The paper studies properties of intrinsically Lipschitz constants in metric spaces.
problem Investigating properties of intrinsically Lipschitz constants.
method Introduced Leibniz and product formulas for intrinsic slope.
result Formulated Leibniz and product formulas for intrinsic slope.
Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.
problem Understanding the relationship between intrinsic and extrinsic metrics and surface area.
method Examined surfaces within a unit ball in R3, provided lower bounds on the ratio in terms of area, and showed non-existence of global lower bounds.
result Found that the ratio of intrinsic to extrinsic metrics has a lower bound in terms of surface area, but no global lower bound exists.
In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spac…
Characterizes intrinsic Lorentzian spaces using midpoint properties.
problem Deciding if a metric is length-based in Lorentzian spaces.
method Adapting midpoint criteria from metric geometry to Lorentzian pre-length spaces.
result Spaces with specific midpoint properties are strictly or merely intrinsic.
New metric measure space theory for Lipschitz constants.
problem Defining and characterizing Cheeger energy in metric measure spaces.
method Adapting Cheeger theory to intrinsically Lipschitz sections.
result Characterization of intrinsic Cheeger energy in terms of relaxed slope.
Introduces intrinsically Lipschitz graphs in metric spaces.
problem Graphs in metric spaces with Lipschitz conditions.
method Focuses on quotient maps and intrinsically Lipschitz sections.
result Compactness, Ahlfors regularity, and extension theorems.
Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
Defines new metric space sections with Ahlfors-David regularity.
problem Defining and analyzing new types of sections in metric spaces.
method Introducing intrinsically quasi-symmetric sections and proving their Ahlfors-David regularity.
result Proves Ahlfors-David regularity for intrinsically quasi-symmetric sections.
Paper analyzes a new Hopf-Lax semigroup in metric spaces.
problem Analyzing a new Hopf-Lax semigroup in metric spaces.
method Using continuous sections of quotient maps and variational problems.
result The 'symmetrized' Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equation.
Metric minimizing surfaces are locally CAT(0) in CAT(0) spaces.
problem Characterizing surfaces with no length nonincreasing deformations.
method Analyzing metric minimizing surfaces in CAT(0) spaces.
result Metric minimizing surfaces in CAT(0) spaces are locally CAT(0).
The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet f…
Defines intrinsically Hölder sections in metric spaces.
problem Characterizing Hölder sections in metric spaces.
method Introducing intrinsically Hölder graphs, proving compactness, regularity, and extension theorems.
result Establishes properties for intrinsically Hölder graphs, including vector space, convex set, and equivalence relation.
Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.
problem Proving Lipschitz conditions in semidirect products of groups without intrinsic dilations.
method Using equivalent conditions and properties of projection maps in metric spaces.
result Proves the same Lipschitz results as in Carnot groups, without intrinsic dilations.
Study minimal discs in metric spaces with a quadratic isoperimetric inequality.
problem Understanding the geometry of minimal discs in metric spaces.
method Associate a compact metric space to each minimal disc, controlling its properties by the isoperimetric inequality.
result The geometry of the associated space can control the shapes of curves and the original space's topology.
Estimates scalar curvature of point clouds without embedding.
problem Estimating scalar curvature of data sets without embedding.
method Intrinsic estimator based on metric structure, consistent and stable.
result Estimator converges to scalar curvature as sample size increases.
A new method classifies almost contact metric manifolds using intrinsic endomorphisms.
problem Difficult classification of almost contact metric manifolds into 2^12 classes.
method Introducing intrinsic endomorphisms S and h, providing a flowchart based on algebraic conditions.
result A more natural classification scheme with fewer classes, including H-parallel manifolds. Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is i…
Characterizes functions in Carnot groups of step 2.
problem Understanding intrinsic Lipschitz functions in Carnot groups.
method Characterization via intrinsic distributional gradients.
result Characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2.
The study finds conditions for scalar-flat metrics on ruled surfaces.
problem Conditions for the existence of scalar-flat metrics on ruled surfaces.
method Analyzes intrinsic number and complex structure of ruled surfaces.
result Scalar-flat metrics exist only for ruled surfaces with genus g≥2 and m(X)>2−2g. We compute persistent homology using an intrinsic metric derived from density.
problem Estimating topological features from high-dimensional data.
method Density-based metric learning for persistent homology.
result Persistent homology converges to intrinsic manifold metric.
Study shows tori metrics converging to flat under specific conditions.
problem Understanding convergence of metrics on tori with non-negative scalar curvature.
method Uniformly conformal metrics and controlled geometry sequences.
result Sequence of metrics converges to flat metric in multiple senses.
Weyl's intrinsic volumes converge to the Euler characteristic of the base manifold under certain metrics.
problem Convergence of intrinsic volumes on Riemannian manifolds.
method Defined a new metric and used it to study the convergence of intrinsic volumes.
result Intrinsic volumes converge to the Euler characteristic of the base manifold.
No isometric holomorphic maps between Teichmüller spaces and bounded domains.
problem Holomorphic isometries between Teichmüller spaces and bounded symmetric domains.
method Analyzing intrinsic metrics and properties of Teichmüller spaces and bounded symmetric domains.
result No isometric holomorphic maps exist in dimensions two or more.
We study the intrinsic geometry of a one-dimensional complex space provided with a Kaehler metric in the sense of Grauert. We show that if K is an upper bound for the Gaussian curvature on the regular locus, then the intrinsic metric has curvature at most K in the sense of Alexandrov.
Abstract proof of non-existence of certain types of almost contact metric structures.
problem Proving the non-existence of specific types of almost contact metric structures on manifolds.
method Using interrelations among components of intrinsic torsion to describe exterior derivatives of relevant forms.
result Proof of non-existence of 132 Chinea and González-Dávila types of almost contact metric structures.
Proves metric equivalence for GL^+_n matrices.
problem Metric equivalence of GL^+_n matrices.
method Exploits conical structure of rank stratification.
result Bilipschitz equivalence of intrinsic and extrinsic metrics.
Study on rolling Stiefel manifolds with specific metrics.
problem Intrinsic and extrinsic rolling of Stiefel manifolds with α-metrics. method Investigation of intrinsic rolling of normal naturally reductive homogeneous spaces, derivation of ODEs for rolling, and explicit solutions.
result Explicit solutions for intrinsic and extrinsic rolling of Stiefel manifolds.
Study on predicting sequences with Gaussian constraints, linking to intrinsic volumes and metric complexity.
problem Predicting sequences almost as well as the best Gaussian distribution with mean in a given subset.
method Expressed minimax regret in terms of intrinsic volumes, established comparison inequality for Wills functional, characterized global covering numbers and local Gaussian widths.
result Sharp estimates on the log-Laplace transform of intrinsic volume sequence for a general nonconvex set.
Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that k-NN regression is also adaptive to intrinsic dimension. …
In this article we present a study of the subspaces of the manifold OscM, the total space of the osculator bundle of a real manifold M. We obtain the induced connections of the canonical metrical N-linear connection determined by the homogeneous prolongation of a Finsler metric to the manifold OscM. We present the rela…
In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality d(x,y)≤σ(d(x,z)+d(z,y)) for some constant σ≥1, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-kn…
I consider compact metric spaces which admit intrinsic isometries to Euclidean d-space. The main result roughly states that the class of these spaces coincides with class of inverse limits of Euclidean d-polyhedra.
MSA compares neural representations' intrinsic geometry for better understanding.
problem Existing similarity measures fail to capture subtle distinctions between neural network solutions.
method Metric similarity analysis (MSA) using Riemannian geometry.
result MSA can disentangle features of neural computations and compare nonlinear dynamics.
New algorithm estimates intrinsic dimension of discrete datasets.
problem Inaccuracies in using continuous methods for discrete datasets.
method Introduced an algorithm to infer intrinsic dimension of discrete spaces.
result Demonstrated accuracy on benchmark datasets and found a small intrinsic dimension in a metagenomic dataset.
Investigates intrinsic Lipschitz sections in nonlinear quotient maps.
problem Analyzing intrinsic Lipschitz sections in non-linear quotient maps.
method Introduced Leibniz formula for intrinsic slope under weaker conditions, used properties of intrinsic dilations in Carnot groups, and provided conditions for sum of sections.
result Found conditions for sum of intrinsically Lipschitz sections in Carnot groups of step 2.
New geometric conditions ensure compactness of ∂ˉ-Neumann problem.
problem Compactness of ∂ˉ-Neumann operator on specific domains. method Introduced new geometric conditions for a class of domains, proving compactness equivalence to boundary properties.
result Compactness of ∂ˉ-Neumann operator equivalent to boundary lack of analytic varieties. This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…
The paper connects different convergence concepts in geometric analysis.
problem Comparing convergence concepts in geometric analysis.
method Relating Lp convergence and volume convergence to Intrinsic Flat and Gromov-Hausdorff convergence. result Conditions for convergence of Riemannian manifolds under specific conditions.
Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all c…
In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and t…
The paper proves stability of manifolds with boundary under volume and distance constraints.
problem Stability of manifolds with boundary under volume and distance constraints.
method Volume preserving intrinsic flat convergence of metrics with boundary constraints.
result The stability of manifolds with boundary under volume and distance constraints is proven.
New method learns intrinsic manifold from observed data for accurate localization.
problem Learning latent intrinsic geometry from observed high-dimensional data.
method Intrinsic Isometric Manifold Learning (IIML) using neural networks for metric estimation.
result IIML achieves accurate localization in ad-hoc sensor networks from imaging data.
Intrinsic formulation of noncommutative geometry for quantum gravity.
problem Formalizing noncommutative differential geometry for quantum gravity.
method Geometric definitions and proofs of noncommutative Ricci curvatures and Bianchi identities.
result Quantum fluctuations and curvatures of (pseudo-) Riemannian metrics are renormalizable.
The paper analyzes finite element methods on manifolds with approximate metrics.
problem Analyzing finite element methods on manifolds with approximate metrics.
method Intrinsic finite element exterior calculus applied to manifolds with Regge metrics.
result Analysis and implementation of a method for computing an approximate Levi-Civita connection form.
The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical…
Null distance metric studies spacetime convergence.
problem Investigate convergence in spacetime geometry.
method Introduced null distance metric for Lorentzian manifolds, proving convergence results.
result Null distance metric leads to distinct limiting behavior under non-uniform convergence of warping functions.
The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.
problem Comparing and analyzing shapes of curves.
method Construction and theoretical properties of quotient elastic metrics, special case of square root velocity metric, numerical approaches for estimation.
result Simplified expression for the square root velocity metric distance.