Classifies 5D homogeneous geometries with specific isotropy properties.
problem Classifying 5D homogeneous geometries with certain isotropy properties.
method Thurston's classification of homogeneous geometries, focusing on irreducible and trivial isotropy representations.
result Identifies 5D geometries with irreducible isotropy as irreducible Riemannian symmetric spaces and those with trivial isotropy as specific solvable Lie groups.
Classifies 5D homogeneous geometries with nontrivial reducible linear isotropy.
problem Classifying 5D homogeneous geometries with specific properties.
method Thorough classification using Thurston's criteria and analysis of linear isotropy representations.
result Found a countably infinite family of geometries diffeomorphic to S3imesS2. Paper classifies 5D Thurston geometries, finding an uncountable family.
problem Classifying 5-dimensional Thurston geometries.
method Summarizes the general classification, providing a full list and examples.
result An uncountable family of geometries discovered.
Extreme black holes with $\SU(2)$ symmetry have a specific near horizon geometry.
problem Understanding the near horizon geometry of extreme black holes with $\SU(2)$ symmetry.
method Analyzing the near horizon geometry of 5D extreme black holes with $\SU(2)$ symmetry.
result The near horizon geometry of these black holes must be that of a Berger sphere.
Consider the nonstandard embedding of SO(3) into SO(5) given by the 5-dimensional irreducible representation of SO(3), henceforth called SO(3)_\ir. In this note, we study the topology and the differential geometry of 5-dimensional Riemannian manifolds carrying such an SO(3)_\ir structure, i.\,e. with a reduction of the…
The study proves the existence and uniqueness of near-horizon geometries for 5D black holes.
problem Existence and uniqueness of near-horizon geometries for 5-dimensional black holes.
method Proved existence and uniqueness of bi-axisymmetric near-horizon geometries with specific topologies and charges.
result Existence and uniqueness of near-horizon geometries for 5D black holes established up to isometry.
Recent work Bobienski-Nurowski on 5-dimensional Riemannian manifolds with an SO(3) structure prompts us to investigate which Lie groups admit such a geometry. The case in which the SO(3) structure admits a compatible connection with torsion is considered. This leads to a classification under special behaviour of the on…
Study finds area inequalities for black hole horizons in 5D minimal supergravity.
problem Bounding the area of black hole horizons in 5D minimal supergravity.
method Established area-angular momentum-charge inequalities for stable marginally outer trapped surfaces with U(1)2 symmetry. result Unique geometries saturating the inequalities correspond to extreme black hole solutions.
We reduce to various absolute parallelisms, namely to certain {e}-structures on manifolds of dimensions 7, 6, 5, the biholomorphic equivalence problem or the intrinsic CR equivalence problem for generic submanifolds M^5 in C^4 of CR dimension 1 and of codimension 3 that are maximally minimal and are geometry-preserving…
Constructs canonical frames for specific distributions, proving maximality and describing germs.
problem Local geometry of rank 2 distributions with specific cube dimensions.
method Uniform construction of canonical absolute parallelism, iterative Cartan deprolongation.
result Automatic maximality condition holds at generic points, covering all cases.
The study examines conditions for weak nearly cosymplectic manifolds to split into products.
problem Understanding the curvature and topology of weak nearly cosymplectic manifolds.
method Analyzes the conditions for splitting and characterizes specific manifolds.
result Conditions for weak nearly cosymplectic manifolds to become Riemannian products are identified.
On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a canonical Cartan connection is constructed after reduction to a certain partially expl…
Survey on four-dimensional Thurston geometries with Riemannian metrics.
problem Understanding homogeneous manifolds and their Riemannian metrics.
method Description and analysis of specific Thurston geometries in four dimensions.
result Exhibition of all Riemannian metrics invariant under group actions.
A flying saucer's flight is mathematically modeled with geometric structures.
problem Modeling the complex flight maneuvers of a flying saucer.
method Imposing nonlinear restrictions on the saucer's velocity to define geometric structures.
result Three types of flat parabolic geometries are defined for the saucer's configuration space.
New families of hyperbolic polyhedra yield infinitely many unique reflection groups.
problem Understanding commensurability classes of compact Coxeter polyhedra in hyperbolic spaces.
method Analyzing families of compact Coxeter polyhedra constructed by Makarov.
result Proves infinitely many commensurability classes in 4- and 5-dimensional hyperbolic spaces.
Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the normal scalar curvature. This property is conformal invariant; hence we study them in the framework of Moebius geometry, and restrict…
Geometric structures on surfaces relate to 2-plane distributions in 5D.
problem Understanding geometric properties of vector bundles and distributions.
method Study of horizontal 2-plane distributions on 5-manifolds.
result Established a connection between surface projective differential geometry and 2-plane distribution growth.
5D contact structures are universally tight via Bourgeois construction.
problem Understanding tightness of 5D contact structures.
method Explicit construction of contact structures on VimesT2. result All constructed contact structures are universally tight.
We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or 5-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact ma…
Gravitational instantons are constructed as superpositions of Atiyah-Hitchin and Taub-NUT geometries.
problem Constructing gravitational instantons from Atiyah-Hitchin and Taub-NUT geometries.
method A gluing construction that captures the superposition of moduli spaces of centred SU(2) monopoles and Taub-NUT manifolds.
result Gravitational instantons are explicitly shown to be superpositions of Atiyah-Hitchin and Taub-NUT geometries.
Positive mass theorem and Yamabe equation on CR manifolds
problem Positive mass theorem and Yamabe equation on CR manifolds
method Positive mass theorem and Yamabe equation on CR manifolds
result Positive mass theorem in 3-dimensional CR geometry
Heegaard diagrams for 5-manifolds help in understanding their structure.
problem Understanding the structure of 5-dimensional manifolds.
method Introducing a version of Heegaard diagrams for 5-dimensional cobordisms and manifolds, showing diffeomorphisms through specific moves.
result Every smooth 5-manifold can be represented by a Heegaard diagram, and diagrams representing diffeomorphic manifolds are related by certain moves.
Almost contact manifolds with B-metric are considered. There are studied three natural connections (i.e. linear connections preserving the structure tensors) determined by conditions for their torsions. These connections are investigated on a family of Lie groups considered as 5-dimensional almost contact B-metric mani…
The six nondegeneracy conditions of geometric nature that are satisfied by the only six possibly existing nondegenerate general classes I, II, III-1, III-2, IV-1, IV-2 of 5-dimensional CR manifolds are shown to be readable instantaneously from their elementarily normalized respective defining graphed equations, without…
Embeddings of surfaces in 5-manifolds are classified by an invariant.
problem Classifying isotopy classes of surface embeddings in 5-manifolds.
method Construction of an invariant defined in terms of homotopy groups.
result Isotopic embeddings of surfaces in 5-manifolds are classified by the invariant.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
Study on a specific type of Lie algebras with Kähler and contact properties.
problem Characterizing and classifying transversely Kähler almost contact metric Lie algebras.
method Analyzing properties of Lie algebras with contact forms and Kähler structures, considering center dimensions and quotient properties.
result Classification of 5-dimensional η-Einstein transversely Kähler almost contact metric Lie algebras.
The paper confirms the existence of 5D regular static vacuum solutions with multiple black holes and Kasner asymptotics.
problem Existence of 5D regular static vacuum solutions with multiple black holes.
method Construction of specific examples with different horizon topologies and analysis of spacetime properties.
result Existence of 5D vacuum solitons with Kasner asymptotics and regular static space-periodic spacetimes.
Study symplectification of rank 2 distributions and their connections.
problem Understanding symplectification and Cartan prolongations of rank 2 distributions.
method Using Tanaka-Morimoto theory and symplectification procedure for rank 2 distributions.
result Demonstrates the existence of normal Cartan connections and iterated prolongations for rank 2 distributions.
Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are constant, then M is isoparametric. Moreover, We give all possible values for squared…
We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor Υ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold (M,g,Υ) with Riemannian metric g and ternary form generate…
5D shrinking Ricci solitons with constant scalar curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons with constant scalar curvature.
method Proving rigidity by showing they are finite quotients of a known space.
result 5D shrinking gradient Ricci solitons with constant scalar curvature are rigid.
In this paper, we write down Seiberg-Witten equations on contact metric manifolds of dimension 5. Any contact metric manifold has a spin^c structure. For Dirac equation we use Dirac type operators associated to the generalized Tanaka-Webster connection on spin^c spinor bundle of a contact metric manifold. For curvature…
We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain associate…
This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice subgroup. We re-examine Alexander's criteria for existence of lattices on solvable…
Calegari's 4-spheres from fibered knots are proven standard.
problem Proving Calegari's 4-spheres from fibered knots are standard.
method 5-dimensional handlebody techniques and mapping class groups of 3-dimensional handlebodies.
result All Calegari's homotopy 4-spheres from fibered knots are diffeomorphic to the standard 4-sphere.
We prove that every homotopy class of almost contact structures on a closed 5-dimensional manifold admits a contact structure.
The paper characterizes ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds.
problem Characterizing ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds. method Analyzing conditions for compressing, balancing, or enlarging ∗-Ricci-Bourguignon on Kenmotsu manifolds; estimating curvature properties; featuring with torse-forming vector fields; providing an example. result Found conditions and curvature properties for ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds. Paper proves existence of minimum energy solutions in 5D contact spin manifolds.
problem Finding minimum energy solutions for CR Yamabe equation in 5D contact spin manifolds.
method Spinorial approach based on a positive mass theorem.
result Existence of minimum energy solutions in 5D contact spin manifolds.
The paper explains how to draw 3D pictures of 5D manifolds.
problem Visualizing 5-dimensional manifolds is impossible due to human limitations.
method Using open books and contact geometry, the paper provides a method to draw 3D representations of 5D manifolds.
result The approach allows for the depiction and analysis of 5-dimensional manifolds.
Study on CR-manifolds reveals symmetry algebra properties.
problem Characterizing symmetry algebras of 5D CR-manifolds.
method Analytic and holomorphic properties examined.
result Symmetry algebras have specific dimensions and structures.
The paper classifies foliations formed by generic coadjoint orbits of specific Lie groups.
problem Classifying foliations formed by generic coadjoint orbits of certain Lie groups.
method Analyzing Lie groups with specific dimensions and nilradicals, proving foliations in the coadjoint representation.
result The family of generic coadjoint orbits forms a measurable foliation in the Lie groups considered.
We study five dimensional geometries associated with the 5-dimensional irreducible representation of GL(2,R). These are special Weyl geometries in signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The reduction is obtained by means of a conformal class of totally symmetric 3-tensors. Among all…
Let M be a CR manifold of hypersurface type, which is Levi degenerate but also satisfying a k-nondegeneracy condition at all points. This might be only if dim M is greater than or equal to 5 and if dim M = 5, then k= 2 at all points. We prove that for any 5-dimensional, uniformly 2-nondegenerate CR manifold M there exi…
The paper connects complex spacetimes to twistor spaces and finds an almost contact structure.
problem Understanding the geometry of complex spacetimes and their relation to twistor spaces.
method Using a correspondence between 5D complex spacetimes and 4D twistor spaces, the paper constructs almost contact structures.
result A 5-dimensional K-contact manifold can be derived from the Ren-Wang twistor space.
Classifies CR-embedded 3-folds in 5D hyperquadrics using moving frames.
problem Classifying CR-embedded 3-folds in 5D hyperquadrics.
method Applying Cartan's moving frames method to classify CR-embedded manifolds.
result Identifies homogeneous CR 3-folds in 5D hyperquadrics.
Study proves rigidity of certain hypersurfaces in 5- and 6-manifolds.
problem Proving rigidity of stable minimal hypersurfaces in 5- and 6-manifolds.
method Nonnegative 3-intermediate Ricci curvature combined with uniformly positive k-triRic curvature.
result No complete noncompact stable minimal hypersurface in a closed 5-dimensional manifold with positive sectional curvature.
The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.
problem Understanding the number of stabilizations required to convert 5D s-cobordisms into product cobordisms.
method Utilizes Gabai's 4D light bulb theorem and Freedman Quinn invariant.
result Determines the number of stabilizations needed for 5D s-cobordisms.