We construct invariants of four-dimensional piecewise-linear manifolds, represented as simplicial complexes, with respect to rebuildings that transform a cluster of three 4-simplices having a common two-dimensional face in a different cluster of the same type and having the same boundary. Our construction is based on t…
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48 results for “four-dimensional simplices”
In this paper we construct and study a new 15-vertex triangulation of the complex projective plane $\CP^2$. The automorphism group of is isomorphic to . We prove that the triangulation is the minimal by the number of vertices triangulation of $\CP^2$ admitting a chess colouring of four-dimens…
Proof confirms all smooth 4D Schoenflies balls are geometrically simple.
problem Geometric simplicity of smooth 4D Schoenflies balls.
method Complete proof based on announced result.
result All smooth four-dimensional Schoenflies balls are geometrically simply connected.
We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. First, we write out form…
This paper quantizes geometry using SL(2,C) Chern-Simons theory and flat connections.
problem Quantizing four-dimensional quantum geometry.
method Using a correspondence between flat connections and four-dimensional simplices, the paper quantizes geometry via complex SL(2,C) Chern-Simons theory.
result The quantum geometrical states are represented by the 3d blocks of analytically continued Chern-Simons theory, and in the semiclassical limit, the three-dimensional Chern-Simons action becomes the discrete Einstein-Hilbert action of a 4-simplex.
We write out some sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex - a triangulation of a manifold. If the sequence turns out to be an acyclic complex then one can construct a manifold…
The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.
problem Proving positivity of simplicial volume for specific types of 4-manifolds.
method Using the Gauss-Bonnet theorem for Riemannian simplices, the paper shows that for closed nonpositively curved 4-manifolds with nonzero Euler characteristic, simplicial volume is positive.
result The paper confirms conjectures about the relationship between simplicial volume and Euler characteristic for four-dimensional manifolds.
Improved formulation of spinfoam quantum gravity with cosmological constant, ensuring all amplitudes are finite and providing semiclassical asymptotics.
problem Ensuring the finiteness of spinfoam amplitudes and providing semiclassical asymptotics for quantum gravity.
method Using state-integral model of PSL(2, ) Chern-Simons theory and implementing simplicity constraint.
result All spinfoam amplitudes are finite and provide semiclassical asymptotics with oscillatory terms related to the Regge action.
The paper explores symmetry properties of four-dimensional Walker manifolds.
problem Characterizing geometric properties of Walker manifolds.
method Investigation of curvature tensors and characterization theorems.
result Characterization theorems obtained for four-dimensional Walker manifolds.
Sizes of spaces of triangulations of 4-manifolds and balanced presentations of the trivial groupmath.GT
The paper finds exponential lower bounds for distances between triangulations and balanced presentations.
problem Finding distances between triangulations and balanced presentations of the trivial group.
method Using Tietze transformations and bistellar transformations, the paper establishes exponential lower bounds for distances.
result Exponential lower bounds for distances between triangulations and balanced presentations.
Study finds all 4D neutral manifolds.
problem Classifying neutral manifolds in four dimensions.
method Examined homogeneous semi-symmetric neutral manifolds.
result Identified all four-dimensional neutral manifolds.
New types of Ricci solitons found in 4D Lorentzian geometry.
problem Understanding Ricci solitons in Lorentzian geometry.
method Analyzing four-dimensional Lie groups for left-invariant Lorentz metrics.
result Any connected and simply connected 4D Lie group admits a left-invariant Lorentz metric that is a Ricci soliton.
The study examines four-dimensional gradient Ricci solitons and their properties.
problem Characterizing four-dimensional complete gradient shrinking Ricci solitons.
method Proving properties and providing curvature estimates for solitons under specific conditions.
result Conditions for four-dimensional complete gradient shrinking Ricci solitons.
The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.
problem Classifying homogeneous hypersurfaces in specific 4D geometries.
method Analyzing isometry groups and applying classification techniques.
result Homogeneous hypersurfaces identified in , and .
In this note we prove that any four-dimensional half conformally flat gradient steady Ricci soliton must be either Bryant's soliton or Ricci flat. We also classify four-dimensional half conformally flat gradient shrinking Ricci solitons with bounded curvature.
Study finds all 4D Lie groups with harmonic curvature.
problem Finding Lie groups with harmonic curvature in 4D.
method Analyzing pp-wave Lie groups and determining harmonic curvature conditions.
result Description of 4D pp-wave Lie groups obtained.
The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
Investigates a new four-dimensional energy related to Willmore energy.
problem Exploring a new conformally invariant energy in four dimensions.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the new energy are smooth and do not include minimal hypersurfaces.
A note on the classification of four-dimensional gradient steady and expanding Ricci solitonsmath.DG
Study classifies 4D Ricci solitons with specific curvature conditions.
problem Classifying 4D gradient steady and expanding Ricci solitons with given curvature properties.
method Asymptotically cylindrical and conical assumptions; half-harmonic and half-nonnegative isotropic curvature conditions.
result Partial classification of 4D gradient expanding Ricci solitons with half-nonnegative isotropic curvature.
The paper classifies 4D gradient Ricci solitons under specific curvature conditions.
problem Classifying four-dimensional gradient Ricci solitons under various conditions.
method Analyzing specific curvature conditions to classify solitons.
result Classification results for 4D gradient Ricci solitons under certain conditions.
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.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to -trees and estimate their dimensions.
result Estimates the dimensions of maximal simplices for both rational and irrational trees.
The paper proves conditions for four-dimensional gradient shrinking solitons to be flat or have specific curvature bounds.
problem Conditions for four-dimensional gradient shrinking solitons to be flat or have specific curvature bounds.
method Proving conditions for four-dimensional gradient shrinking solitons using Ricci curvature and Weyl curvature.
result Conditions for four-dimensional gradient shrinking solitons to be flat or have specific curvature bounds.
The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.
problem Conditions for Riemannian connections and semi-simplicity of Lie algebras.
method Using almost product structures and spray, the paper provides necessary and sufficient conditions for these properties.
result Equivalence of semi-simplicity of Lie algebras to derived ideal coincidence, interiority of derivations, and adjoint representation semi-simplicity.
Study on a specific obstruction in four-dimensional geometry.
problem Singular Yamabe obstruction of a hypersurface.
method Derived various explicit formulas from the original definition.
result Relate formulas to literature and prove elementary.
Four-dimensional, oriented Lie algebras which satisfy the tame-compatible question of Donaldson for all almost complex structures on are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are…
We completely classify the algebraic Ricci solitons of four-dimensional pseudo-Riemannian generalized symmetric spaces.
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to double covering, as G/L where G is a simply connected solvable Lie group and L is a …
New criteria judge combinatorial equivalence of polytopes to products of simplices.
problem Determining combinatorial equivalence of polytopes to products of simplices.
method Combination of combinatorial, geometric, and topological conditions inspired by toric topology.
result New criteria for judging combinatorial equivalence of polytopes to products of simplices.
Study classifies Lie groups with specific metric properties.
problem Investigating metrics on Lie groups with zero Schouten-Weyl tensor.
method Complete classification through Lie algebra structure constants.
result Complete classification of metric Lie groups.
A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable g…
We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…
Classifies special types of 4D spaces.
problem Classifying non-reductive 4D spaces.
method Classification based on conformal Einstein manifolds.
result Classification of non-reductive 4D homogeneous conformally Einstein manifolds.
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 pseudo-hyperbolic space has finite volume.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
Study on smoothness of 4D Willmore-type hypersurfaces.
problem Investigating smoothness of critical points of a 4D Willmore-type energy.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the energy are smooth.
Rigidity for 4D Willmore submanifolds with boundary.
problem Understanding critical points of Willmore energy with boundary conditions.
method Proving a 4-Willmore equation and establishing curvature estimates.
result Four dimensional Willmore submanifolds with totally geodesic boundary are umbilic.
The paper is devoted to the investigation of four-dimensional Kahler manifolds admitting non-affine H-projective mappings. We find all such manifolds which are non-Einstein. In the paper also Kahler manifolds admitting infinitesimal H-projective transformations are determined. It is proved that the class of Kahler mani…
Study four-dimensional Ricci flow using branching curves.
problem Characterize Type I singularities in four-dimensional Ricci flow.
method Associate one-parameter families of curves to points in the product of two projective lines.
result Characterize singularity models in four-dimensional Ricci flow.
Paper proves biharmonic hypersurfaces in nonzero space form have constant mean curvature.
problem Proving constant mean curvature for biharmonic hypersurfaces.
method Careful analysis of Gauss and Codazzi equations.
result Positive answer to Balmus-Montaldo-Oniciuc's conjecture for four dimensional hypersurfaces.
The four-dimensional sphere is uniquely rigid in terms of scalar curvature.
problem Proving the uniqueness of the four-dimensional sphere in terms of scalar curvature.
method Combining harmonic map heat flow and Ricci flow to rule out non-isometric maps.
result A smooth map of non-zero degree from a four-dimensional manifold to the unit four-sphere is an isometry.
Study on harmonic maps from projective plane to 4D sphere, showing empty spaces for even degrees.
problem Investigating harmonic maps from projective plane to 4D sphere.
method Using twistor lifts to analyze spaces of harmonic maps.
result Spaces are empty for even harmonic degree and path-connected for harmonic degree less than 6.
We consider four dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension two. These foliations produce local complex-valued harmonic morphisms.
Study on scalar curvature decay in four-dimensional steady solitons.
problem Behavior of scalar curvature at infinity on four-dimensional steady solitons.
method Analysis of scalar curvature decay rate and asymptotic cone properties.
result Linear scalar curvature decay away from edges, stronger inequality if scalar curvature vanishes at infinity.
Critical metrics for the quadratic curvature functional on complete four-dimensional manifoldsmath.DG
Critical metrics on four-dimensional manifolds are either Einstein or product of two-dimensional manifolds.
problem Classifying critical metrics of a curvature functional on complete four-dimensional manifolds.
method Analyzing the curvature operator and energy condition to prove metric properties.
result Complete four-dimensional manifolds with finite energy are either Einstein or product of two-dimensional manifolds.
Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.
We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.