Study local geometries of symmetric affine surfaces.
problem Characterize local geometries of locally symmetric affine surfaces.
method Classify 6 non-isomorphic local geometries using Opozda's result and classify them up to linear isomorphism.
result 6 non-isomorphic local geometries identified and classified.
Introduces locally conformally symplectic and Kähler geometry.
problem None explicitly stated, focuses on background and applications.
method Background introduction and demonstration of applications.
result Illustrates applications of these geometries in physics.
Study on symmetries in geometric structures, finding unique symmetries imply affine symmetry.
problem Investigating symmetries in geometric structures.
method Analyzing (local) automorphisms of parabolic geometries.
result Many parabolic geometries have at most one generalized geodesic symmetry at points with non-zero harmonic curvature.
Constructs Cartan geometries from automorphism behaviors.
problem Determining Cartan geometries from automorphism local behavior.
method Introduces a construction for Cartan geometries capturing automorphism local behavior.
result The sprawl uniquely characterizes Cartan geometries with equivalent local behavior.
Surveying higher prequantum geometry and its applications.
problem Prequantizing local field theory locally and invariantly.
method Abstract cohesive homotopy theory.
result Existence of a solution in higher differential geometry.
Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
A framework compares image representations based on local geometry.
problem Comparing image representations based on global structure overlooks local differences.
method Quantify local geometry using Fisher information matrix and optimize differentiation with principal distortions.
result Identifies differences in local sensitivities between models.
We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γto its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivar…
Study of symmetric affine surfaces with non-vanishing torsion.
problem Characterizing geometries of symmetric affine surfaces with torsion.
method Complete classification of local geometries assuming parallel torsion and local homogeneity for non-parallel torsion.
result Classification of local geometries for symmetric affine surfaces with torsion, generalizing previous results.
3D Ricci flow controls local geometry for a time.
problem Controlling the geometry of a ball in 3D Ricci flow.
method Proving local geometric control persists under Ricci flow.
result Local C/t decay of curvature tensor.
We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence…
New proof shows normal sub-Riemannian extremals locally minimize energy.
problem Understanding why normal sub-Riemannian extremals locally minimize energy.
method Provides a new proof without explicit use of local optimality and geometry.
result Explicitly shows the relation between the geometry and local optimality.
Derives localization formulas in Batalin-Vilkovisky formalism.
problem Localization in Batalin-Vilkovisky formalism.
method Equivariant localization formulas in Batalin-Vilkovisky formalism.
result Derives localization formulas in Batalin-Vilkovisky formalism.
Study how local differential geometry affects global algebraic geometry of curves.
problem Understand the relationship between local and global algebraic geometry of curves.
method Investigate the web-structure of algebraic curves and its differential geometry.
result Local differential geometry determines global algebraic geometry of Fano manifolds up to biregular equivalences.
Extends moment map concept to locally conformally Kähler manifolds.
problem No specific problem stated; extends existing concept.
method Extends classical moment map interpretation to locally conformally Kähler geometry.
result Scalar curvature as moment map in locally conformally Kähler geometry.
Book introduces principles of LCK geometry for complex manifold students.
problem Understanding locally conformally Kahler manifolds.
method Explains the concept and properties of LCK manifolds.
result Provides insights into the structure and behavior of LCK manifolds.
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…
Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimen…
We prove that the Hilbert geometry of a convex domain in Rn has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of Rn. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We…
The paper develops quaternionic toric geometry and classifies local actions.
problem Classifying local quaternionic torus actions on manifolds.
method Develops local Qn-actions, introduces invariants, and studies tetraplectic structures. result Classifies local quaternionic torus actions up to homeomorphism.
The Laplace spectrum uniquely identifies five out of eight metrically maximal three-dimensional geometries.
problem Characterizing compact locally homogeneous three-manifolds using their Laplace spectra.
method Analyzing geometric structures and their spectral properties.
result For five out of eight metrically maximal three-dimensional geometries, compact locally homogeneous three-manifolds are uniquely determined by their spectra.
In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to di…
Researchers tackle the globalization problem of locally cosymplectic Hamiltonian dynamics.
problem Globalization problem of locally cosymplectic Hamiltonian dynamics.
method Investigate the geometry of locally conformally cosymplectic manifolds and provide a geometric Hamilton-Jacobi theory.
result Provide a geometric Hamilton-Jacobi theory on locally conformally cosymplectic manifolds.
Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is gen…
The paper proves global invertibility for certain local diffeomorphisms and biholomorphisms in higher dimensions.
problem Global invertibility of local diffeomorphisms and biholomorphisms in higher dimensions.
method The approach uses conformal geometry, complex analysis, elliptic PDEs, and topology.
result The main theorem guarantees global invertibility for specific local diffeomorphisms and biholomorphisms in higher dimensions.
Study local invariants and geometry of sub-Laplacian on H-type foliations.
problem Characterize the geometry and invariants of H-type foliations.
method Use Bott connection, scalar curvature, and new invariant to analyze sub-Riemannian geometry.
result Express second heat invariant as a linear combination of scalar curvature and new invariant.
We introduce and discuss (local) symmetries of geometric structures. These symmetries generalize the classical (locally) symmetric spaces to various other geometries. Our main tools are homogeneous Cartan geometries and their explicit description. This allows us to describe the structure of symmetric geometric structur…
Generalizes tools for studying collapsed manifolds to new geometry.
problem Studying collapsed manifolds with bounded sectional curvature.
method Generalizes fibration and stability theorems for compact group actions on manifolds with local bounded Ricci covering geometry.
result Two generalized results used in Xiaochun Rong's work on almost flat manifolds.
Local Hardy spaces defined for Riemannian manifolds with bounded geometry.
problem Defining Hardy spaces for Riemannian manifolds with specific curvature conditions.
method Using local Riesz transforms and atomic Goldberg-type spaces.
result Atomic Hardy spaces and local Hardy spaces are equivalent on Riemannian manifolds with bounded geometry.
BN refines local partition geometry in piecewise-affine networks during training.
problem Understanding the effect of BN on the function realized during training in piecewise-affine networks.
method Analyzing the geometry of switching hyperplanes and affine-region partition conditioned on a mini-batch.
result BN increases expected local partition refinement in ReLU and piecewise-affine networks.
Novel approach uses quasi-conformal geometry for OSA classification from cephalometry.
problem Classifying obstructive sleep apnea (OSA) based on craniofacial profiles.
method Quasi-conformal geometry for local deformation analysis of 15 landmark points in lateral cephalograms.
result Proposed model achieves 92.5% testing accuracy.
Study different types of G_2-structures on Lie groups.
problem Characterize and find examples of G_2-structures on Lie groups.
method Analyze invariant exact locally conformal closed G_2-structures on simply connected Lie groups.
result Complete characterization of invariant exact locally conformal closed G_2-structures on simply connected Lie groups.
Surveying hermitian integral geometry, the paper describes new kinematic formulas for complex space forms.
problem Understanding curvature measures and valuations on complex spaces.
method Analyzing valuations and curvature measures on complex space forms, deriving kinematic formulas.
result New local kinematic formulas for hermitian geometry, containing more information than global formulas.
Study large-scale geometry of infinite type surface mapping class groups.
problem Classify surfaces based on mapping class group properties.
method Coarse geometry, using Rosendal's framework.
result Classification of surfaces based on group properties.
This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemanni…
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.
Given a complete and (locally) cartesian closed category U, it is shown that the category of functors from the category of Weil algebras to the category U is (locally, resp.) cartesian closed. The corresponding axiomatization for differential geometry based upon Weil functors is then given.
The paper studies geometric properties of a specific type of submanifolds in Kaehler manifolds.
problem Analyzing the geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds.
method The study extends Chen's inequality for CR-warped product submanifolds and investigates equality cases.
result Several results extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an aut…
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invari…
Local supertwistors help study 6D conformal supergravity.
problem Understanding conformal supergravity in 6D.
method Local supertwistor formalism with a superconformal connection.
result Derived geometry of (1,0) and (2,0) supergravity multiplets.
In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an n-harmonic coordinate system and normalizing the determi…
Proper holomorphic isometries between Bergman domains are biholomorphisms.
problem Characterizing isometries between Bergman domains.
method New method from Information Geometry.
result Proper holomorphic local isometries are biholomorphisms.
A moduli space is constructed for affine homogeneous spaces using their local geometry.
problem Describing and classifying affine homogeneous spaces locally and globally.
method Local description via curvature, torsion, and connection; algebraic variety construction; Spencer cohomology for infinitesimal deformations.
result A moduli space M(glV) is constructed for affine homogeneous spaces of dimension dimV. A new dynamical method calculates shear-bend coordinates for surfaces.
problem Computing shear-bend coordinates for twisted SL2C local systems.
method Dynamics-based approach to abelianization of local systems.
result A dynamical recipe for shear-bend parameterization.
New geometries defined for string models, filling gaps in the literature.
problem Developing mathematical structures for string models.
method Defining E-metric-connection geometries and locality structures.
result Unified framework for metric-affine and generalized geometries.
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove…