Researchers find non-diagonal Einstein metrics in various signatures.
problem Finding non-diagonal four-dimensional cohomogeneity-one Einstein metrics in different signatures.
method Explicitly seeking and constructing new examples of non-diagonal Einstein metrics, particularly in neutral signature.
result Construct new examples of neutral signature non-diagonal Bianchi type VIII Einstein metrics with self-dual Weyl tensor.
T. Saito and M. Teragaito asked whether Berge knots of type VII are hyperbolic, and showed that some infinite sequences of the knots are hyperbolic. We show that Berge knots of types VII and VIII are hyperbolic except the known sequence of torus knots. We used the Reidemeister torsions. As a result, the Alexander polyn…
For the simply connected compact exceptional Lie group E8, we determine the structure of subgroup (E8)σ,σ′ of E8 which is the intersection (E8)σ∩(E8)σ′. Then the space E8/(E8)σ,σ′ is the exceptional Z2×Z2- symmetric space of type EVIII-VIII-VIII, and that we…
Machine learning identifies Shakespeare and Fletcher's contributions to Henry VIII.
problem Determining the relative contributions of Shakespeare and Fletcher in Henry VIII.
method Combined analysis of vocabulary and versification with machine learning techniques.
result Supports canonical division and new modifications of Henry VIII's authorship.
Method constructs Bryant surfaces in hyperbolic space.
problem Constructing Bryant type surfaces in hyperbolic space.
method Bianchi-Calo type construction method
result Constructs Bryant type surfaces in hyperbolic space.
No minimal chart of type (7) exists.
problem Characterizing minimal charts of specific types.
method Analyzing charts based on edge labels and white vertex counts.
result There is no minimal chart of type (7).
The rational homotopy type of certain manifolds is determined by their cohomology and Bianchi-Massey tensor.
problem Determining the rational homotopy type of (n-1)-connected manifolds.
method Defining the Bianchi-Massey tensor and using it to prove the rational homotopy type is determined by cohomology and tensor.
result The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3 is determined by their cohomology and Bianchi-Massey tensor.
Study on triaxial Bianchi IX metrics and Ricci flow singularity.
problem Understanding singularity formation in triaxial Bianchi IX metrics under Ricci flow.
method Investigates sufficient conditions for Type I singularity in triaxial Bianchi IX metrics on foliated manifolds.
result Generalizes previous studies on rotationally symmetric and triaxial metrics.
We characterize Bianchi-Bäcklund transformations of surfaces of positive constant Gauss curvature in terms of dressing actions of the simplest type on the space of harmonic maps.
We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper-symplectic structures in dimension four.
The aim of this paper is to describe the local Bianchi identities for an h-normal Γ-linear connection of Cartan type ∇Γ on the first-order jet space J1(R,M). In this direction, we present the local expressions of the adapted components of the torsion and curvature d-tensors produced by ∇Γ and we gi…
We investigate the set of spacetime general coordinate transformations (G.C.T.) which leave the line element of a generic Bianchi Type Geometry, quasi-form invariant; i.e. preserve manifest spatial Homogeneity. We find that these G.C.T.'s, induce special time-dependent automorphic changes, on the spatial scale factor m…
Completed classification of Einstein spaces with specific metric properties.
problem Classifying Einstein spaces with a specific type of Stackel metric.
method Invariant under three-parameter abelian group of motions, completed classification of vacuum and electrovacuum spaces.
result Complete list of metrics for Einstein spaces in privileged coordinate systems.
Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensiona…
An invariant description of Bianchi Homogeneous (B.H.) 3-spaces is presented, by considering the action of the Automorphism Group on the configuration space of the real, symmetric, positive definite, 3×3 matrices. Thus, the gauge degrees of freedom are removed and the remaining (gauge invariant) degrees, are th…
Study proves existence of Kähler-Einstein metrics and Ricci flat Kähler metrics in 4-manifolds.
problem Existence of Kähler-Einstein metrics and Ricci flat Kähler metrics in 4-manifolds.
method Proves existence through cohomogeneity one triaxial Kähler-Einstein metrics and generalized PDEs for Ricci flat Kähler metrics.
result Proves existence of complete cohomogeneity one triaxial Kähler-Einstein metrics and local existence of Ricci flat Kähler metrics.
New Bianchi-convex sets generalize Ricci flow maximum principle.
problem Generalizing maximum principle for Ricci flow.
method Introducing Bianchi-convex sets.
result Hamilton's maximum principle extended to Bianchi-convex sets.
In this paper we describe the local Ricci and Bianchi identities for an h-normal N-linear connection DΓ(N) on the dual 1-jet space J^{1*}(T,M). To reach this aim, we firstly give the expressions of the local distinguished (d-) adapted components of torsion and curvature tensors produced by DΓ(N), and then we analyze th…
We study an analogue of the classical Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation. We show how to construct the Bianchi-Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutabili…
Lie groups form specific 3D manifolds with unique properties.
problem Understanding geometric structures on Lie groups.
method Constructing 3D almost contact B-metric manifolds using Lie groups.
result Identified specific classes and types of Lie algebras.
We construct new examples of torsional heterotic backgrounds using duality with orientifold flux compactifications. We explain how duality provides a perturbative solution to the type I/heterotic string Bianchi identity. The choice of connection used in the Bianchi identity plays an important role in the construction. …
Study degenerate Bianchi transformations for pseudo-spherical submanifolds in 5D space.
problem Characterize three-dimensional pseudo-spherical submanifolds with degenerate Bianchi transformations.
method Complete description through holonomically degenerate Bianchi transformations.
result Obtained a complete description of degenerate pseudo-spherical submanifolds.
10D IIA Superspace is put on shell by imposing duality-symmetric Bianchi identities on super-flux densities.
problem Dimensional reduction of 11D supergravity to 10D IIA
method Cyclification of 11D supergravity
result Full 10D IIA supergravity is put on shell with duality-symmetric Bianchi identities.
Study of extended Bianchi-Cartan-Vranceanu spaces in seven dimensions.
problem Understanding geometric properties of extended Bianchi-Cartan-Vranceanu spaces.
method Investigation of Levi-Civita connection, Ricci curvatures, Killing fields, and geodesics in seven-dimensional EBCV spaces.
result Characterization of geometric properties in extended Bianchi-Cartan-Vranceanu spaces.
The paper discusses a new method for constructing two-step Darboux transforms of isothermic surfaces.
problem Constructing two-step Darboux transforms of isothermic surfaces.
method Sym-type construction using parallel sections of the associated family.
result All two-step Darboux transforms of an isothermic surface are given without further integration.
New 3D Lie group structures identified in specific B-metric manifolds.
problem Classifying 3D almost contact B-metric manifolds.
method Constructing manifolds using Lie groups and classifying them.
result Identified new classes of 3D almost contact B-metric manifolds.
We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the …
Study modular forms in spectral action of Bianchi IX gravitational instantons.
problem Arithmetic and modular properties of spectral action for Bianchi IX gravitational instantons.
method Parametrization of gravitational instantons in terms of theta functions, modular function analysis, isospectrality of Dirac operators.
result Each Seeley-de Witt coefficient gives rise to vector-valued and ordinary modular functions of weight 2.
Transformed quadrics from 2D to higher dimensions.
problem Generalizing quadric transformations to higher dimensions.
method Bianchi's Hazzidakis transformation method.
result Generalization to higher dimensional quadrics.
Bianchi proves Mumford conjecture using branched covers.
problem Mumford conjecture in algebraic topology.
method Moduli spaces of branched covers.
result Proof of Mumford conjecture.
Researchers found explicit formulae for special solitons in 3D spaces.
problem Understanding Ricci solitons in specific 3D Lorentzian spaces.
method Analyzing homogeneous Ricci solitons on Bianchi-Cartan-Vranceanu spaces.
result Explicit formulae for Ricci solitons were derived.
Researchers created group presentations for specific Bianchi groups.
problem Finding explicit group presentations for certain Bianchi groups.
method Applied Macbeath's theorem to a large horoball of the Bianchi groups as discrete subgroups of H3. result Produced a complete list of group presentations for specified Bianchi groups.
The paper studies hyperbolic equations in a spacetime foliation, proving existence and uniqueness.
problem Existence and uniqueness of solutions for first-order linear hyperbolic systems in a double null foliation.
method Proves global existence and uniqueness for first-order linear hyperbolic systems with initial data on a past null hypersurface.
result Derives a novel algebraic constraint for tensorfields satisfying the linearized Bianchi equations.
Intrinsic formulation of noncommutative geometry for quantum gravity.
problem Formalizing noncommutative differential geometry for quantum gravity.
method Geometric definitions and proofs of noncommutative Ricci curvatures and Bianchi identities.
result Quantum fluctuations and curvatures of (pseudo-) Riemannian metrics are renormalizable.
Study heat kernel coefficients in Bianchi IX gravity models.
problem Analyzing gravitational wave properties in Bianchi IX models.
method Algebraic geometry and motives applied to heat kernel coefficients.
result Coefficients are algebro-geometric periods of motives.
Study subgroups of Bianchi groups with effective methods.
problem Characterize subgroups of Bianchi groups with specific properties.
method Effectivise Agol-Long-Reid arguments to find C-special subgroups.
result Find 20-sheeted cover of figure-eight knot complement.
Characterizes Filippov n-algebroids using connections and formulas.
problem Generalizing Lie algebroids to Filippov n-algebroids.
method Introducing Filippov connections and transforming the Jacobi identity into the Bianchi-Filippov identity.
result Expressed the n-ary bracket using a torsion-free formula.
We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms…
Proves Strong Cosmic Censorship conjecture for specific spacetimes.
problem Proving the Strong Cosmic Censorship conjecture for certain spacetimes.
method Analyzing curvature invariants and expansion-normalised variables.
result Proven for orthogonal Bianchi class B perfect fluids and vacuum spacetimes.
Quantum computing uses Bianchi groups to build gates.
problem Building universal quantum gates from Bianchi groups.
method Using subgroups of Bianchi groups to derive quantum gate generators.
result Demonstrated the use of Bianchi groups for quantum computing.
Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.
problem Embedding closed totally geodesic hyperbolic 2-orbifolds in Bianchi orbifolds.
method Analyzing Bianchi orbifolds H3/PSL(2,Od) for large d. result Existence of at least cd closed embedded totally geodesic hyperbolic 2-orbifolds for large d. A geometric interpretation of curvature and torsion of linear transports along paths is presented. A number of (Bianchi type) identities satisfied by these quantities are derived. The obtained results contain as special cases the corresponding classical ones concerning curvature and torsion of linear connections.
The paper revisits Bianchi identities for Riemann and Weyl tensors using advanced algebraic analysis.
problem Proving Bianchi identities for Riemann and Weyl tensors in arbitrary dimensions.
method Using Spencer cohomology and differential systems, the paper proves Bianchi identities for Riemann and Weyl tensors in arbitrary dimensions.
result The paper discovers that the number of generating Bianchi identities for the Weyl tensor varies depending on the dimension, with 9 second-order identities in n=4 and 5 third-order identities in n=3.
Weak Bianchi identity found for spacetimes with timelike singularities.
problem Weak Bianchi identity for spacetimes with curvature singularities.
method Found sufficient conditions for a weak version of the second Bianchi identity.
result Identified a class of spherically symmetric static spacetimes with timelike singularities for which the identity holds.
The study describes maximal Fuchsian subgroups of a specific Bianchi group and computes their covolumes.
problem Characterizing maximal Fuchsian subgroups of a Bianchi group and computing their covolumes.
method Explicit description of conjugacy classes of maximal Fuchsian subgroups using quaternion algebras.
result Explicit description and covolumes of maximal Fuchsian subgroups.
Higher curvature corrections to Bianchi identities conflict with exceptional generalised geometry.
problem Incompatibility of higher curvature corrections with exceptional generalised geometry.
method Analysis of higher derivative corrections to Bianchi identities in supergravity and M-theory.
result Higher curvature corrections imply an L∞ algebra for the gauge field, terminating at level ten. In a recent comment (Johansen A 2003 An alternative view, Quant. Finance 3: C6-C7, cond-mat/0302141), Anders Johansen has criticized our methodology and has questioned several of our results published in [Sornette D and Zhou W-X 2002 The US 2000-2002 market descent: how much longer and deeper? Quant. Finance 2: 468-81,…
New representations for discrete surfaces derived from dual transforms.
problem Constructing discrete surfaces in differential geometry.
method Using Ω-dual transform and lightlike Gauss maps in Laguerre geometry. result All discrete linear Weingarten surfaces arise via Weierstrass-type representations.