We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L^2 Riemannian metric - so called because it induces an L^2 topology on each tangent space. It turns out that this top…
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48 results for “metric topology”
Paper introduces a complete metric topology for low energy spaces.
problem Defining a topology for low energy spaces with prescribed singularity.
method Introduces a completely metrizable topology stronger than capacity convergence.
result Low energy spaces have a natural completely metrizable topology.
Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.
problem Understanding entropy changes in flows near hyperbolic metrics.
method Analysis of geodesic flow on Riemannian manifolds with variable negative curvature.
result Topological entropy strictly decreases along normalized Ricci flow near hyperbolic metrics.
New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl curvature.
The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under perturbations.
method Study of topological entropy on Riemannian metrics with topology.
result Metrics with contractible closed geodesics have robust entropy.
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
The study connects manifold topology to metrics with positive intermediate curvature.
problem Understanding the relationship between manifold topology and metrics with positive intermediate curvature.
method Formulated a conjecture and proved it for specific dimensions and conditions.
result Closed, aspherical 6-manifolds cannot admit metrics with positive 4-intermediate curvature.
New examples of degenerating metrics on R^4 found.
problem Degenerating Poincaré-Einstein metrics on R^4.
method Riemannian ansatz of Debever and Deimaniański, constructing families.
result Continuous families of metrics with cusps and unexpected degenerations.
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.
The paper highlights issues in fixed point claims in digital topology.
problem Flaws in published assertions about fixed points in digital metric spaces.
method Continues a series of studies examining these flaws.
result Identifies and discusses problems in fixed point claims.
We describe an explicit metric that induces the Chabauty topology on the space of closed subsets of a proper metric space M.
Galatius and Randal-Williams defined a topology on the set of closed submanifolds of . Bökstedt and Madsen proved that a -version of this topology is metrizable by showing that it is regular and second countable. Using that the scanning map of a topological sheaf on manifolds is an embedding, we giv…
Corrects incorrect assertions about fixed points in digital topology.
problem Incorrect or incorrectly proven assertions about fixed points in digital metric spaces.
method Analysis of existing assertions and proofs.
result Identifies and corrects errors in published assertions.
Mixing of the Mineyev flow, orbital counting and Poincaré series for strongly hyperbolic metricsmath.DS
Study on counting orbits and Poincaré series for specific hyperbolic metrics.
problem Counting orbits and analyzing Poincaré series for strongly hyperbolic metrics.
method Combining ergodic theory techniques with topological flows and symbolic dynamics.
result Obtained orbital counting results and described the domain of analyticity for Poincaré series.
The paper explores metrics on tree moduli spaces and a new topological group.
problem Continuous metrics on tree moduli spaces.
method Proposes and analyzes continuous piecewise-smooth metrics.
result Observes a new abelian topological group structure.
Magnitude is not continuous but may be stable for most finite metric spaces.
problem Stability of magnitude invariant in finite metric spaces.
method Investigates the continuity properties of magnitude with respect to Gromov-Hausdorff topology.
result Magnitude is nowhere continuous but may be generically continuous.
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
Paper finds conditions for two geodesics on complex manifolds.
problem Existence of two distinct closed geodesics on manifolds with infinite fundamental group.
method Topological and metric conditions for existence of geodesics in Riemannian and Finsler metrics.
result Generic Finsler metrics have two distinct closed geodesics.
Generalizes Thurston's asymmetric metric to flat metrics.
problem Defining an asymmetric metric on flat metrics.
method Defined an asymmetric metric on the space of unit-area flat metrics.
result Discussed two different topologies from the asymmetry.
New TDA approach using Finsler metrics.
problem Traditional TDA concepts and methods.
method Introducing Finsler metrics for TDA.
result Relevance of Finsler metrics to TDA.
Survey on uniformization of metric surfaces, including fractal and topological manifolds.
problem Uniformization of metric surfaces homeomorphic to 2D topological manifolds.
method Various uniformization theorems, including quasisymmetric and quasiconformal approaches.
result Uniformization results for metric spheres and arbitrary metric surfaces.
We consider a smooth closed surface of fixed genus with a Riemannian metric of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for is greater than or equal to the topological entropy for the metric of constant negative curvatu…
New tool for summarizing time-varying data shapes.
problem Understanding dynamic data shapes.
method Introducing crocker stacks for time-varying metric spaces.
result Demonstrated utility in parameter identification task.
The paper proves metrizability and dynamics of Weil bundles.
problem Metrizability and dynamics of Weil bundles in differential geometry.
method Investigation of metrizability and dynamics of Weil bundles for smooth compact manifolds and Weil algebras.
result A canonical, complete, weighted metric \(\mathfrak{d}_w\) on \(M^\mathbf{A}\) that encodes geometry and deformations.
We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…
A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…
Study shows non-orientable manifolds restrict signature-changing metrics globally.
problem Global obstructions to signature-changing metrics on non-orientable manifolds.
method Explicit geometric constructions based on Möbius strip topology.
result Radical of signature-changing metrics cannot be everywhere transverse.
This paper formalizes state similarity metrics for reinforcement learning.
problem Leveraging state similarity for reinforcement learning in continuous-state systems.
method Introducing a unified formalism for defining topologies through metrics.
result Established a hierarchy of metrics and demonstrated their theoretical implications.
Geometry without Topologymath.MG
The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set …
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …
Teaches Dirac operators for geometry and topology.
problem Interactions between geometry and topology.
method Families of Dirac operators and index theorems.
result Applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.
Incorrect fixed point assertions in digital topology are discussed.
problem Incorrect or poorly stated fixed point assertions in digital topology.
method Discussion of problematic publications in digital metric spaces.
result Clarification of incorrect fixed point assertions.
Dunkl connections on complex plane don't preserve metrics.
problem Preserving metrics with Dunkl connections on \(\mathbb{C}^2\).
method Analysis of the topology of spherical tori with conical points.
result General Dunkl connections on \(\mathbb{C}^2\) do not preserve non-zero Hermitian forms.
Fixed point assertions in digital topology are often incorrect or poorly stated.
problem Fixed points in digital metric spaces
method Discussing publications with bad assertions
result Identifying and correcting errors in fixed point assertions
Geometry of Datamath.MG
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
This paper introduces a new metric for deep learning networks based on their classification performance.
problem The mystery and black-box nature of deep learning networks.
method Proposes a new distance measure based on the probabilistic performance of deep learning networks.
result The proposed metric space is compact and coincides with the quotient topological space.
Study confirms conjecture on extremal length of hyperbolic metrics.
problem Determining the extremal length of hyperbolic metrics on Riemann surfaces.
method Analyzes the topology of closed hyperbolic Riemann surfaces to find extremal lengths.
result Extremal length is topology-dependent and has a specific upper bound.
Paper shows Ricci flow with surgery converging to Taub-NUT metric.
problem Existence of Ricci flow with surgery and topology change.
method Ricci flow with surgery and local topology change.
result Flow converges to Taub-NUT metric on \mathbb{R}^4.
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
The Spheres of Solmath.DG
Study on Sol's metric spheres and their properties.
problem Characterizing metric spheres in Sol.
method Detailed analysis of cut locus and Riemannian exponential map.
result Metric spheres in Sol are topological spheres with precise singular points.
Solves modified Schouten tensor problems in conformal metric classes.
problem Prescribed problems for modified Schouten tensors in conformal classes of metrics.
method Uniform ellipticity confirmation under topological and functional constraints.
result Extends results from previous work on smooth complete metrics.
Study evaluates synthetic data augmentation for small datasets, highlighting inconsistencies in traditional metrics.
problem Inconsistent validation of synthetic data generated for small sample sizes.
method Proposes a normalized Bottleneck distance metric to evaluate synthetic tabular data.
result Common metrics like propensity scoring and MMD fail for small datasets, showing instability and high variability.
Computational method approximates homology groups of compact metric spaces.
problem Computing homology groups of compact metric spaces.
method Inversely sequence of finite topological spaces, inverse limit, homeomorphic copy, strong deformation retract.
result Approximates homology groups of compact metric spaces.
Study finds topological restrictions on 4-manifolds with uniformly positive scalar curvature.
problem Understanding which 4-manifolds can have metrics with uniformly positive scalar curvature.
method Topological obstructions and metric constructions on specific 4-manifolds.
result Existence of uncountably many exotic 's without such metrics and topological uniqueness of certain metrics.
Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed non-degenerate 2-form. Although in symplectic topology a famous bi-invariant metric, the …
Study finds open manifolds without complete metrics with positive scalar curvature.
problem Topological obstruction to positive scalar curvature on open manifolds.
method Defined Schoen-Yau-Schick and weak Schoen-Yau-Schick manifolds to prove the absence of complete metrics with positive scalar curvature.
result Proved no complete metric with positive scalar curvature on open Schoen-Yau-Schick manifolds.