Metric transforms make Euclidean half lines hyperbolic, preserving geodesic properties.
problem Characterizing metric transforms that make Euclidean half lines hyperbolic.
method Characterization of metric transforms φ \varphi φ such that ( [ 0 , ∞ ) , ∣ ⋅ ∣ φ ) ([0,\infty),|\cdot|_\varphi) ([ 0 , ∞ ) , ∣ ⋅ ∣ φ ) is Gromov hyperbolic. result Metric transform rigidity for roughly geodesic Gromov hyperbolic spaces.
Complex Legendre transforms for Kähler metrics solve a geometric problem.
problem Finding local symmetries for the Mabuchi metric on Kähler metrics.
method Introducing complex Legendre transforms on Kähler metrics.
result Explicit local isometric symmetries for the Mabuchi metric are found.
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.
For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conform…
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Paper studies Riesz transform stability under metric perturbations.
problem Stability of Riesz transform boundedness under metric perturbations.
method Derives conditions for stability of L p L^p L p -boundedness of Riesz transform. result Provides counter-examples for instability of Riesz transform boundedness.
Two-dimensional metrics related by conformal transformations are also Randers.
problem Understanding the relationship between conformally related Douglas metrics.
method Analyzing two-dimensional metrics and their conformal transformations.
result Two-dimensional conformally related Douglas metrics are specifically Randers.
A theorem transforms Lorentzian to signature-changing metrics.
problem Signature-changing manifolds and their initial conditions.
method Transformation prescription to change metrics.
result Transformation Theorem linking Lorentzian to signature-changing metrics.
Paper simplifies transforms for elastic metrics on curve shapes.
problem Improving the performance of shape analysis algorithms.
method Extending coordinate transformations to a family of isometries.
result Existence of optimal matchings over the diffeomorphism group.
Paper proves conformal transformation of special ( α , β ) (α, β) ( α , β ) -metrics to Douglas spaces.
problem Douglas spaces of second kind with ( α , β ) (α, β) ( α , β ) -metrics and their conformal transformations. method Analyzes and proves conformal transformation of specific ( α , β ) (α, β) ( α , β ) -metrics to Douglas spaces of second kind. result Proves that certain ( α , β ) (α, β) ( α , β ) -metrics are conformally transformed to Douglas spaces of second kind. Study X-ray transform on conic spaces, proving injectivity under certain conditions.
problem Injectivity of geodesic X-ray transform on conic metrics.
method Injectivity under non-trapping and no conjugate point assumptions.
result Injectivity of geodesic X-ray transform for asymptotically conic metrics.
New spin frame transformations affect Dirac equations without preserving metric structures.
problem Extending spin structures to spin manifolds with fixed signature.
method Defining spin frames and studying their effects on connections and Dirac equations.
result New transformations affect Dirac equations more generally than usual spin transformations.
Let (M,g) be a Riemannian manifold with an isometric action of the Lie group G. Let g_G be a left invariant metric on G. Consider the diagonal G action on the product M × G M \times G M × G with the metric g+g_G. In this paper we calculate the formula for the metric h on the quotient space ( M × G ) / G (M \times G) / G ( M × G ) / G ; the map from g to h…
Study of ray transforms on spherically symmetric manifolds with low regularity metrics.
problem Injectivity of X-ray transforms and broken ray transforms on spherically symmetric manifolds with low regularity metrics.
method Introduced and studied generalized Abel transforms to handle low regularity settings.
result Injectivity results for X-ray and broken ray transforms on spherically symmetric manifolds with C 1 , 1 C^{1,1} C 1 , 1 metrics. The paper develops methods to generate invariant quantities in Metric-Affine Geometry.
problem Developing methods to generate invariant quantities in Metric-Affine Geometry.
method The paper introduces a theorem to generate invariant quantities under transformations of the affine connection, proving invariance conditions.
result Theorem establishing conditions for invariance of functionals under transformations of the affine connection.
Graph scattering transforms are stable to metric perturbations of network topology.
problem Stability of graph data representations under metric perturbations.
method Extending scattering transforms to network data using multiresolution graph wavelets and graph convolutions.
result Graph scattering transforms are stable to metric perturbations of the underlying network topology.
New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.
problem Existence of essential conformal transformations in pseudo-Riemannian manifolds.
method Construction of compact locally conformally pseudo-Kähler manifolds with essential conformal transformations.
result Found compact examples of pseudo-Kähler manifolds with essential conformal transformations that are not conformally flat.
Madelung transform connects quantum and fluid dynamics via symplectic geometry.
problem Relating quantum mechanics and fluid dynamics equations.
method Proved Madelung transform as a Kähler map between wave functions and cotangent bundles.
result Madelung transform is a symplectomorphism and isometry between wave function space and density space.
Scattering transforms adapted for non-Euclidean domains using diffusion wavelets.
problem Stability of data representations in non-Euclidean domains.
method Generalization of scattering transforms to non-Euclidean domains using diffusion wavelets and diffusion maps.
result Stability of the representation to metric perturbations of the domain.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L 2 L^2 L 2 for metrics with finitely differentiable tensor. Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.
problem Proving uniqueness of Poincaré type cscK metric with singularity at a smooth divisor.
method Holomorphic transformations, asymptotic behavior analysis, fixed point problem.
result Unique Poincaré type cscK metric with singularity at a smooth divisor is unique up to holomorphic transformations.
Paper introduces new Finsler metrics preserved under projective transformations.
problem Developing new projective invariant in Finsler geometry.
method Formulated weakly generalized Douglas-Weyl ( W − G D W ) (W-G D W) ( W − G D W ) equation to generalize Finsler metrics. result Introduces new subclasses of Finsler metrics: generalized weakly-Weyl and generalized i l d e D ilde{D} i l d e D -metrics. Small neural networks embed arbitrary metric spaces into Gaussian mixtures.
problem Embedding arbitrary metric spaces into a fixed space with low distortion.
method Probabilistic transformers of small depth and width.
result Embeddings with low metric distortion for various metric spaces.
We give a generalization of the Penrose transform on Hermitian manifolds with metrics locally conformally equivalent to Bochner-Kähler metrics. We also give an explicit formula for the inverse transform. This paper is a generalization of "The Twistor correspondence of the Dolbeault complex over $\C^n$ " (dg-ga/9501004) …
Injective X-ray transform on Heisenberg group for regular functions.
problem Injectivity of X-ray transform on sub-Riemannian manifolds.
method Group Fourier Transform and analysis of taming metrics.
result Sufficiently regular functions on Heisenberg group are determined by their line integrals.
The paper explores simple and relatively simple transformation groups and their universal coverings.
problem Understanding the structure of universal coverings of transformation groups.
method Study of relatively simple groups and generalization of Tsuboi's metric space.
result Tsuboi's metric space of H a m ~ ( M , ω ) \widetilde{\mathrm{Ham}}(M, ω) Ham ( M , ω ) is not quasi-isometric to the half line. Study constructs transverse metrics using transformations commuting with elliptic operators.
problem Existence of transverse metrics in foliation theory.
method Applying the Average Method to construct a transverse metric.
result Pseudogroup of local transformations equicontinuous and quasi-analytic.
Yamabe solitons defined on specific Sasaki-like manifolds.
problem Defining Yamabe solitons on Sasaki-like almost contact B-metric manifolds.
method Contact conformal transformation of manifold components to define Yamabe solitons.
result Explicit 5-dimensional Lie group example of a Yamabe soliton.
Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.
problem Optimizing total σ 2 σ_2 σ 2 -curvature on spheres with positive scalar curvature. method Analyzes metrics conformal to the standard sphere, uses Sobolev norms to measure closeness.
result Near-minimizers of total σ 2 σ_2 σ 2 -curvature are almost the standard metric (up to Möbius transformations). The Legendre transform and its generalizations, originally found in supersymmetric sigma-models, are techniques that can be used to give constructions of hyperkahler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli spa…
Study on determining metrics from minimal surface areas, extending earlier work.
problem Determining a Riemannian manifold with boundary from minimal surface areas.
method Linearized forward operator of minimal surface transform, double fibration transforms, Bolker condition.
result Invertibility of the minimal surface transform on analytic manifolds.
Unique extremal Kähler metric found near a divisor.
problem Uniqueness of extremal Kähler metric near a smooth divisor.
method Analyzes Poincaré type extremal Kähler metric with cusp singularity.
result Uniqueness of extremal Kähler metric up to holomorphic transformations.
An analogue of the correspondence between GL(k)-conjugacy classes of matricial polynomials and line bundles is given for K-conjugacy classes, where K is one of the following: maximal parabolic, maximal torus, GL(k-1) embedded diagonally. The generalised Legendre transform construction of hyperkaehler metrics is studied…
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Match van Stockum dust to vacuum metrics with a single parameter.
problem Matching van Stockum dust to vacuum metrics.
method 1-parametric family of non-static Papapetrou vacuum metrics, Ehlers and Kramer--Neugebauer transformations.
result Explicit examples of matching, including Bonnor metric and Lanczos--van Stockum dust metric.
We show that for a closed Riemannian manifold the quotient of the group of projective transformations by the group of isometries contains at most two elements unless the metric has constant positive sectional curvature or every projective transformation is an affine transformation.
This research examines anisotropic conformal transformations of pseudo-Finsler surfaces.
problem Investigating anisotropic conformal transformations of conic pseudo-Finsler surfaces.
method Using modified Berwald frame, the study finds necessary and sufficient conditions for anisotropic conformal transformations and analyzes geometric properties.
result Necessary and sufficient conditions for anisotropic conformal transformations of pseudo-Finsler surfaces are derived.
COMID-SADL learns adaptive metrics for nonstationary data.
problem Learning metrics when constraint generation is nonstationary.
method COMID-SADL, an adaptive, online approach.
result Significant performance improvements over existing methods.
A metric defined on body moduli space.
problem Defining a metric on the moduli space of bodies.
method Quotient space by integral affine transformations.
result Identifies moduli space of Delzant polytopes with symplectic toric manifolds.
Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
problem Exploring connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
method Surveying and reviewing recent works, transforming complex Monge-Ampère equations, and proving the YTD conjecture.
result Established a transformation from irregular Sasaki-Einstein metrics to g g g -solitons on quasi-regular quotients. Study recovers Lorentzian metrics from boundary data, proving local rigidity.
problem Recovering a Lorentzian metric from scattering data on a boundary.
method Analyzes jet and real analyticity of metrics near lightlike points.
result Metric can be recovered up to gauge transformations near lightlike strictly convex points.
Research explores flat subspaces in complex projective manifolds using Okounkov bodies.
problem Existence of flat subspaces in complex projective manifolds.
method Utilizes the generalised Legendre transform to the Okounkov body and a result by Schwer--Lytchak.
result Sufficient conditions for the existence of flat subspaces are identified.
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.
In this paper, we introduce and investigate a general transformation or change of Finsler metrics, which is referred to as a generalized β β β -conformal change: L ( x , y ) ⟶ L ‾ ( x , y ) = f ( e σ ( x ) L ( x , y ) , β ( x , y ) ) . L(x,y) \longrightarrow\overline{L}(x,y) = f(e^{σ(x)}L(x,y),β(x,y)). L ( x , y ) ⟶ L ( x , y ) = f ( e σ ( x ) L ( x , y ) , β ( x , y )) . This transformation combines both β β β -change and conformal change in a general setting. T…
We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H 2 H^2 H 2 -metric without zero order terms. We find isometries (called R R R -transforms) from some of these spaces i…
Unified method for studying geometric structures of pseudo-Riemannian manifolds.
problem Study of geometric structures on pseudo-Riemannian manifolds.
method Introduce homogeneous pairs and use conformal transformations to analyze sectional curvature and geodesics.
result Unified approach to studying geometric structures on various moduli spaces.
Surveying inverse problems on manifolds with boundaries.
problem Inverse problems for geometric structures on manifolds with boundaries.
method Analyzes linear and non-linear geometric inverse problems.
result Recent results on geodesic X-ray transforms and Riemannian metrics.
Local X-ray transform works well near boundaries in hyperbolic spaces.
problem Injectivity of X-ray transform near boundaries for hyperbolic metrics.
method Local injectivity proof for geodesic X-ray transform on asymptotically hyperbolic manifolds.
result Local injectivity near a boundary point for X-ray transform in dimensions 3 and higher, up to O ( ρ 5 ) O(ρ^5) O ( ρ 5 ) .