin component.
method Study metric degeneration on a surface in pseudo-hyperbolic space.
result Establish closure in projectivized geodesic currents space.
Researchers study horospherical transforms on hyperboloids.
problem Analyzing harmonic analysis on pseudo-hyperbolic spaces.
method Investigate horospherical transform and its inversion in 3 hyperboloid examples.
result Horospherical inversion formulas can be derived from classical Radon inversion.
It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and…
A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called w-variables. In this paper, we consider the case when pinched octahedra appear as a b…
Researchers found a spinorial representation for surfaces in 3D Lorentzian spaces.
problem Representation of surfaces in Lorentzian spaces.
method Spinorial representation approach.
result New correspondences between different types of surfaces in 3D spaces.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces formed by pseudo null, partially null, and null curves.
method Obtained parametric expressions and geometric invariants of canal hypersurfaces.
result Characterizations of tubular hypersurfaces in E14. The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces in Lorentz-Minkowski 4-space.
method Analyzing geometric invariants and conditions for canal hypersurfaces.
result Characterizations of canal hypersurfaces, including flatness and minimality conditions.
We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an elem…
The study characterizes canal hypersurfaces formed by non-null curves with parallel frame in Minkowski space-time.
problem Characterizing canal hypersurfaces in Minkowski space-time.
method Obtained canal hypersurfaces by pseudo hyperspheres or pseudo hyperbolic hyperspheres with parallel timelike normal vector field.
result Gaussian curvature, mean curvature, and principal curvatures of canal hypersurfaces were derived.
Warning signs about the developing economic crisis in Greece were present in the growth rate of the Gross Domestic Product (GDP) and in the growth of the GDP well before the economic collapse. The growth rate was strongly unstable. On average, in less than 50 years, it decreased 10-folds but after reaching a low minimu…
Paper introduces a hyperbolic Gaussian distribution for better learning in hierarchical data.
problem Learning hierarchical data in hyperbolic space.
method Developed a novel hyperbolic distribution for gradient-based learning.
result Demonstrated improved learning on various datasets.
Study classifies hypersurfaces in 4D Lorentz-Minkowski space using specific operators.
problem Classifying tubular hypersurfaces in 4D Lorentz-Minkowski space.
method Analysis of Gauss map and linearized operators L1 and L2. result Classifications of hypersurfaces with specific types of Gauss maps.