SU(2) flat connections link to 3D geometry with cosmological constant.
problem Mapping flat connections on Riemann surfaces to 3D twisted geometry.
method Relating flat connection quantities to geometrical quantities in discrete 3D space.
result Moduli space of SU(2) flat connections generalizes phase space of twisted geometry.
Study on symmetric CR geometries of hypersurface type, showing they are either flat or homogeneous.
problem Characterizing symmetric CR geometries of hypersurface type.
method Analyzing CR transformations and symmetries, constructing examples.
result Non-flat non-degenerate symmetric CR geometries of hypersurface type are either flat or homogeneous.
The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.
problem Understanding projective and Carrollian geometries at infinity for Ricci flat Einstein manifolds.
method Developed a new type of Cartan geometry based on non-effective homogeneous models for projective geometry.
result Carrollian geometries are determined by the projective compactification data of Ricci flat Einstein manifolds.
Study of flat geometry of swallowtail submersions.
problem Classifying submersions preserving the swallowtail.
method Classifying submersions from (R3,0) to (R,0) up to diffeomorphisms. result Derived flat geometry from contact with planes and singularities of height function.
Study on deformations of holomorphic Cartan geometries, focusing on flat cases.
problem Deformation of holomorphic Cartan geometries on complex manifolds.
method Computing infinitesimal deformations and analyzing the forgetful map.
result The forgetful map from infinitesimal deformations of a flat holomorphic Cartan geometry to the underlying flat principal bundle is an isomorphism.
Study involutions on flat Seifert manifolds and their Borsuk-Ulam indices.
problem Characterizing involutions on Seifert manifolds with flat geometry.
method Determine all free involutions and compute Borsuk-Ulam indices.
result All free involutions and Borsuk-Ulam indices for flat Seifert manifolds are identified.
Notes on flat pseudo-Riemannian manifolds, focusing on their characterization and properties.
problem Characterizing flat pseudo-Riemannian manifolds and their properties.
method Survey of basic concepts in affine and Riemannian geometry, characterization of flat manifolds, and analysis of Lie groups.
result Characterization and properties of flat pseudo-Riemannian Lie groups and their metrics.
We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of 2N dimensional Ricci flat (pseudo-) Riemannian geometries.
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
We construct a series of examples of non--flat non--homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.
The paper studies the geometry of flat manifolds with controlled holonomy.
problem Investigating the geometry of asymptotically flat manifolds with specific properties.
method Analyzes torus fibrations and Hitchin-Thorpe inequalities for Ricci-flat 4-manifolds.
result Proves that certain flat metrics on 4-manifolds are isometric to Euclidean or Taub-NUT.
The paper studies geometric PDEs for flatness on Riemannian manifolds.
problem Understanding flatness in geometric PDEs.
method Study geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness.
result Introduce new Theorems about flatness in Differential Geometry.
The paper studies a generalized Pythagorean theorem on dually flat spaces via toric geometry.
problem Understanding the geometry of dually flat spaces and their toric Kähler manifolds.
method Introducing a dually flat structure and Bregman divergence on the boundary of toric Kähler manifolds.
result A continuity and generalized Pythagorean theorem for the divergence on the boundary.
Warren's metric makes C2 flat, with known geodesics and volumes.
problem Describing the geometry of C2 with a specific metric. method Defined a Kähler metric on C2 and showed its flatness. result Warren's metric makes C2 a flat manifold. The paper characterizes hypersurfaces in curved spaces using their geometry.
problem Geometric characterization of hypersurfaces in curved spaces.
method Extrinsic geometry analysis of conformally and radially flat hypersurfaces.
result Classification of hypersurfaces in terms of rotation and semi-parallel hypersurfaces.
New tractor geometry derived from asymptotically flat spacetimes.
problem Understanding the geometry of spacetimes near their boundaries.
method Derived null-tractor bundle from interior spacetime geometry, proved connections' uniqueness, and expressed results in BMS coordinates.
result Tractor connection encodes mass and angular momentum in 3D, and asymptotic shear in higher dimensions.
Study classifies half conformally flat GQE manifolds of signature (2,2).
problem Classifying half conformally flat generalized quasi-Einstein manifolds.
method Analysis and examples provided.
result Natural affine quasi-Einstein equation derived.
I give a theory of Moebius-flat hypersurfaces in n-dimensional projective space, analogous to that in conformal geometry. This unifies the classes of hypersurfaces with flat induced conformal structure (n > 3) and a classically studied class of surfaces (n = 3). I extend an example of Akivis-Konnov, and use polynomial …
Holomorphic branched Cartan geometry defined on complex manifolds.
problem Defining holomorphic branched Cartan geometry on complex manifolds.
method Using Atiyah bundle and foliations, defining transversely flat branched complex projective geometry.
result Holomorphic branched Cartan geometries on compact manifolds are flat.
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
Study calculates intersection forms of almost-flat 4-manifolds.
problem Understanding the intersection forms of almost-flat 4-manifolds.
method Calculation of intersection forms for all 4-dimensional almost-flat manifolds.
result Intersection forms of all 4-dimensional almost-flat manifolds have been calculated.
We compare the flat geometry associated to a quadratic differential with the hyperbolic geometry associated to the underlying Riemann surface. We show that if a curve is contained in a thick subsurface, then its hyperbolic length is comparable to its flat length times the flat size of the subsurface.
This paper classifies minimal complex surfaces with Levi-Civita Ricci-flat metrics.
problem Classifying minimal complex surfaces with specific geometric properties.
method Study of compact complex manifolds with Levi-Civita Ricci-flat metrics.
result Minimal complex surfaces with Levi-Civita Ricci-flat metrics are Kähler Calabi-Yau surfaces and Hopf surfaces.
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-s…
Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.
problem Investigating curvature in location-scale-shape models under Wasserstein metric.
method Introduced location-scale-shape model and investigated its geometry.
result Location-scale-shape model is intrinsically flat but extrinsically curved in Wasserstein geometry.
New equivalence found for flat vector bundles without extra conditions.
problem Flat vector bundles over compact Riemannian manifolds.
method Extended Corlette and Donaldson's result to arbitrary vector bundles.
result Equivalence of harmonic metrics and semi-simpleness for arbitrary vector bundles.
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or…
The paper explores a duality between conformally flat metrics and hyperbolic geometry.
problem Locally conformally flat metrics and their relationship to hyperbolic geometry.
method Analyzes the Gauss-Codazzi equations and their duals in hyperbolic space.
result Identifies a unique solution for B^ when g^ is locally conformally flat. Study on non-flat two-plectic geometry of six-sphere and its Hamiltonian dynamics.
problem Non-flat two-plectic geometry of six-sphere and Hamiltonian dynamics.
method Explicitly proving non-flatness and showing infinitesimal automorphisms via g2. result Explicit solutions of Hamilton-de Donder-Weyl equations with one- and two-dimensional sources.
Study on complex tori foliations and flat geometries.
problem Understanding turbulent foliations on compact complex tori.
method Defined and analyzed smooth turbulent foliations on compact complex tori.
result All transversely holomorphic Cartan geometries are flat.
New geometric variant of factorization homology for conformally flat manifolds.
problem Defining invariants of conformally flat manifolds.
method Introducing a metric-dependent geometric variant of factorization homology.
result Left Kan extensions of conformally flat d-disk algebras define invariants of conformally flat manifolds. The abstract discusses instantons on flat spaces and provides explicit constructions.
problem Understanding instantons on flat spaces.
method Explains and provides explicit constructions of instantons on R4, R7, R8, and Hn. result Natural generalizations of instantons on flat R4 to R7 and R8. Detect spacetime curvature without rulers and clocks in 3D.
problem Detecting spacetime curvature without traditional measurement tools.
method Generalized results from 2D to 3D spacetime, proving well-stitched spacetime for conformally flat cases.
result A 3D spacetime is well-stitched if and only if it is conformally flat, providing a tool for curvature detection.
New Ricci flat Kähler metrics found on complex symmetric spaces.
problem Finding Ricci flat Kähler metrics on complex symmetric spaces.
method Using an explicit asymptotic model with interpretation of geometry at infinity in the wonderful compactification.
result Obtained new Ricci flat Kähler metrics on complex symmetric spaces of rank two.
The flat geometry of the I1 singularity: (x,y)↦(x,xy,y2,y3)math.DG Study the flat geometry of a specific singularity in 4D space.
problem Understanding the flat geometry of a least degenerate singularity in 4D space.
method Obtained a generic normal form invariant under diffeomorphisms and isometries, classified submersions preserving the image, studied the height function.
result Obtained a generic normal form for the I1 singularity invariant under diffeomorphisms and isometries. Flat minimal hypersurfaces in 4D space are always flat.
problem Understanding stable minimal hypersurfaces in 4D space.
method Proving stability and completeness lead to flatness.
result Complete, stable minimal hypersurfaces in 4D are flat.
Affirmative answer to flat holomorphic Cartan geometries on complex tori.
problem Whether all flat holomorphic Cartan geometries on complex tori are translation invariant.
method Using complex affine Lie groups, we show that all holomorphic Cartan geometries on complex tori are translation invariant.
result Holomorphic Cartan geometries on complex tori are translation invariant.
The paper classifies and studies conformally flat hypersurfaces in 4D space.
problem Understanding conformally flat hypersurfaces in 4D space.
method Using Möbius geometry, the paper classifies and investigates the global behavior of these hypersurfaces.
result Examples of conformally flat hypersurfaces include cones, cylinders, and rotational hypersurfaces over surfaces with constant Gaussian curvature.
Defines Lewy curves in para-CR geometry and characterizes their path geometries.
problem Characterizing path geometries defined by para-CR Lewy curves.
method Definition and characterization of para-CR Lewy curves in various dimensions.
result Lewy curves determine the para-CR structure up to sign in flat cases.
We propose a Lie geometric point of view on flat fronts in hyperbolic space as special omega-surfaces and discuss the Lie geometric deformation of flat fronts.
Local flatness theorem for paraquaternionic contact structures.
problem Local flatness of paraquaternionic contact manifolds.
method Defined paraquaternionic contact conformal curvature tensor and showed local flatness condition.
result Paraquaternionic contact conformal curvature vanishing implies local flatness.
Short introduction to discrete flat fronts in hyperbolic space with a Weierstrass representation proof.
problem Understanding discrete flat fronts in hyperbolic space.
method Proving a Weierstrass representation for discrete flat fronts.
result Any discrete flat front in the mixed area sense admits a Weierstrass representation.
We establish 2-jet determinacy for the symmetry algebra of the underlying structure of any (complex or real) parabolic geometry. At non-flat points, we prove that the symmetry algebra is in fact 1-jet determined. Moreover, we prove 1-jet determinacy at any point for a variety of non-flat parabolic geometries - in parti…
The paper extends Ricci curvature in Finsler geometry and finds conditions for specific metrics.
problem Characterizing and finding conditions for specific metrics in Finsler geometry.
method Introducing weighted projective Ricci curvature and analyzing Randers and Kropina metrics.
result Conditions for metrics to have weighted projective Ricci flat curvature.
It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtai…
We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …
The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.
problem Understanding conformal properties of cubic metrics with isotropic scalar curvature.
method Analyzing the conformal flatness and isotropic scalar curvature of cubic metrics.
result Cubic metrics with weakly isotropic scalar curvature must be Minkowski metrics.