We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponentia…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
Trend · papers per month
48 results for “Riemannian Metric”
Introduces new metric for Riemannian metrics, extending unbalanced optimal transport.
problem Extending unbalanced optimal transport to Riemannian metrics.
method Dynamic and static formulations of unbalanced optimal transport on Riemannian metrics.
result Wasserstein--Ebin metric provides a new Riemannian structure on the space of Riemannian metrics.
Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.
problem Approximating sub-Riemannian structures for analysis.
method Constructing Riemannian metrics tailored to sub-Riemannian structures and studying spectral convergence.
result Riemannian volumes converge to Popp's volume and spectral convergence of Laplace operators is studied.
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodes…
Riemannian metrics on orbifolds are equivalent to diffeological ones.
problem Equivalence of Riemannian and diffeological orbifolds.
method Framework of Riemannian diffeology and analysis of 2-metrics.
result Riemannian metrics on orbifolds are equivalent to diffeological ones.
Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.
problem Understanding the space of all locally elliptic and bounded Riemannian metrics on manifolds.
method Introduced an extended metric space and proved its properties.
result Proved the space of rough Riemannian metrics is complete and connected.
The study examines non-continuous Riemannian metrics on manifolds and their infinitesimal properties.
problem Investigating non-continuous Riemannian metrics and their infinitesimal structure.
method Constructing examples of metric measure spaces with discontinuous metrics.
result Examples show failure of infinitesimal Hilbertian or quasi-Riemannian properties.
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannia…
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…
Survey on metrics on compact Lie groups.
problem None explicitly stated; focuses on metrics.
method Left-invariant semi-Riemannian metrics.
result Survey of existing metrics.
Consider the sum of the first eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree to be thos…
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Understanding Riemannian metrics on lens spaces and their geometric properties.
method Geometric control theory methods applied to axisymmetric metrics.
result Cut loci and cut times converge to sub-Riemannian structure's values.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Analyzing Riemannian metrics on lens spaces.
method Geometric control theory methods.
result Cut loci and cut times converge to sub-Riemannian structure's cut locus and time.
Introduces a new framework for Riemannian diffeology.
problem No specific problem stated; focuses on a new framework.
method Uses tangent functor and metric from Iglesias-Zemmour to establish weak Riemannian diffeological spaces.
result Establishes a category of weak Riemannian diffeological spaces and shows induced pseudodistance is a distance under technical conditions.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…
Completeness of Riemannian metrics: an application to the control of constrained mechanical systemseess.SY
We improve Riemannian metrics for constrained systems control.
problem Controlling mechanical systems with configuration constraints.
method Constructing complete Riemannian metrics by modifying incomplete ones.
result A controller can be found to satisfy a design criterion.
Study rigidity in low-regularity Riemannian and semi-Riemannian metrics.
problem Rigidity problems for low-regularity metrics.
method Proves Cheeger-Gromoll splitting theorem and flatness criterion for semi-Riemannian metrics of regularity.
result Obtains isometry of higher regularity than Lipschitz.
This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^…
The paper studies Riemannian metrics on tangent Lie groups using two left-invariant metrics.
problem Exploring Riemannian structures on tangent Lie groups.
method Defining a new left-invariant Riemannian metric on the tangent Lie group using two left-invariant metrics and symplectic forms.
result Explicit formulas for the Levi-Civita connection, tensor curvature, and sectional curvature of the new metric in terms of the original metrics.
Riemannian metric learning improves data representation across various fields.
problem Traditional distance metrics fail to capture intrinsic data geometry.
method Leverages differential geometry to model data on Riemannian manifolds.
result Demonstrates remarkable success in diverse domains.
The paper proves Weyl projective rigidity for sub-Riemannian metrics and shows genericity of such metrics.
problem Investigating the Weyl projective rigidity of sub-Riemannian metrics.
method Analytic and smooth category proofs for specific distributions with minimal order complex abnormal extremals.
result Genericity of Weyl projectively rigid sub-Riemannian metrics and distributions.
Characterizes self-isometries of Riemannian metrics on compact manifolds.
problem Understanding isometries of Riemannian metrics.
method Characterization of self-isometries and proof of isometry conditions.
result Two Riemannian metric spaces are isometric if and only if their manifolds are diffeomorphic.
In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. A…
We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold , the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the -topology, , in the set of metrics of …
The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using an one-to-one correspondence between para-holomorph…
Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on for .
result Any conformal metric to the round metric on for can be a limit of metrics with positive scalar curvature.
A new metric is created on a special bundle.
problem Creating a metric on a specific type of bundle.
method Lifting a metric and almost symplectic form to a supermanifold.
result A super-Sasaki metric is constructed on the antitangent bundle.
Study uniquely determines Riemannian metric derivatives from boundary data.
problem Determining Riemannian metric derivatives from boundary data.
method Computing the full symbol of the elastic Dirichlet-to-Neumann map.
result The elastic Dirichlet-to-Neumann map uniquely determines all partial derivatives of the Riemannian metric on the boundary.
Two pseudo-Riemannian metrics and are geodesically equivalent, if they share the same (unparameterized) geodesics. We give a complete local description of such metrics which solves the natural generalisation of Beltrami problem for pseudo-Riemannian metrics.
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
The paper finds maximal metrics on Euclidean spaces.
problem Finding maximal elements in moduli spaces of Riemannian metrics.
method Defining a preorder on moduli space by isometry groups and identifying maximal elements.
result Constructs many examples of maximal metrics on Euclidean spaces.
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geome…
Eisenhart's theorem extended to sub-Riemannian metrics on specific Lie algebras.
problem Extending Eisenhart's theorem to sub-Riemannian metrics on step 2 distributions.
method Introducing ad-surjective step 2 nilpotent Lie algebras and extending Eisenhart's theorem.
result The theorem holds for sub-Riemannian metrics on ad-surjective step 2 distributions.
Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metricsmath.OC
New formulas for Riemannian gradient and Hessian on manifold metrics.
problem Evaluate Riemannian gradient and Hessian for various metrics on manifolds.
method Explicit formulas derived from Levi-Civita connection and projection.
result Derives new metrics and optimization frameworks on manifolds.
We consider a -dimensional differentiable manifold with two circulant structures -- a Riemannian metric and an additional structure, whose third power is the identity. The structure is compatible with the metric such that an isometry is induced in any tangent space of the manifold. Further, we consider an associated…
Classifies all flat Riemannian metrics on the plane, including complete and incomplete cases.
problem Classifying all flat Riemannian metrics on the plane.
method Examined conformal metrics of the form \( e^{2\varphi}g_0 \) where \( \varphi \) is a harmonic function.
result All flat Riemannian metrics on the plane, including complete and incomplete cases, arise from Riemann surfaces.
The study finds obstructions to certain Riemannian metrics using Lorentzian geometry.
problem Finding obstructions to curvature distinguished Riemannian metrics.
method Dual Lorentzian metrics and Penrose's plane wave limit.
result Necessary local conditions for certain Riemannian metrics.
With a f-left-invariant Riemannian metric on a Lie group , we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor . In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Rie…
In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Ran…
Length metrics can be closely approximated by conformally flat metrics.
problem Approximating length metrics with conformally flat metrics.
method Uniform approximation of length metrics by conformally flat Riemannian metrics.
result Any length metric on \(\mathbb{R}^d\) can be uniformly approximated by conformally flat Riemannian metrics.
Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free action of a compact Lie group. We then extend a well-known result of Nomizu and Ozek…
Classifies Riemannian manifolds with specific torsion properties.
problem Classifying Riemannian manifolds with parallel, non-twistorial torsion.
method Classifies complete simply connected Riemannian manifolds with a metric connection having parallel torsion, non-zero vectorial component, and zero twistorial component.
result Classifies complete simply connected Riemannian manifolds with the specified torsion properties.
This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics …
Riemannian Metric Bundlemath.DG
Study Riemannian metric bundles and their connections to K-theory.
problem Understanding geometry and topology of manifolds with Riemannian metrics.
method Develop rigorous theory of Riemannian metric bundles and apply to K-theory.
result Contribute to deeper understanding of manifold geometry and topology.
Study pseudo-Riemannian metrics on Jordan superalgebras.
problem No specific problem stated; focus on metrics.
method Coadjoint orbit-like construction for pseudo-Euclidean Jordan superalgebras.
result Investigated canonical pseudo-Riemannian metrics.
We study compact complex 3-manifolds admitting holomorphic Riemannian metrics. We prove a uniformization result: up to a finite unramified cover, such a manifold admits a holomorphic Riemannian metric of constant sectionnal curvature.