Weak harmonic Weyl metrics found on all 4D closed manifolds.
problem Finding canonical metrics on 4D closed manifolds.
method Critical points of a quadratic functional involving the divergence of the Weyl tensor.
result Every 4D closed manifold admits a unique weak harmonic Weyl metric.
The paper studies automorphisms of Weyl manifolds and constructs modified contact Weyl diffeomorphisms.
problem Analyzing the automorphisms of Weyl manifolds associated with symplectic structures.
method Investigates the automorphisms of Weyl manifolds corresponding to Poincaré-Cartan classes and constructs modified contact Weyl diffeomorphisms.
result Construction of modified contact Weyl diffeomorphisms and analysis of automorphisms of Weyl manifolds.
New Finsler metrics constructed from GDW-metrics.
problem Exploring new Finsler metrics within the GDW-metric class. method Constructing new sub-classes of GDW-metrics. result Presented illustrative examples of new Finsler metrics.
Paper introduces new Finsler metrics preserved under projective transformations.
problem Developing new projective invariant in Finsler geometry.
method Formulated weakly generalized Douglas-Weyl (W−GDW) equation to generalize Finsler metrics. result Introduces new subclasses of Finsler metrics: generalized weakly-Weyl and generalized ildeD-metrics. Motivated by the study of Weyl structures on conformal manifolds admitting parallel weightless forms, we define the notion of conformal product of conformal structures and study its basic properties. We obtain a classification of Weyl manifolds carrying parallel forms, and we use it to investigate the holonomy of the a…
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
This is a survey on quaternion Hermitian Weyl (locally conformally quaternion Kähler) and hyperhermitian Weyl (locally conformally hyperkähler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a stu…
This paper explores new Finsler metrics and their connections to generalized Sakaguchi's Theorem.
problem Understanding the properties of Finsler metrics and their projective invariants.
method Developed and examined weakly-Weyl and generalized weakly-Weyl Finsler metrics.
result Equivalence of weakly-Weyl and W-quadratic spherically symmetric Finsler metrics. Constructs generalized Frobenius manifolds for specific Weyl groups.
problem Creating structures for orbit spaces of Weyl groups.
method Applying a previously established construction method to specific Weyl groups.
result Generalized Frobenius manifold structures constructed for Aℓ,Bℓ,Cℓ and Dℓ. New groups and manifolds from Weyl groups, with Frobenius structures.
problem Understanding new extended affine Weyl groups and their properties.
method Developed new Weyl groups and constructed Frobenius manifold structures.
result Existence of Frobenius manifold structures on orbit spaces of new Weyl groups.
This article examines the coincidence of the projective and conformal Weyl tensors associated to a given connection D. The connection may be a general Weyl connection associated to a conformal class of metrics [g]. The main result for n>3 is that the Weyl tensors coincide iff D is the Levi-Civita connection of an Einst…
The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity. In previous papers it has been proposed an equivalent form of Einstein's equation, which extends i…
The study of Killing tensor fields in Weyl's class of stationary and axially symmetric space-times.
problem Understanding the space of Killing tensor fields of valence 3 in Weyl's class.
method Analyzing the static vacuum sub-class of stationary and axially symmetric space-times, proving reducibility of valence-3 Killing tensors.
result The space of Killing tensor fields of valence 3 in Weyl's class is reducible to Killing vector fields and quadratic Killing tensor fields.
If a smooth compact 4-manifold M admits a Kaehler-Einstein metric g of positive scalar curvature, Gursky showed that its conformal class [g] is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, [g] also minimizes of the …
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
Characterizes Spinor groups using cohomology operations.
problem Understanding the cohomology and invariants of Spinor groups.
method Constructing integral cohomology rings and characteristic classes.
result Determined the ring of integral Weyl invariants of Spin(n).
We extend the Siu--Beauville theorem to a certain class of compact Kaehler--Weyl manifolds, proving that they fiber holomorphically over hyperbolic Riemannian surfaces whenever they satisfy the necessary topological hypotheses. As applications we obtain restrictions on the fundamental groups of such Kaehler--Weyl manif…
HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions …
New Finsler metrics with constant flag curvature defined using Weyl-type curvature tensor.
problem Characterizing Finsler metrics with constant flag curvature.
method Defining a Weyl-type curvature tensor and constructing projectively related Finsler metrics.
result Construction of new families of Finsler metrics with constant flag curvature.
The study finds obstructions for certain Weyl curvature tensors on manifolds.
problem Can manifolds admit metrics with purely electric or magnetic Weyl tensors?
method Analyzes algebraic curvature tensors and their Pontryagin classes on scalar product spaces.
result Obstructions to the existence of metrics with PE or PM Weyl tensors in top-degree cohomology.
I show that solutions of the SU(infinity) Toda field equation generating a fixed Einstein-Weyl space are governed by a linear equation on the Einstein-Weyl space. From this, obstructions to the existence of Toda solutions generating a given Einstein-Weyl space are found. I also give a classification of Einstein-Weyl sp…
The quadric ansatz solves dKP equations in arbitrary dimensions, leading to Einstein-Weyl structures.
problem Characterizing solutions of the dispersionless KP equation in arbitrary dimensions.
method Quadric ansatz for the dKP equation, constructing Einstein-Weyl spaces.
result Explicit new family of Einstein-Weyl spaces constructed and characterized.
3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
Maps dBKP solutions to MS system solutions, defining Einstein-Weyl structures.
problem Constructing solutions and structures for dBKP and MS systems.
method Map construction and spectral characterisation of reductions.
result Defines Einstein-Weyl structures for dBKP and BMS systems.
Paper extends Weyl's lemma to RCD(K,N) spaces.
problem Applying Weyl's lemma to RCD(K,N) metric measure spaces.
method Extending Weyl's lemma to RCD(K,N) spaces and proving applications.
result Local regularity of solutions for Poisson equations and Liouville-type results for harmonic functions.
Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces G/P with G semisimple and P parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions of scales, clos…
New metric reduces Weyl's energy in manifold connected sums.
problem Minimizing Weyl's energy in connected sums of manifolds.
method Interplay of Bach-flat and locally conformally flat metrics, topology of Z. result Existence of a metric with lower Weyl energy than gM. The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.
problem Understanding spectral encodings under Weyl growth conditions.
method Analyzing geometric Weyl bulk-density exponent and proving spectral rigidity.
result The geometric Weyl bulk-density exponent (d−2)/2 rigidifies spectral encodings in the O-regularly varying class, leading to unique admissible exponents and scaling laws. The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry of the foliations …
By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction of hyperbolic manifolds of very small volume in up to 8 dimensions.
Study magnetic Dirac operator on manifolds, proving Weyl law and eta bound.
problem Analyzing the semi-classical Dirac operator with magnetic potentials.
method Almost analytic continuations, Birkhoff normal form, local index theory.
result Sharp local Weyl law and bound on eta invariant.
New insights into 3D PDEs via Einstein-Weyl geometry.
problem Understanding second-order PDEs in 3D with Einstein-Weyl conformal structure.
method Analyzing solutions of second-order dispersionless integrable PDEs in 3D, relating them to Einstein-Weyl geometry.
result The covector w can be expressed in terms of the equation for generic second-order PDEs, providing a dispersionless integrability test.
Study projective and almost conformally symplectic structures on manifolds.
problem Relations between projective and almost conformally symplectic structures.
method Single almost conformally symplectic connection with totally trace-free torsion.
result Generalizes Fedosov structures and encodes variability of connections in projective class.
New calculus for invariant differential operators in parabolic geometries.
problem Understanding invariant differential operators for parabolic geometries.
method Developed a universal calculus to construct all affine invariants of Weyl connections.
result A natural procedure to determine affine invariants of Weyl connections.
The paper extends Weyl formulae for Schrödinger operators with singular potentials.
problem Analyzing the spectral behavior of Schrödinger operators with critically singular potentials.
method Generalizations of classical Weyl formulae, extending results by Avakumović, Levitan, and Hörmander.
result Obtained O(λn−1) bounds for the error term in the Weyl formula under minimal assumptions. A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its …
Researchers extend topological classification to knotted semimetals in 3D.
problem Classifying semimetals with knotted nodal lines.
method Using cohomology and Mayer-Vietoris sequence to account for nodal line semimetals with space-time inversion symmetry.
result Manifest proof of Weyl charge cancellation condition for Z_2 monopole charge.
Extends Weyl geometry from conformal to Weyl manifolds using ambient metrics.
problem Generalizing ambient constructions to Weyl manifolds.
method Introduces Weyl-ambient metric and Weyl-Fefferman-Graham gauge; shows Weyl-ambient space induces Weyl geometry; defines Weyl-connection and Weyl structure.
result Weyl-ambient construction for Weyl manifolds provides a well-defined initial value problem.
Segre quartic surfaces linked to minitwistor spaces with Einstein-Weyl structures.
problem Understanding the relationship between Segre quartic surfaces and minitwistor spaces.
method Using Penrose correspondence and detailed investigation of dual varieties.
result Determined the degrees and structure of components of dual varieties.
Study of recurrent Lorentzian Weyl spaces with detailed local and global structures.
problem Characterizing and classifying non-closed Lorentzian Weyl manifolds.
method Analyzing curvature tensors, Lie group actions, and differential invariants.
result Locally homogeneous non-closed recurrent Lorentzian Weyl manifolds are classified.
For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of t…
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
The paper proves compactness of scalar-flat metrics on manifolds with umbilic boundary.
problem Finding scalar-flat metrics with specific boundary conditions.
method Analyzing the conformal class of metrics on compact manifolds with umbilic boundaries.
result Scalar-flat metrics are a compact set under certain conditions involving the Weyl tensor.
New metrics defined in Finsler geometry with specific properties.
problem Understanding the properties of Finsler metrics and their subclasses.
method Introducing the generalized Berwald projective Weyl metric and proving properties of the class of generalized Douglas metrics.
result All GDW metrics with vanishing Landsberg curvature are of R-quadratic type. Researchers found a new upper limit for isoperimetric quotient in scalar-flat conformal classes.
problem Finding the maximum isoperimetric quotient in scalar-flat conformal classes.
method Analyzing Riemannian manifolds with scalar-flat metrics and boundary conditions.
result The supremum of the isoperimetric quotient is strictly greater than the Euclidean space's best constant.
We apply the theory of Weyl structures for parabolic geometries developed by A. Cap and J. Slovak in to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot-Carathéodory metric in the conformal class. The result of this computation has appli…
Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.
problem Characterizing Lorentzian Weyl spin manifolds with weighted parallel spinors.
method Analyzing holonomy algebras and introducing special coordinates.
result Local forms and examples of Lorentzian Weyl spin manifolds with weighted parallel spinors.
Recently, an extension of teleparallelism to a Weyl geometry which allows us to easily establish conformal invariance and "geometrize" electromagnetism has been presented. In this paper, I extend a result which concerns the existence of the Schwarzschild solution to a particular class of this extension. In addition, I …