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Journal articles on the topic 'Riemannian metric and distance'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Xie, Ying Hong, and Cheng Dong Wu. "Study on the Comparison of Two Metrics Used in the Fields of Object Tracking." Applied Mechanics and Materials 483 (December 2013): 419–22. http://dx.doi.org/10.4028/www.scientific.net/amm.483.419.

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The paper researches on the comparison of two metrics methods under Riemannian metric and under Log-Euclidean metric respectively. Firstly, Experiments are done for comparing the distance and mean values worked out under the two metrics. And the time required for computing under these two metrics is also shown. Lastly, the two methods are applied to the field of image tracking. The performance of the two methods is compared, and the time required for tracking each frame is gained. Experiments results show that the two methods can gain similar distance and mean values, and similar tracking resu

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2

Sanjay, Kumar. "Statistical features on Riemannian Manifold." MATHEMATICS EDUCATION LVIII, no. 2, June 2024 (2024): 10–22. https://doi.org/10.5281/zenodo.13855976.

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             <em>  Manifold are fundamental structures in different geometry. A smooth manifold with a metric is called a Riemannian Manifold. A connected Riemannian manifold carries the structure of metric space where distance function is the arc length of minimizing geodesic. In this paper, we consider finite dimensional manifold with a Riemannian metric as the basic structure. Based on this metric, we develop the notion of mean value, and covariance matrix of a random element, normal law mahalanobis distance and x2 test.    </em>

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Mahamane, Saminou Ali, Hassirou Mouhamadou, and Mahaman Bazanfare. "Geodesically Complete Lie Algebroid." British Journal of Mathematics & Computer Science 22, no. 5 (2017): 1–12. https://doi.org/10.9734/BJMCS/2017/34009.

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In this paper we introduce the notion of geodesically complete Lie algebroid. We give a Riemannian distance on the connected base manifold of a Riemannian Lie algebroid. We also prove that the distance is equivalent to natural one if the base manifold was endowed with Riemannian metric. We obtain Hopf Rinow type theorem in the case of transitive Riemannian Lie algebroid, and give a characterization of the connected base manifold of a geodesically complete Lie algebroid.

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4

Rylov, Yuri A. "Geometry without topology as a new conception of geometry." International Journal of Mathematics and Mathematical Sciences 30, no. 12 (2002): 733–60. http://dx.doi.org/10.1155/s0161171202012243.

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A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the T-geometric one. T-geometry is the simplest geometric conception (essentially, only finite point sets are investigated) and simultaneously, it is the most general one. It is insensitive to the s

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5

Ho, Pei-Ming. "Riemannian Geometry on Quantum Spaces." International Journal of Modern Physics A 12, no. 05 (1997): 923–43. http://dx.doi.org/10.1142/s0217751x97000694.

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An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective space and the two-sheeted space.

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Correa, Cleber Souza, Thiago Braido Nogueira de Melo, and Diogo Machado Custódio. "The Alpha Group Tensorial Metric." Revista Brasileira de História da Matemática 24, no. 48 (2024): 51–57. http://dx.doi.org/10.47976/rbhm2024v24n4851-57.

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The Alpha Group is an abstract geometry group in R4. The way it was conceived allows a new interpretation of the structure of hypercomplex space with a new geometry and spatial topology, and a meaning for the geometric representation of R4 space to infinity. Therefore, it has been described as the tensorial metric formula in the Alpha Group. It will be shown that the Riemannian and Euclidean distance metrics between infinitesimal surfaces are represented as special cases of the metric of the Alpha group.

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Mönkkönen, Keijo. "Boundary rigidity for Randers metrics." Annales Fennici Mathematici 47, no. 1 (2021): 89–102. http://dx.doi.org/10.54330/afm.112492.

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 If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.

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Stojisavljević, Vukašin, and Jun Zhang. "Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics." International Journal of Mathematics 32, no. 07 (2021): 2150040. http://dx.doi.org/10.1142/s0129167x21500403.

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We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text]. The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text]. Our main focus is on the space of unit codisc bundles of orientable surfaces of positiv

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Beem, John K., and Paul E. Ehrlich. "Geodesic completeness and stability." Mathematical Proceedings of the Cambridge Philosophical Society 102, no. 2 (1987): 319–28. http://dx.doi.org/10.1017/s0305004100067347.

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A (connected) Riemannian manifold M is geodesically complete (i.e. each geodesic may be extended to a geodesic with domain ( −∞, + ∞)) iff, as a metric space under the induced Riemannian distance function, M is Cauchy complete ([12], p. 138). Furthermore, all compact Riemannian manifolds are complete. On the other hand, compact pseudo-Riemannian manifolds exist which are not geodesically complete. For example Fierz and Jost[8] have constructed incomplete metrics of the form 2dx dy + h dy2 on T2. Furthermore, Williams [16] has shown that geodesic completeness may fail to be stable for pseudo-Ri

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Fang, Yong, and Patrick Foulon. "On Finsler manifolds of negative flag curvature." Journal of Topology and Analysis 07, no. 03 (2015): 483–504. http://dx.doi.org/10.1142/s1793525315500181.

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One of the key differences between Finsler metrics and Riemannian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p, q) is not necessarily equal to d(q, p). In this paper, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan–Hadamard manifolds. In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let φ be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then its cohomological press

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Chen, Da, Jean-Marie Mirebeau, and Laurent D. Cohen. "Vessel tree extraction using radius-lifted keypoints searching scheme and anisotropic fast marching method." Journal of Algorithms & Computational Technology 10, no. 4 (2016): 224–34. http://dx.doi.org/10.1177/1748301816656289.

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Geodesic methods have been widely applied to image analysis. They are particularly efficient to extract a tubular structure, such as a blood vessel, given its two endpoints in a 2D or 3D medical image. We address here a more difficult problem: the extraction of a full vessel tree structure given a single initial root point, by growing a collection of keypoints or new initial source points, connected by minimal geodesic paths. In this article, those keypoints are iteratively added, using a new detection criteria, which utilize the weighted geodesic distances with respect to a radius-lifted Riem

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Wang, Jing, Huafei Sun, and Simone Fiori. "Empirical Means on Pseudo-Orthogonal Groups." Mathematics 7, no. 10 (2019): 940. http://dx.doi.org/10.3390/math7100940.

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The present article studies the problem of computing empirical means on pseudo-orthogonal groups. To design numerical algorithms to compute empirical means, the pseudo-orthogonal group is endowed with a pseudo-Riemannian metric that affords the computation of the exponential map in closed forms. The distance between two pseudo-orthogonal matrices, which is an essential ingredient, is computed by both the Frobenius norm and the geodesic distance. The empirical-mean computation problem is solved via a pseudo-Riemannian-gradient-stepping algorithm. Several numerical tests are conducted to illustr

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Suh, Yoon-Je, and Byung Hyung Kim. "Riemannian Embedding Banks for Common Spatial Patterns with EEG-based SPD Neural Networks." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 1 (2021): 854–62. http://dx.doi.org/10.1609/aaai.v35i1.16168.

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Modeling non-linear data as symmetric positive definite (SPD) matrices on Riemannian manifolds has attracted much attention for various classification tasks. In the context of deep learning, SPD matrix-based Riemannian networks have been shown to be a promising solution for classifying electroencephalogram (EEG) signals, capturing the Riemannian geometry within their structured 2D feature representation. However, existing approaches usually learn spatial-temporal structures in an embedding space for all available EEG signals, and their optimization procedures rely on computationally expensive

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Brown, Adam R., Michael H. Freedman, Henry W. Lin, and Leonard Susskind. "Universality in long-distance geometry and quantum complexity." Nature 622, no. 7981 (2023): 58–62. http://dx.doi.org/10.1038/s41586-023-06460-3.

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AbstractIn physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class1. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds2 by their long-distance properties. We show that many metrics on low-dimensional Lie groups have markedly different short-distance properties but nearly identical distance functions at long distances, and provide evidence that this phenomenon is even more robust in high dimensions. An application

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Veronelli, Giona. "Scalar Curvature via Local Extent." Analysis and Geometry in Metric Spaces 6, no. 1 (2018): 146–64. http://dx.doi.org/10.1515/agms-2018-0008.

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AbstractWe give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between (n + 1) points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.

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Engelstein, Max, Robin Neumayer, and Luca Spolaor. "Quantitative stability for minimizing Yamabe metrics." Transactions of the American Mathematical Society, Series B 9, no. 13 (2022): 395–414. http://dx.doi.org/10.1090/btran/111.

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On any closed Riemannian manifold of dimension n ≥ 3 n\geq 3 , we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false.

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Bidabad, Behroz, and Maryam Sepasi. "On complete Finsler spaces of constant negative Ricci curvature." International Journal of Geometric Methods in Modern Physics 17, no. 03 (2020): 2050041. http://dx.doi.org/10.1142/s0219887820500413.

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Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

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Tamang, Sagar K., Ardeshir Ebtehaj, Peter J. van Leeuwen, Dongmian Zou, and Gilad Lerman. "Ensemble Riemannian data assimilation over the Wasserstein space." Nonlinear Processes in Geophysics 28, no. 3 (2021): 295–309. http://dx.doi.org/10.5194/npg-28-295-2021.

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Abstract. In this paper, we present an ensemble data assimilation paradigm over a Riemannian manifold equipped with the Wasserstein metric. Unlike the Euclidean distance used in classic data assimilation methodologies, the Wasserstein metric can capture the translation and difference between the shapes of square-integrable probability distributions of the background state and observations. This enables us to formally penalize geophysical biases in state space with non-Gaussian distributions. The new approach is applied to dissipative and chaotic evolutionary dynamics, and its potential advanta

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Uhlmann, Gunther, and Jenn-Nan Wang. "Boundary determination of a Riemannian metric by the localized boundary distance function." Advances in Applied Mathematics 31, no. 2 (2003): 379–87. http://dx.doi.org/10.1016/s0196-8858(03)00017-4.

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Liu, Jianxiang. "Study on the Space Geodesy." Highlights in Science, Engineering and Technology 38 (March 16, 2023): 1079–88. http://dx.doi.org/10.54097/hset.v38i.5998.

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Geodesic is a kind of shortest path among all curves in a metric space, which originally appeared in the Gaussian period. Since then, it was extensively applied in various branches of mathematics and physics, such as Riemannian geometry, digital geometry, Einstein’s relativity, etc. This thesis mainly discusses geodesic definitions in Euclidean space and those in smooth manifolds after introducing the basic theory of smooth manifolds. At first, the thesis applies three ways to define geodesics on a surface, including the geodesic curvature method, the shortest distance method, and the relation

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Iosevich, Alex, Krystal Taylor, and Ignacio Uriarte-Tuero. "Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds." Mathematics 9, no. 15 (2021): 1802. http://dx.doi.org/10.3390/math9151802.

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Let M be a compact d-dimensional Riemannian manifold without a boundary. Given a compact set E⊂M, we study the set of distances from the set E to a fixed point x∈E. This set is Δρx(E)={ρ(x,y):y∈E}, where ρ is the Riemannian metric on M. We prove that if the Hausdorff dimension of E is greater than d+12, then there exist many x∈E such that the Lebesgue measure of Δρx(E) is positive. This result was previously established by Peres and Schlag in the Euclidean setting. We give a simple proof of the Peres–Schlag result and generalize it to a wide range of distance type functions. Moreover, we exten

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Williams, Simon, Arthur George Suvorov, Zengfu Wang, and Bill Moran. "The Information Geometry of Sensor Configuration." Sensors 21, no. 16 (2021): 5265. http://dx.doi.org/10.3390/s21165265.

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In problems of parameter estimation from sensor data, the Fisher information provides a measure of the performance of the sensor; effectively, in an infinitesimal sense, how much information about the parameters can be obtained from the measurements. From the geometric viewpoint, it is a Riemannian metric on the manifold of parameters of the observed system. In this paper, we consider the case of parameterized sensors and answer the question, “How best to reconfigure a sensor (vary the parameters of the sensor) to optimize the information collected?” A change in the sensor parameters results i

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Bauer, Martin, Martins Bruveris, Philipp Harms, and Peter W. Michor. "Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation." Annals of Global Analysis and Geometry 41, no. 4 (2011): 461–72. http://dx.doi.org/10.1007/s10455-011-9294-9.

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Honda, Shouhei. "Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces." Analysis and Geometry in Metric Spaces 6, no. 1 (2018): 129–45. http://dx.doi.org/10.1515/agms-2018-0007.

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Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distan

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Bauer, Martin, Nicolas Charon, Tom Needham, and Mao Nishino. "Path constrained unbalanced optimal transport." Nonlinearity 38, no. 7 (2025): 075019. https://doi.org/10.1088/1361-6544/ade21d.

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Abstract Dynamical formulations of optimal transport (OT) frame the task of comparing distributions as a variational problem which searches for a path between distributions minimizing a kinetic energy functional. In applications, it is frequently natural to require paths of distributions to satisfy additional conditions. Inspired by this, we introduce a model for dynamical OT which incorporates constraints on the space of admissible paths into the framework of unbalanced OT, where the source and target measures are allowed to have a different total mass. Our main results establish, for several

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Taylor, Stephen. "Clustering Financial Return Distributions Using the Fisher Information Metric." Entropy 21, no. 2 (2019): 110. http://dx.doi.org/10.3390/e21020110.

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Information geometry provides a correspondence between differential geometry and statistics through the Fisher information matrix. In particular, given two models from the same parametric family of distributions, one can define the distance between these models as the length of the geodesic connecting them in a Riemannian manifold whose metric is given by the model’s Fisher information matrix. One limitation that has hindered the adoption of this similarity measure in practical applications is that the Fisher distance is typically difficult to compute in a robust manner. We review such complic

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Barilari, Davide, Ugo Boscain, and Daniele Cannarsa. "On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 9. http://dx.doi.org/10.1051/cocv/2021104.

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Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.

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Pinele, Julianna, João E. Strapasson, and Sueli I. R. Costa. "The Fisher-Rao Distance between Multivariate Normal Distributions: Special Cases, Boundsand Applications." Entropy 22, no. 4 (2020): 404. http://dx.doi.org/10.3390/e22040404.

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The Fisher–Rao distance is a measure of dissimilarity between probability distributions, which, under certain regularity conditions of the statistical model, is up to a scaling factor the unique Riemannian metric invariant under Markov morphisms. It is related to the Shannon entropy and has been used to enlarge the perspective of analysis in a wide variety of domains such as image processing, radar systems, and morphological classification. Here, we approach this metric considered in the statistical model of normal multivariate probability distributions, for which there is not an explicit expr

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Gao, Wenxu, Zhengming Ma, Weichao Gan, and Shuyu Liu. "Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry." Entropy 23, no. 9 (2021): 1117. http://dx.doi.org/10.3390/e23091117.

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Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based o

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Pereira, R., X. Mestre, and D. Gregoratti. "Asymptotics of Distances Between Sample Covariance Matrices." IEEE TRANSACTIONS ON SIGNAL PROCESSING 72 (January 1, 2024): 1460–74. https://doi.org/10.1109/TSP.2024.3368771.

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This work considers the asymptotic behavior of the distance between two sample covariance matrices (SCM). A general result is provided for a class of functionals that can be expressed as sums of traces of functions that are separately applied to each covariance matrix. In particular, this class includes very conventional metrics, such as the Euclidean distance or Jeffrery's divergence, as well as a number of other more sophisticated distances recently derived from Riemannian geometry considerations, such as the log-Euclidean metric. In particular, we analyze the asymptotic behavior of this cla

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PORTESI, MARIELA, ANGEL L. PLASTINO, and FLAVIA PENNINI. "INFORMATION GEOMETRY AND PHASE TRANSITIONS." International Journal of Modern Physics B 20, no. 30n31 (2006): 5250–53. http://dx.doi.org/10.1142/s0217979206036338.

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We present, from an information theoretic viewpoint, an analysis of phase transitions and critical phenomena in quantum systems. Our study is based on geometrical considerations within the Riemannian space of thermodynamic parameters that characterize the system. A metric for the space can be derived from an appropriate definition of distance between quantum states. For this purpose, we consider generalized α-divergences that include the standard Kullback–Leibler relative entropy. The use of other measures of information distance is taken into account, and the thermodynamic stability of the sy

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Oviedo, Harry. "Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold." Mathematics 11, no. 11 (2023): 2414. http://dx.doi.org/10.3390/math11112414.

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In this paper, we consider the problem of minimizing a continuously differentiable function on the Stiefel manifold. To solve this problem, we develop a geodesic-free proximal point algorithm equipped with Euclidean distance that does not require use of the Riemannian metric. The proposed method can be regarded as an iterative fixed-point method that repeatedly applies a proximal operator to an initial point. In addition, we establish the global convergence of the new approach without any restrictive assumption. Numerical experiments on linear eigenvalue problems and the minimization of sums o

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Karmanova, M. B. "Метрические характеристики классов компактных множеств на группах Карно с сублоренцевой структурой". Владикавказский математический журнал 26, № 3 (2024): 56–64. http://dx.doi.org/10.46698/d9212-8277-5800-l.

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We consider classes of mappings of Carnot groups that are intrinsically Lipschitz and defined on compact subsets, and describe the metric characteristics of their images under the condition that a~sub-Lorentzian structure is introduced on the image. This structure is a sub-Riemannian generalization of Minkowski geometry. One of its features is the unlimitedness of the balls constructed with respect to the~intrinsic distance. In sub-Lorentzian geometry, the study of spacelike surfaces whose intersections with such balls are limited, is of independent interest. If the mapping is defined on an op

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Duan, Xiaomin, Xueting Ji, Huafei Sun, and Hao Guo. "A Non-Iterative Method for the Difference of Means on the Lie Group of Symmetric Positive-Definite Matrices." Mathematics 10, no. 2 (2022): 255. http://dx.doi.org/10.3390/math10020255.

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A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invariant Riemannian metrics have a perfect theoretical framework and avoid the drawbacks of the Euclidean inner product, their complex formulas also lead to sophisticated and time-consuming algorithms. To make up for this limitation, log-Euclidean metrics with simpler formulas and faster calculations are employed in this manuscript. Our new approach is t

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Lim, Yongdo. "Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices." Canadian Journal of Mathematics 56, no. 4 (2004): 776–93. http://dx.doi.org/10.4153/cjm-2004-035-5.

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AbstractWe explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold Sym(n, ℝ)++ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold Sym(p, ℝ)++ × Sym(q, ℝ)++ block diagonally embedded in Sym(n, ℝ)++ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when p ≤ 2 or q ≤ 2.

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GUILLARMOU, COLIN, and MARCO MAZZUCCHELLI. "Marked boundary rigidity for surfaces." Ergodic Theory and Dynamical Systems 38, no. 4 (2016): 1459–78. http://dx.doi.org/10.1017/etds.2016.94.

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We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

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Besson, Gérard, Gilles Courtois, and Sylvestre Gallot. "Minimal entropy and Mostow's rigidity theorems." Ergodic Theory and Dynamical Systems 16, no. 4 (1996): 623–49. http://dx.doi.org/10.1017/s0143385700009019.

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Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we definewhere B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sen

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Duan, Xiaomin, Huafei Sun, and Linyu Peng. "Riemannian Means on Special Euclidean Group and Unipotent Matrices Group." Scientific World Journal 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/292787.

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Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and

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Peng, Zhen, Hongyi Li, Di Zhao, and Chengwei Pan. "Reducing the Dimensionality of SPD Matrices with Neural Networks in BCI." Mathematics 11, no. 7 (2023): 1570. http://dx.doi.org/10.3390/math11071570.

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In brain–computer interface (BCI)-based motor imagery, the symmetric positive definite (SPD) covariance matrices of electroencephalogram (EEG) signals with discriminative information features lie on a Riemannian manifold, which is currently attracting increasing attention. Under a Riemannian manifold perspective, we propose a non-linear dimensionality reduction algorithm based on neural networks to construct a more discriminative low-dimensional SPD manifold. To this end, we design a novel non-linear shrinkage layer to modify the extreme eigenvalues of the SPD matrix properly, then combine the

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Neilson, Peter, Megan Neilson, and Robin Bye. "A Riemannian Geometry Theory of Three-Dimensional Binocular Visual Perception." Vision 2, no. 4 (2018): 43. http://dx.doi.org/10.3390/vision2040043.

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We present a Riemannian geometry theory to examine the systematically warped geometry of perceived visual space attributable to the size–distance relationship of retinal images associated with the optics of the human eye. Starting with the notion of a vector field of retinal image features over cortical hypercolumns endowed with a metric compatible with that size–distance relationship, we use Riemannian geometry to construct a place-encoded theory of spatial representation within the human visual system. The theory draws on the concepts of geodesic spray fields, covariant derivatives, geodesic

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Banks, Jessica E. "Is a straight line the shortest path?" Mathematical Gazette 102, no. 553 (2018): 1–12. http://dx.doi.org/10.1017/mag.2018.2.

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Is the shortest path from A to B the straight line between them? Your first response might be to think it's obviously so. But in fact you know that it's not quite that straightforward. Your sat-nav knows it's not that straightforward. It asks whether you would like it to find the shortest route or the fastest route, because finding the best path depends on knowing what exactly you mean by ‘long’. Likewise, if you're on a walk in the mountains, there's a good chance you'd rather follow the path around the head of the valley, rather than heading down the steep slope and up the other side.The sam

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Le, Huiling. "Locating Fréchet means with application to shape spaces." Advances in Applied Probability 33, no. 2 (2001): 324–38. http://dx.doi.org/10.1017/s0001867800010818.

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We use Jacobi field arguments and the contraction mapping theorem to locate Fréchet means of a class of probability measures on locally symmetric Riemannian manifolds with non-negative sectional curvatures. This leads, in particular, to a method for estimating Fréchet mean shapes, with respect to the distance function ρ determined by the induced Riemannian metric, of a class of probability measures on Kendall's shape spaces. We then combine this with the technique of ‘horizontally lifting’ to the pre-shape spheres to obtain an algorithm for finding Fréchet mean shapes, with respect to ρ, of a

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Arimoto, Suguru, Morio Yoshida, Masahiro Sekimoto, and Kenji Tahara. "A Riemannian-Geometry Approach for Modeling and Control of Dynamics of Object Manipulation under Constraints." Journal of Robotics 2009 (2009): 1–16. http://dx.doi.org/10.1155/2009/892801.

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A Riemannian-geometry approach for modeling and control of dynamics of object manipulation under holonomic or non-holonomic constraints is presented. First, position/force hybrid control of an endeffector of a multijoint redundant (or nonredundant) robot under a holonomic constraint is reinterpreted in terms of “submersion” in Riemannian geometry. A force control signal constructed in the image space of the constraint gradient is regarded as a lifting (or pressing) in the direction orthogonal to the kernel space. By means of the Riemannian distance on the constraint submanifold, stability of p

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Lecian, Orchidea Maria. "The new Generalized Schwarzschild-spacetimes trivial Ricci solitons and the new smooth metric space." Journal of AppliedMath 3, no. 4 (2025): 2901. https://doi.org/10.59400/jam2901.

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The Ricci flow of the Generalized-Schwarzschild spacetimes is newly studied. The soliton configurations are newly stated as trivial Ricci soliton of (Generalized)-Schwarzschild spacetimes. The new smooth metric space is written; the majorization theorem for the distance is given. The application of harmonic maps is presented. The definition of topological soliton as a Schwarzschild soliton of complete Riemannian manifold is newly provided with. New theorems about Generalized-Schwarzschild solitons which are extended from those about the Kaehler solitons are proven; the new theorems are given,

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Vossos, Spyridon, Elias Vossos, and Christos G. Massouros. "The Relation between General Relativity’s Metrics and Special Relativity’s Gravitational Scalar Generalized Potentials and Case Studies on the Schwarzschild Metric, Teleparallel Gravity, and Newtonian Potential." Particles 4, no. 4 (2021): 536–76. http://dx.doi.org/10.3390/particles4040039.

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This paper shows that gravitational results of general relativity (GR) can be reached by using special relativity (SR) via a SR Lagrangian that derives from the corresponding GR time dilation and vice versa. It also presents a new SR gravitational central scalar generalized potential V=V(r,r.,ϕ.), where r is the distance from the center of gravity and r.,ϕ. are the radial and angular velocity, respectively. This is associated with the Schwarzschild GR time dilation from where a SR scalar generalized potential is obtained, which is exactly equivalent to the Schwarzschild metric. Thus, the Prece

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NILSSON, OLA, MARTIN REIMERS, KEN MUSETH, and ANDERS BRUN. "A NEW ALGORITHM FOR COMPUTING RIEMANNIAN GEODESIC DISTANCE IN RECTANGULAR 2-D AND 3-D GRIDS." International Journal on Artificial Intelligence Tools 22, no. 06 (2013): 1360020. http://dx.doi.org/10.1142/s0218213013600208.

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We present a novel way to efficiently compute Riemannian geodesic distance over a two- or three-dimensional domain. It is based on a previously presented method for computation of geodesic distances on surface meshes. Our method is adapted for rectangular grids, equipped with a variable anisotropic metric tensor. Processing and visualization of such tensor fields is common in certain applications, for instance structure tensor fields in image analysis and diffusion tensor fields in medical imaging. The included benchmark study shows that our method provides significantly better results in anis

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Luo, Yihao, Shiqiang Zhang, Yueqi Cao, and Huafei Sun. "Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing." Entropy 23, no. 9 (2021): 1214. http://dx.doi.org/10.3390/e23091214.

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The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design a

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BESSA, G. P., and J. F. MONTENEGRO. "On Cheng's eigenvalue comparison theorem." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 3 (2008): 673–82. http://dx.doi.org/10.1017/s0305004107000965.

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AbstractWe observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls [4] is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics $g_{\kappa}$ on $[0,r]\times \mathbb{S}^{3}$, non-isometric to the standard metric canκ of constant se

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Himpel, Benjamin. "Geometry of Music Perception." Mathematics 10, no. 24 (2022): 4793. http://dx.doi.org/10.3390/math10244793.

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Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of all chords is shown to be a Whitney stratified space. Each stratum is a Riemannian manifold which naturally yields a geodesic distance across strata. The resulting metric is compatible with voice-leading satisfying the triangle inequality. The geometric model allows for rigorous studies of psychoacoustic quantities such as roughness and harmonicity as heigh

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Sahu, Pradip Kumar. "Optimal Trajectory Planning of Industrial Robots using Geodesic." IAES International Journal of Robotics and Automation (IJRA) 5, no. 3 (2016): 190. http://dx.doi.org/10.11591/ijra.v5i3.pp190-198.

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<p>This paper intends to propose an optimal trajectory planning technique using geodesic to achieve smooth and accurate trajectory for industrial robots. Geodesic is a distance minimizing curve between any two points on a Riemannian manifold. A Riemannian metric has been assigned to the workspace by combining its position and orientation space together in order to attain geodesic conditions for desired motion of the end-effector. Previously, trajectory has been planned by considering position and orientation separately. However, practically we cannot plan separately because the manipulat

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