Weyl-type theorems extended to Galilei and Carroll geometries.
problem Extending Weyl's theorem to non-relativistic and ultra-relativistic spacetimes.
method Defining and analyzing conformal and projective structures in Galilei and Carroll geometries.
result Torsion-free connections in Galilei and Carroll geometries are uniquely determined by their projective structures.
Study of potential Carroll structures and special Carrollian manifolds for null hypersurfaces.
problem Understanding intrinsic geometry of null hypersurfaces.
method Initiate study of potential Carroll structures and explore their relationship to special Carrollian manifolds.
result Initiate the study of potential Carroll structures and their relationship to special Carrollian manifolds.
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.
Introduces Carrollian Lie algebroids to handle singular Carrollian geometries.
problem Handling singular Carrollian geometries within standard Carrollian geometry.
method Introduces Carrollian Lie algebroids to study singular Carrollian geometries.
result Established the existence of compatible connections on Carrollian Lie algebroids.
The paper classifies intrinsic torsion in various spacetime structures.
problem Classifying intrinsic torsion in different spacetime structures.
method Review and classification of intrinsic torsion in galilean, Carrollian, Aristotelian, and Bargmannian spacetime structures.
result Found 16 classes for Aristotelian structures and 27 for Bargmannian structures.
Gauging procedure constructs lagrangians for carrollian gravity.
problem Constructing lagrangians for carrollian gravity.
method Gauging procedure applied to Klein pairs corresponding to homogeneous spaces.
result Generalizes first-order lagrangians for four-dimensional maximally symmetric carrollian spaces.
The Carrollian superplane is constructed as a supermanifold generalization of the Carrollian plane.
problem Constructing the Carrollian superplane as a supermanifold.
method Intrinsic construction of the Carrollian superplane as a supermanifold generalization of the Carrollian plane, defining Carroll spinors, and showing it as a principal R1∣2-bundle. result Novel N=2 Carrollian supersymmetry transformations are generated. Explains non-lorentzian theories and their dynamics.
problem Understanding non-lorentzian kinematics and dynamics.
method Review of kinematical spacetimes, construction of particle dynamics actions, discussion of gravity theories and field theories.
result Introduction and analysis of non-lorentzian gravity and field theories.
Study BGG operators on homogeneous conformal geometries.
problem Finding solutions to BGG operators on homogeneous conformal geometries.
method Invariant calculus for algebraic computations.
result Explicit solutions found for conformal Killing tensors, forms, and spinors.
The paper applies potential theory to conformal geometry, proving theorems and dimension estimates.
problem Understanding the behavior of solutions near singularities in conformal geometry.
method Linear and nonlinear potential theory applied to conformal geometry problems.
result Established Huber's type theorems and Hausdorff dimension estimates for conformal 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…
Explains conformal symmetry with examples in geometry and analysis.
problem None explicitly stated; focuses on introduction.
method Introduction based on examples of Yamabe operator and its applications.
result Illustrates conformal symmetry in geometry and analysis.
This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal…
The paper classifies and studies conformal variations of submanifolds.
problem Classifying and understanding conformal variations of submanifolds.
method Develops a Fundamental theorem and a rigidity theorem for Euclidean submanifolds.
result Fundamental theorem and rigidity theorem for Euclidean submanifolds.
Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension …
Curved loxodromes on spheres are explained and their ODE derived.
problem Understanding curved analogues of compass-bearing curves on spheres.
method Explained curved loxodromes and derived the fifth order invariant ODE.
result Derived the fifth order invariant ODE for loxodromes.
We study the conformal geometry of timelike curves in the (1+2)-Einstein universe, the conformal compactification of Minkowski 3-space defined as the quotient of the null cone of R2,3 by the action by positive scalar multiplications. The purpose is to describe local and global conformal invariants of time…
The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kähler geometry is "Locally conformal Kähl…
We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…
The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as p…
The study characterizes geometries of hypersurfaces in warped product and conformal manifolds.
problem Characterizing the geometry of hypersurfaces in warped product and conformal manifolds.
method Using higher fundamental forms and conformal metrics, the study characterizes the geometries of hypersurfaces in warped product and conformal manifolds.
result Higher conformal fundamental forms play a critical role in the characterization of the geometry of hypersurfaces in conformal manifolds.
Characterizes conformal classes of tori using differential geometry.
problem Classifying conformal classes of tori in complex dimension 1.
method Basic differential geometry methods, contrasting with Hopf tori.
result Complete characterization of conformal classes of product and standard flat tori.
Embedding theorem for tractor bundles applied to conformal geometry.
problem Embedding theorem for tractor bundles in Cartan geometries.
method Extension of Gromov-Zimmer embedding theorem to tractor bundles.
result Rigidity result for conformal actions of special pseudo-unitary groups.
We answer to the question whether a system of the 3rd order ODEs describes geodesics of a conformal structure. We construct a functor from a category of conformal geometries to a category of Cartan geometries associated to the 3rd order ODEs systems. Explicit formulas which define the family of all equations on conform…
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. In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli-Silvestre and a class of conformally covariant operators in conformal geometry.
Enhances conformal geometry in higher dimensions with infinite-dimensional algebra.
problem Finite-dimensional group of conformal transformations in higher dimensions.
method Derived deformation theory of ambitwistor space of complex null-geodesics.
result Infinite-dimensional dg-Lie algebra incorporating symmetries and conformal structure deformations.
Researchers study the conformal geometry of bivariate Gaussian manifolds.
problem Exploring the conformal structure of Fisher-Rao metric on statistical manifolds.
method Determined invariants of the conformal structure of the Fisher-Rao metric on the bivariate Gaussian manifold.
result The conformal holonomy group is SO0(1,6) for generic random variables, but SO0(1,4) for independent ones. The paper explores projective structures on curves and their applications in conformal geometry.
problem Finding qualitative information about solutions of Hill equations.
method Detailed description of projective structures and their isomorphism classes, correcting previous inaccuracies.
result The Yamabe problem for curves has no general solutions in a conformal/Möbius ambient space.
We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …
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…
New conformal geometry method solves Einstein-Weyl equations.
problem Solving Einstein-Weyl equations in 4D spacetimes.
method Combining conformal and complex geometry techniques.
result Reduced Einstein-Weyl equations to a single conformally invariant scalar equation.
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.
The note is about some nonlinear curvature conditions which arise naturally in conformal geometry.
The aim of the present paper is to investigate conformal changes in absolute parallelism geometry. We find out some new conformal invariants in terms of the Weitzenböck connection and the Levi-Civita connection of an absolute parallelism space.
In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral conformal invariants on 4-manifolds and geometric applications. The development was he…
We use the general theory developed in our article arXiv:1208.5510 in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor conn…
The study characterizes surfaces with specific harmonic properties in pseudo-conformal geometry.
problem Characterizing surfaces with harmonic properties in pseudo-conformal geometry.
method Investigating sphere congruences, quasi-umbilical surfaces, and constant mean curvature surfaces.
result Generically, Bryant's quartic differential is divergence free if and only if the surface is superconformal or orthogonal to a harmonic congruence of spheres.
Study reveals geometric context of second-order superintegrable systems.
problem Understanding second-order superintegrable systems and their Weylian geometry.
method Re-examined second-order maximally conformally superintegrable Hamiltonian systems, revealing their Weyl structure.
result Extended conformal superintegrability to Weyl structures, interpreting systems as semi-Weyl structures.
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.
New classification of hypersurfaces with conformal variations.
problem Classifying hypersurfaces with conformal infinitesimal variations.
method Analyzing hypersurfaces in conformal geometry, extending previous work by Cartan and Sbrana.
result The class of hypersurfaces with conformal infinitesimal variations is larger than previously known.
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.
This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This i…
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.
We use the CR geometry of the standard hyperquadric in complex projective three-space to give a detailed twistor description of conformal foliations in Euclidean three-space.
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.