New Finsler metrics derived from pedal curves.
problem Constructing new Finsler metrics.
method Using pedal curves or surfaces of other curves/surfaces.
result Generalization of slope metric.
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48 results for “curved metrics”
Curve shortening in metric-affine plane shrinks convex curves to points.
problem Shortening curves in non-Euclidean spaces.
method Curve shortening flow in metric-affine plane with geometric conditions.
result Closed convex curves in metric-affine plane shrink to points in finite time.
Study on completeness of Sobolev metrics on manifold-valued curves.
problem Completeness of Sobolev metrics on spaces of manifold-valued curves.
method Analysis of reparametrization invariant Sobolev metrics of order .
result Sobolev immersions are metrically and geodesically complete for several important cases of metrics.
New curves generalize flat metrics from quadratic to q-differentials.
problem Determining flat metrics from curve lengths.
method Introduced q-simple curves to generalize results from quadratic to q-differentials.
result Lengths of q-simple curves uniquely determine non-positively curved Euclidean cone metrics induced by q-differentials.
Completeness of Sobolev metrics on curve spaces proven.
problem Proving completeness of Sobolev metrics on curve spaces.
method Analyzing Sobolev metrics with nonconstant coefficients on curve spaces.
result Necessary and sufficient conditions for metric completeness provided.
Study disproves conjecture about metric completion of curve spaces.
problem Completeness properties of spaces of immersed curves with reparametrization-invariant metrics.
method Examined Sobolev-type metrics on real-valued immersed curves, demonstrating multiple distinct limit points.
result Metric completion of spaces of immersed open curves includes multiple distinct limit points, not a single point as previously conjectured.
Curve shortening flow is not unique on certain metrics.
problem Non-uniqueness of curve shortening flow on specific metrics.
method Formulated a uniqueness conjecture and constructed a non-static solution.
result Curve shortening flow is not unique on a non-flat metric on the plane.
Study constructs Frenet curves using semi-symmetric metric connection.
problem Understanding Frenet curves in various manifolds.
method Using semi-symmetric metric connection to construct Frenet frames and curvatures.
result Examples of semi-symmetric Frenet curves in Euclidean, Sasakian, and Kenmotsu manifolds.
The paper finds two types of metric lines in curve spaces.
problem Classifying metric lines in jet spaces of curves.
method Established the existence of two families of metric lines in the 2-jet space of plane curves.
result Found precise criteria for identifying metric lines in sub-Riemannian geodesics.
New solitons found in curve metric space.
problem Elastic metric on curve spaces for shape analysis.
method Reparametrization-invariant Sobolev metric extension, geodesic equation analysis.
result Geodesics are soliton solutions for elastic metric.
New method matches curves using varifolds and Sobolev metrics.
problem Matching unparametrized curves efficiently.
method Combines varifold-based inexact matching with second order Sobolev metrics.
result Shows improved shape analysis of mosquito wings.
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
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.
We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this …
New Einstein metrics found on complex manifolds.
problem Locally symmetric metrics on complex manifolds.
method Construction of manifolds with specific curvature properties.
result Infinitely many manifolds with negatively curved Einstein metrics but no locally symmetric metrics.
We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.
The study finds flag-wise positively curved metrics on compact manifolds.
problem Finding flag-wise positively curved metrics in Finsler geometry.
method Generic Finslerian perturbation and Lie group techniques.
result Examples of compact manifolds with flag-wise positively curved metrics.
New metrics on curve spaces split tangent bundles in various ways.
problem Characterize metrics on curve spaces that split tangent bundles in prescribed ways.
method Study reparametrization invariant metrics on the space of parametrized curves, focusing on tangent bundle splittings.
result Characterized all metrics that induce any prescribed splitting of the tangent bundle.
Study angles between curves in metric measure spaces.
problem Define and analyze the angle between curves in metric measure spaces.
method Introduce a new notion of angle and prove the cosine formula on spaces.
result The new notion of angle is compatible with classical notions in Riemannian manifolds and Alexandrov spaces.
On properties of the sets of positively curved Riemannian metrics on generalized Wallach spacesmath.DG
Curved metrics on Wallach spaces bounded by curves under flow.
problem Properties of positively curved Riemannian metrics on Wallach spaces.
method Normalized Ricci flow analysis on specific Wallach spaces.
result Set of metrics forms bounded curves asymptotically.
Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the s…
The paper studies cylinder curves in flat metrics with q > 2.
problem Characterizing behaviors of embedded cylinder curves in flat metrics with q > 2.
method Constructing examples and proving properties of cylinder curves.
result Embedded cylinder curves form a finite diameter subset of the curve complex when the surface is fully punctured and the metric has a specific form.
Five-point Toponogov theoremmath.DG
Condition for embedding metric spaces into curved manifolds.
problem Embedding conditions for metric spaces in curved manifolds.
method If-and-only-if condition on five-point metric spaces.
result Five-point metric spaces admit embeddings into nonnegatively curved Riemannian manifolds.
Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.
problem Investigating geometric properties of immersed curves with fractional Sobolev metrics.
method Analyzing Riemannian metrics on spaces of immersed curves, proving completeness and geodesic properties.
result Fractional Sobolev metrics are geodesically complete for .
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
Study shows nontrivial homotopy groups for negatively curved metrics on hyperbolic manifolds.
problem Understanding the homotopy groups of Teichmüller spaces for negatively curved metrics.
method Analyzes Teichmüller spaces of negatively curved metrics on hyperbolic manifolds, proving nontrivial homotopy groups for some dimensions.
result Proves existence of nontrivial rational homotopy groups and elements of infinite order in π_i B Diff(M).
New proof for stable reduction theorem using Kähler-Einstein metrics.
problem Proving the stable reduction theorem for curves over punctured curves.
method Using Kähler-Einstein metrics on fibers to obtain limiting stable curves.
result A new analytic proof of the stable reduction theorem for curves over punctured curves.
Lorentz meets Lipschitzmath.DG
Maximal causal curves for Lipschitz metrics are either lightlike or timelike.
problem Characterizing maximal causal curves for Lipschitz metrics.
method Analyzing the parametrization and geodesic equation for maximal causal curves in terms of Filippov solutions.
result Maximal causal curves for Lipschitz metrics are either everywhere lightlike or everywhere timelike.
A new metric-based principal curve method learns 1D manifolds from spatial data.
problem Learning 1D manifolds from spatial data.
method Metric-based Principal Curve (MPC) approach.
result The method effectively learns the shape of 1D manifolds from synthetic and real datasets.
New Einstein metrics found in curved spaces.
problem Finding Einstein metrics in curved spaces.
method Analyzing almost-Einstein metrics to find genuine Einstein metrics.
result Negative curvature preserved in Einstein metrics.
We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order are metrically complete on the space of Sobolev immersions of the same regularity and that any two curves i…
Teichmüller space of negatively curved metrics on Complex Hyperbolic Manifolds is not contractiblemath.GT
The Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible.
problem Proving the non-contractibility of Teichmüller space for complex hyperbolic manifolds.
method Analyzing the Teichmüller space of negatively curved metrics on complex hyperbolic manifolds.
result The Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible.
Study of null φ-slant curves in specific 3D manifolds.
problem Characterizing null φ-slant curves in 3D normal almost contact B-metric manifolds.
method Analyzing the geometric properties and Frenet frames of φ-slant null curves.
result Existence of a unique Frenet frame for non-geodesic φ-slant null curves.
Limit curve theorems for incomplete metric spaces and the null distance on Lorentzian manifoldsmath.DG
Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.
problem Control of Lorentzian lengths of limit curves in incomplete metric spaces.
method Proves limit curve theorem for incomplete metric spaces and applies to null distance.
result Strong control on Lorentzian lengths of limit curves in Sormani and Vegas' null distance.
The paper studies properties of optimal metrics associated to curves on surfaces.
problem Investigating properties of optimal metrics associated to curves on surfaces.
method Starting from a filling curve and a separating curve, constructing a two integer parameter family of curves and deriving coarse length bounds and qualitative properties of their associated optimal metrics.
result There are infinitely many pairs of filling curves with distinct inf invariants but the same self-intersection number.
It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…
The paper extends pressure metrics to punctured surfaces and proves their properties.
problem Constructing pressure metrics for Teichmüller spaces of punctured surfaces.
method Extending pressure metrics to Teichmüller spaces of surfaces with punctures, proving real analyticity and convexity of Manhattan curves.
result Derives the pressure metric by varying Manhattan curves.
We show that the space of negatively curved metrics of a closed negatively curved Riemannian -manifold, , is highly non-connected.
Metrics on shape space are used to describe deformations that take one shape to another, and to determine a distance between them. We study a family of metrics on the space of curves, that includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics …
Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We consider two conformally equivalent metrics for which the distances between curves are nontrivial. We show that in the case of the simpler of the two metrics, the only minim…
The paper introduces a metric for curves on manifolds, enabling distance and geodesic calculations.
problem Computing distances and geodesics between curves on Riemannian manifolds.
method Reparametrization invariant metric using SRV function, induced Sobolev metric, geodesic equations.
result Geodesic shooting method for optimal curve deformation, Jacobi fields characterization.
Analyzes limits of flat metrics on complex curves.
problem Understanding limits of flat metrics on complex curves.
method Criterion based on piecewise affine weight function on intersection complex.
result Collapsed limits are metric graphs, non-collapsed are collections of curves.
Study geodesics in constrained curve spaces, including elastic curves and concentric circles.
problem Geodesics in constrained curve spaces with Sobolev metrics.
method Intrinsic and constructive approaches.
result Construct geodesics in elastic curve and concentric circle spaces.
Survey on rigidity problems in negatively curved spaces.
problem Rigidity problems in negatively curved locally symmetric spaces.
method Review of known results and open problems.
result Exploration of topological properties of negatively curved metrics.
Characterizes K-polystability for projective bundles over curves.
problem Existence of extremal Kähler metrics on projective bundles.
method Relative K-polystability and decomposition of vector bundles.
result Existence of extremal Kähler metrics linked to K-polystability.
We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's almost-positively curved metric on S2xS3 extends to a nonnegatively curved metr…
Proposes MCC-F1 curve for better binary classification evaluation.
problem Misleading performance evaluations with ROC and PR curves for imbalanced data.
method Introduces MCC-F1 curve combining MCC and F1 score.
result MCC-F1 curve provides clearer classifier differentiation.
The paper presents new representations and spherical indicatrices of Bertrand curves in Lie groups.
problem Understanding geometric properties of Bertrand curves in Lie groups.
method New representations and spherical indicatrices of Bertrand curves in three Lie groups with bi-invariant metrics are derived.
result Relations between spherical indicatrices and new representations of Bertrand curves are established.