The conformal geometry of surfaces in the conformal space is studied. We classify the space-like surfaces in with vanishing conformal form up to conformal equivalence.
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The paper extends the study of conformally flat spaces to four dimensions.
problem Understanding conformal transformations and flatness in four-dimensional Finsler spaces.
method Extending the study of three-dimensional conformally flat Landsberg and Berwald spaces to four dimensions.
result Conditions for a four-dimensional conformally flat Landsberg space to become a Berwald space.
Study on conformal transformations of Cahen-Wallach spaces, focusing on fixed points and discontinuous groups.
problem Characterizing conformal transformations of Cahen-Wallach spaces.
method Analyzing conformal transformations of indecomposable Lorentzian symmetric spaces, focusing on fixed points and discontinuous groups.
result Essential conformal transformations of conformally curved Cahen-Wallach spaces have fixed points, and such transformations cannot centralize essential homotheties.
Simple conformally recurrent spaces are identified as pp-waves.
problem Characterizing conformally recurrent space-times.
method Analyzing dimension n>3 space-times.
result Simple conformally recurrent space-times are conformally recurrent pp-waves.
Study classifies space-like hypersurfaces with parallel Blaschke tensor in conformal space.
problem Classifying hypersurfaces with parallel Blaschke tensor.
method Classifies hypersurfaces up to conformal equivalence.
result All such hypersurfaces are classified.
There is a Lorenzian group acting on the conformal space . We study the regular submanifolds in the conformal space and construct general submanifold theory in the conformal space . Finally we give the first variation formula of the Willmore volume functional of subma…
This paper classifies Blaschke para-umbilical hypersurfaces in conformal space.
problem Classifying Blaschke para-umbilical hypersurfaces in conformal space.
method Classification through mathematical analysis and extension of previous classifications.
result Classification of Blaschke para-umbilical hypersurfaces in conformal space.
The paper examines conditions for conformal Ricci solitons on generalized ()-space forms.
problem Conditions for conformal Ricci solitons on generalized ()-space forms.
method Derivation of conditions for solitons to be shrinking, steady, or expanding in terms of conformal pressure p.
result Conditions for a Ricci semi-symmetric generalized ()-space form to form an Einstein manifold when equipped with a conformal Ricci soliton.
Study on conformal transformations in metric measure spaces.
problem Preserving curvature bounds under conformal transformations.
method Analysis of Sobolev space, differential structure, and curvature-dimension condition.
result First result showing preservation of lower curvature bounds under perturbation.
Synthetic approach to conformal transformations in metric and Lorentzian spaces.
problem Defining consistent conformal transformations in spaces of low regularity.
method Introducing conformal transformations in metric and Lorentzian spaces, focusing on Lorentzian pre-length spaces.
result Established a consistent notion of conformal length and proved its properties.
The paper characterizes conformal vector fields in -spaces via PDEs.
problem Characterizing conformal vector fields in -spaces.
method Using partial differential equations (PDEs) to characterize conformal vector fields.
result Properties of conformal vector fields in certain -spaces are demonstrated.
A space has the maximal number of CKVs if and only if it is conformally flat.
problem Determining the necessity of conformal flatness for maximal CKVs.
method Analyzing properties of conformally flat spaces and CKVs.
result Conformal flatness is a necessary and sufficient condition for maximal CKVs.
Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.
problem Characterize and classify biharmonic conformal immersions into a conformally flat 3-space.
method Characterization of totally umbilical surfaces, method to produce biharmonic immersions, classification of maps, construction of examples.
result Construct many examples of biharmonic conformal immersions, including proper immersions and isometric ones.
The article defines conditions for a manifold to be conformal to an Einstein space.
problem Determining when a manifold is conformal to an Einstein space.
method Algorithmic conditions based on the metric tensor and the Weyl endomorphism.
result General necessary and sufficient conditions for a pseudo-Riemannian manifold to be conformal to an Einstein space.
Classifies conformal actions on essential 3D Riemannian spaces.
problem Classifying conformal actions on essential 3D Riemannian spaces.
method Determines representations of Lie groups acting with cohomogeneity one or zero, up to conjugacy.
result Orbits specified up to conformal equivalence.
A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the no…
Study of conformal mappings in pseudo-Finsler spaces, extending results from pseudo-Riemannian geometry.
problem Understanding conformal mappings in pseudo-Finsler spaces and their properties.
method Extending results from pseudo-Riemannian geometry and proposing a technique to reduce problems involving pseudo-Finslerian conformal vector fields.
result Conformal groups of flat pseudo-Finsler spaces can be richer than flat Finslerian and pseudo-Euclidean conformal groups.
Paper studies conformal vector fields in specific Finsler spaces.
problem Characterizing and understanding conformal vector fields in spaces.
method Using principles to characterize and determine local structure of conformal vector fields under curvature conditions.
result Construction of non-homothetic conformal vector fields on locally projectively Randers spaces.
The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.
problem Characterizing biharmonic conformal hypersurfaces in Riemannian manifolds.
method Deriving biharmonic equations and proving properties of conformal immersions.
result Properties of biharmonic conformal hypersurfaces in space forms.
Direct proof for all conformally flat isoparametric submanifolds in Euclidean space.
problem Classifying conformally flat isoparametric submanifolds in Euclidean space.
method Direct proof approach.
result Complete classification of conformally flat isoparametric submanifolds of Euclidean space.
New -connection characterizes 4D spaces conformal to Einstein spaces.
problem Characterizing 4D spaces conformal to Einstein spaces.
method Introducing -connection, a Weyl connection that preserves conformal invariance.
result Characterizes non-degenerate spaces conformal to Einstein spaces.
Paper explores conformal immersions of Kaehler manifolds into Euclidean space.
problem Understanding conformal immersions of Kaehler manifolds.
method Used techniques from S. Chion and M. Dajczer for hyperbolic space immersions.
result Proved properties of conformal immersions into Euclidean space.
Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answe…
The paper defines and analyzes conformal trajectories in 3D space forms.
problem Understanding trajectories in curved 3D spaces.
method Defined conformal trajectories and studied their properties in , , and .
result Conformal trajectories in and have constant curvature and torsion.
Proposes feature conformal prediction for broader application in semantic feature spaces.
problem Establishing valid prediction intervals in semantic feature spaces.
method Extends conformal prediction to semantic feature spaces using deep representation learning.
result Feature conformal prediction outperforms regular conformal prediction under mild assumptions.
The study of conformal biharmonic maps and hypersurfaces in various spaces.
problem Understanding the properties and behavior of conformal biharmonic maps and hypersurfaces.
method Investigation of the conformal bienergy functional and its critical points, focusing on hypersurfaces in spheres and hyperbolic spaces.
result Identification and classification of conformal biharmonic hypersurfaces in spheres and hyperbolic spaces, including stability analysis.
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.
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
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 -disk algebras define invariants of conformally flat manifolds.
We investigate the Fefferman spaces of conformal type which are induced, via parabolic geometry, by the quaternionic contact (qc) manifolds introduced by O.Biquard. Equivalent characterizations of these spaces are proved: as conformal manifolds with symplectic conformal holonomy of the appropriate signature; as pseudo-…
The paper constructs arbitrary order conformally invariant operators in higher spin spaces.
problem Classifying and constructing conformally invariant differential operators in higher spin spaces.
method Explicit expressions and convolution type operators, intertwining operators, and representation theory.
result Explicit expressions and properties of conformally invariant differential operators in higher spin spaces.
We study grassmannians associated with a linear space with a nondegenerate hermitian form. The geometry of these grassmannians allows us to explain the relation between a (pseudo-)riemannian projective geometry and the conformal structure on its ideal boundary (absolute). Such relation encompasses, for instance, the us…
The paper classifies conformal solitons in pseudo-Euclidean spaces.
problem Classifying conformal solitons in pseudo-Euclidean spaces.
method Classification through pseudo-Riemannian hypersurfaces and position vector fields.
result Complete classification of conformal solitons in pseudo-Euclidean spaces.
Study on non-Berwaldian Landsberg spaces using conformal transformation.
problem Existence and properties of non-Berwaldian Landsberg spaces.
method Using conformal transformation to analyze Berwald and Landsberg tensors.
result Necessary condition for a Landsberg space to be Berwaldian is given.
The study classifies spaces with specific conformal vector fields.
problem Characterizing closed vacuum static spaces with non-Killing conformal vector fields.
method Provided characterizations and established an identity involving the characteristic function.
result Derived a rigidity theorem and classified spaces with the vector field.
Twistor lifts and factorization for conformal maps from a surface to the Euclidean four-spacemath.DG
Explains conformal maps using twistor lifts and provides a factorization method.
problem Understanding and analyzing conformal maps from a surface to Euclidean 4-space.
method Uses twistor lifts and differential factorization to explain and analyze conformal maps.
result Obtains a local factorization of the differential of a conformal map.
New approach classifies conformal Killing vector fields for FLRW space-time.
problem Classifying conformal Killing vector fields for FLRW space-time.
method Introduced new perspective on conformal Killing vector fields for FLRW space-time, considering three cases for the conformal factor.
result Nine conformal vector fields on FLRW, six of which are Killing and the rest non-Killing.
Study of how small changes in triangulated surfaces affect their geometry.
problem Understanding how small changes in triangulated surfaces affect their geometry.
method Investigates infinitesimal conformal deformations of triangulated surfaces in Euclidean space.
result There is a one-to-one correspondence between infinitesimal conformal deformations and infinitesimal isometric deformations of the stereographic image on the sphere.
For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may…
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.
This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter …
New proof shows all conformal fields are Killing on specific spaces.
problem Infinitesimal conformal rigidity on Damek-Ricci spaces.
method Formulated as PDEs, analyzed locally and directly.
result Constructive proof of rigidity without global methods.
CAT(0) spaces can be deformed in various ways.
problem Deformation of CAT(0) spaces.
method Suitable conformal changes.
result CAT(0) spaces admit non-trivial deformations.
Unified approach to conformal and modular invariants on surfaces.
problem Constructing a general family of conformal invariants on surfaces.
method Using an identification of Teichmüller space and rigged moduli space, and analytic work on harmonic functions.
result Unified conformal and modular invariants can be viewed as generalized modular invariants and functions on the rigged moduli space.
Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metricsmath.DG
Study on 3-metrics and their evolutions in conformally flat spaces.
problem Understanding the space of conformally flat 3-metrics with the Guichard condition.
method Evolution of orthogonal 2-metrics and determination of conformally flat 3-metrics.
result One-parameter family of conformally flat 3-metrics derived from 2-metrics.
We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Ja…
The study classifies conformal biharmonic and k-polyharmonic maps between space forms.
problem Classifying conformal biharmonic and k-polyharmonic maps between space forms.
method Proving conditions for proper biharmonic and k-polyharmonic maps between space forms.
result Proper k-polyharmonic conformal maps exist if and only if the dimension is 2k.
Classification of specific pseudo-Riemannian manifolds.
problem Classifying conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds.
method Complete classification through mathematical analysis.
result Conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds are either de Sitter or anti-de Sitter spacetimes.