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Journal articles on the topic 'Methods of global Riemannian geometry'

Author: Grafiati

Published: 18 April 2022

Last updated: 30 July 2025

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1

STEINBAUER, R., and M. KUNZINGER. "GENERALISED PSEUDO-RIEMANNIAN GEOMETRY FOR GENERAL RELATIVITY." International Journal of Modern Physics A 17, no. 20 (2002): 2776. http://dx.doi.org/10.1142/s0217751x0201203x.

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The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen2 excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems1,5,8. Under the influence of these applications in general relativity the nonlinear theory

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Keningson, Jonathan. "Mathematical foundation of High-Dimensional Data Analysis: Leveraging Topology and Geometry for Enhanced Model Interpretability in AI." International Journal of Scientific Research and Management (IJSRM) 12, no. 11 (2024): 546–57. http://dx.doi.org/10.18535/ijsrm/v12i11.m01.

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One of the most important challenges for modern AI and machine learning is the analysis of high-dimensional data. Traditional methods face serious complications in such cases due to high complexity of datasets: the curse of dimensionality, overfitting, and lack of transparency of model behavior. In this paper, we adopt a novel approach to analyze high-dimensional data; topological and geometric techniques will be exploited, taking advantage of better model interpretability and deeper insights into the structure. Precisely, we discuss Topological Data Analysis, mainly Persistent Homology (Edels

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Zhang, Jianhai, Zhiyong Feng, Yong Su, and Meng Xing. "Bayesian Covariance Representation with Global Informative Prior for 3D Action Recognition." ACM Transactions on Multimedia Computing, Communications, and Applications 17, no. 4 (2021): 1–22. http://dx.doi.org/10.1145/3460235.

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For the merits of high-order statistics and Riemannian geometry, covariance matrix has become a generic feature representation for action recognition. An independent action can be represented by an empirical statistics over all of its pose samples. Two major problems of covariance include the following: (1) it is prone to be singular so that actions fail to be represented properly, and (2) it is short of global action/pose-aware information so that expressive and discriminative power is limited. In this article, we propose a novel Bayesian covariance representation by a prior regularization me

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Uschmajew, André, and Bart Vandereycken. "On critical points of quadratic low-rank matrix optimization problems." IMA Journal of Numerical Analysis 40, no. 4 (2020): 2626–51. http://dx.doi.org/10.1093/imanum/drz061.

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Abstract The absence of spurious local minima in certain nonconvex low-rank matrix recovery problems has been of recent interest in computer science, machine learning and compressed sensing since it explains the convergence of some low-rank optimization methods to global optima. One such example is low-rank matrix sensing under restricted isometry properties (RIPs). It can be formulated as a minimization problem for a quadratic function on the Riemannian manifold of low-rank matrices, with a positive semidefinite Riemannian Hessian that acts almost like an identity on low-rank matrices. In thi

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BELLUCCI, STEFANO, and BHUPENDRA NATH TIWARI. "ON REAL INTRINSIC WALL CROSSINGS." International Journal of Modern Physics A 26, no. 30n31 (2011): 5171–209. http://dx.doi.org/10.1142/s0217751x11054917.

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We study moduli space stabilization of a class of BPS configurations from the perspective of the real intrinsic Riemannian geometry. Our analysis exhibits a set of implications towards the stability of the D-term potentials, defined for a set of Abelian scalar fields. In particular, we show that the nature of marginal and threshold walls of stabilities may be investigated by real geometric methods. Interestingly, we find that the leading order contributions may easily be accomplished by translations of the Fayet parameter. Specifically, we notice that the various possible linear, planar, hyper

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Petersen, Peter. "Aspects of global Riemannian geometry." Bulletin of the American Mathematical Society 36, no. 03 (1999): 297–345. http://dx.doi.org/10.1090/s0273-0979-99-00787-9.

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BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.

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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explana

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González-Dávila, J. C., M. C. González-Dávila, and L. Vanhecke. "Invariant submanifolds in flow geometry." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 62, no. 3 (1997): 290–314. http://dx.doi.org/10.1017/s1446788700001026.

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AbstractWe begin a study of invariant isometric immersions into Riemannian manifolds (M, g) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those (M, g) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided.

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Stavrinos, Panayiotis, and Sergiu I. Vacaru. "Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables." Universe 7, no. 4 (2021): 89. http://dx.doi.org/10.3390/universe7040089.

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We study new classes of generic off-diagonal and diagonal cosmological solutions for effective Einstein equations in modified gravity theories (MGTs), with modified dispersion relations (MDRs), and encoding possible violations of (local) Lorentz invariance (LIVs). Such MGTs are constructed for actions and Lagrange densities with two non-Riemannian volume forms (similar to two measure theories (TMTs)) and associated bimetric and/or biconnection geometric structures. For conventional nonholonomic 2 + 2 splitting, we can always describe such models in Finsler-like variables, which is important fo

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Kapralov, Nikolai, Zhanna Nagornova, and Natalia Shemyakina. "Classification Methods for EEG Patterns of Imaginary Movements." Informatics and Automation 20, no. 1 (2021): 94–132. http://dx.doi.org/10.15622/ia.2021.20.1.4.

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The review focuses on the most promising methods for classifying EEG signals for non-invasive BCIs and theoretical approaches for the successful classification of EEG patterns. The paper provides an overview of articles using Riemannian geometry, deep learning methods and various options for preprocessing and "clustering" EEG signals, for example, common-spatial pattern (CSP). Among other approaches, pre-processing of EEG signals using CSP is often used, both offline and online. The combination of CSP, linear discriminant analysis, support vector machine and neural network (BPNN) made it possi

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Mikeš, Josef, Vladimir Rovenski, Sergey Stepanov, and Irina Tsyganok. "Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds." Mathematics 9, no. 9 (2021): 927. http://dx.doi.org/10.3390/math9090927.

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In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow.

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Rovenski, Vladimir, Sergey Stepanov, and Irina Tsyganok. "A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings." Axioms 10, no. 4 (2021): 333. http://dx.doi.org/10.3390/axioms10040333.

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This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.

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Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Mechanical Design 117, B (1995): 87–92. http://dx.doi.org/10.1115/1.2836475.

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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable sele

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Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Vibration and Acoustics 117, B (1995): 87–92. http://dx.doi.org/10.1115/1.2838681.

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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable sele

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Woodward, L. M. "GLOBAL RIEMANNIAN GEOMETRY (Ellis Horwood Series: Mathematics and Its Applications)." Bulletin of the London Mathematical Society 17, no. 2 (1985): 194–96. http://dx.doi.org/10.1112/blms/17.2.194.

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Brozos-Vázquez, M., and P. Gilkey. "The global geometry of Riemannian manifolds with commuting curvature operators." Journal of Fixed Point Theory and Applications 1, no. 1 (2006): 87–96. http://dx.doi.org/10.1007/s11784-006-0001-6.

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Dr Manju Bala. "Singularities and Metric Structures in Sub-Riemannian Geometries with Applications to Control Theory." International Journal of Scientific Research in Science, Engineering and Technology 12, no. 3 (2025): 359–63. https://doi.org/10.32628/ijsrset251248.

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Sub-Riemannian geometry extends classical Riemannian frameworks by defining metrics only on constrained directions within manifolds, naturally modeling systems with nonholonomic constraints. This paper investigates the nature and impact of singularities—points where the geometric structure or metric degenerates—on the local and global properties of sub-Riemannian manifolds. We analyze metric behavior near singularities through nilpotent approximations and study their influence on geodesic existence, uniqueness, and stability, with particular emphasis on abnormal geodesics. Further, we explore

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BRANDT, HOWARD E. "ASPECTS OF THE RIEMANNIAN GEOMETRY OF QUANTUM COMPUTATION." International Journal of Modern Physics B 26, no. 27n28 (2012): 1243004. http://dx.doi.org/10.1142/s0217979212430047.

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A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2n) for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the g

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Dunajski, Maciej. "Null Kähler Geometry and Isomonodromic Deformations." Communications in Mathematical Physics 391, no. 1 (2021): 77–105. http://dx.doi.org/10.1007/s00220-021-04270-0.

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AbstractWe construct the normal forms of null-Kähler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear on the space of Bridgeland stability conditions on a Calabi–Yau threefold. Using twistor methods we show that, in dimension four—where there is a connection with dispersionless integrability—the cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs.

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Marotta, Vincenzo Emilio, and Richard J. Szabo. "Algebroids, AKSZ Constructions and Doubled Geometry." Complex Manifolds 8, no. 1 (2021): 354–402. http://dx.doi.org/10.1515/coma-2020-0125.

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Abstract We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how the

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Ammann, Bernd, Bernhard Hanke, and Anna Sakovich. "Analysis, Geometry and Topology of Positive Scalar Curvature Metrics." Oberwolfach Reports 21, no. 1 (2024): 483–566. http://dx.doi.org/10.4171/owr/2024/9.

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Riemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity. To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity. This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange.

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Majidov, Ikhtiyor, and Taegkeun Whangbo. "Efficient Classification of Motor Imagery Electroencephalography Signals Using Deep Learning Methods." Sensors 19, no. 7 (2019): 1736. http://dx.doi.org/10.3390/s19071736.

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Single-trial motor imagery classification is a crucial aspect of brain–computer applications. Therefore, it is necessary to extract and discriminate signal features involving motor imagery movements. Riemannian geometry-based feature extraction methods are effective when designing these types of motor-imagery-based brain–computer interface applications. In the field of information theory, Riemannian geometry is mainly used with covariance matrices. Accordingly, investigations showed that if the method is used after the execution of the filterbank approach, the covariance matrix preserves the f

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Boyom, Michel Nguiffo. "Linear Gauge and the Linearization Problem for Webs." Journal of the Tensor Society 8, no. 01 (2007): 1–16. http://dx.doi.org/10.56424/jts.v8i01.10562.

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Regarding the theory of foliation, some aspects of the theory of Riemannian foliations have been brought in completion by the Molino theory. Such a structure is defined by some finite dimensional Lie subalgebra of the Lie algebra of transverse vector fields. The problem I am interested in is more modeste. It is to get sufficient conditions for a smooth manifold admitting foliations with transverse (pseudo) Riemannian metrics. The investigation is inspired by both methods of information geometry and the Hessian geometry.

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Bolsinov, A. V., V. S. Matveev, and A. T. Fomenko. "Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry." Sbornik: Mathematics 189, no. 10 (1998): 1441–66. http://dx.doi.org/10.1070/sm1998v189n10abeh000346.

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Bracken, Paul. "Integrable Equations and Their Evolutions Based on Intrinsic Geometry of Riemann Spaces." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–16. http://dx.doi.org/10.1155/2009/210304.

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The intrinsic geometry of surfaces and Riemannian spaces will be investigated. It is shown that many nonlinear partial differential equations with physical applications and soliton solutions can be determined from the components of the relevant metric for the space. The manifolds of interest are surfaces and higher-dimensional Riemannian spaces. Methods for specifying integrable evolutions of surfaces by means of these equations will also be presented.

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Al-Mashhadani, Zubaidah, Nasrin Bayat, Ibrahim F. Kadhim, Renoa Choudhury, and Joon-Hyuk Park. "The Efficacy and Utility of Lower-Dimensional Riemannian Geometry for EEG-Based Emotion Classification." Applied Sciences 13, no. 14 (2023): 8274. http://dx.doi.org/10.3390/app13148274.

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Electroencephalography (EEG) signals have diverse applications in brain-computer interfaces (BCIs), neurological condition diagnoses, and emotion recognition across healthcare, education, and entertainment domains. This paper presents a robust method that leverages Riemannian geometry to enhance the accuracy of EEG-based emotion classification. The proposed approach involves adaptive feature extraction using principal component analysis (PCA) in the Euclidean space to capture relevant signal characteristics and improve classification performance. Covariance matrices are derived from the extrac

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Park, JuneYoung, YuMi Lee, Tae-Joon Kim, and Jang-Hwan Choi. "Riemannian Geometric-based Meta Learning." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 19 (2025): 19839–47. https://doi.org/10.1609/aaai.v39i19.34185.

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Meta-learning, or "learning to learn," aims to enable models to quickly adapt to new tasks with minimal data. While traditional methods like Model-Agnostic Meta-Learning (MAML) optimize parameters in Euclidean space, they often struggle to capture complex learning dynamics, particularly in few-shot learning scenarios. To address this limitation, we propose Stiefel-MAML, which integrates Riemannian geometry by optimizing within the Stiefel manifold, a space that naturally enforces orthogonality constraints. By leveraging the geometric structure of the Stiefel manifold, we improve parameter expr

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Shuqfa, Zaid, Abdelkader Nasreddine Belkacem, and Abderrahmane Lakas. "Decoding Multi-Class Motor Imagery and Motor Execution Tasks Using Riemannian Geometry Algorithms on Large EEG Datasets." Sensors 23, no. 11 (2023): 5051. http://dx.doi.org/10.3390/s23115051.

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The use of Riemannian geometry decoding algorithms in classifying electroencephalography-based motor-imagery brain–computer interfaces (BCIs) trials is relatively new and promises to outperform the current state-of-the-art methods by overcoming the noise and nonstationarity of electroencephalography signals. However, the related literature shows high classification accuracy on only relatively small BCI datasets. The aim of this paper is to provide a study of the performance of a novel implementation of the Riemannian geometry decoding algorithm using large BCI datasets. In the study, we apply

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Rovenski, Vladimir. "Integral Formulas for Almost Product Manifolds and Foliations." Mathematics 10, no. 19 (2022): 3645. http://dx.doi.org/10.3390/math10193645.

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Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals d

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Makarenko N.G., ChoYong-beom, and Esenaliev A. B. "RIEMANNIAN METRIC FOR TEXTURE RECOGNITION." PHYSICO-MATHEMATICAL SERIES, no. 6 (December 15, 2018): 23–27. http://dx.doi.org/10.32014/2018.2518-1726.13.

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The article discusses the recognition of textures on digital images by methods of computational topology and Riemannian geometry. Topological properties of patterns are represented by segments (barcodes) obtained by filtering by the level of photometric measure. Beginning of barcode encodes level at which topological property appears (connected component and/or “hole”), and its end - level at which the property disappears. Barcodes are conveniently parameterized by coordinates of their ends in rectangular coordinate system “birth” and “death” of topological property. Such representation in for

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Vasantha, D. M. "GEOMETRICAL METHODS IN THERMODYNAMICS." International Journal of Advances in Engineering & Scientific Research 12, no. 2 (2025): 01–10. https://doi.org/10.5281/zenodo.15106462.

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<em>This study jumps right into the unexpected mix of curved-space ideas and the rules of heat and energy—using Riemannian geometry and twisty conformal transformations to shed new light on thermodynamic behavior. It brings together hard, number-driven data from energy systems with softer, more intuitive stretches of geometric insight, generally showing that looking at processes like entropy and energy spread in a curved-space light opens up insights that standard methods might easily overlook. In some cases the geometrical approach yields a refreshed snapshot of thermodynamic states whi

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Varano, Valerio, Stefano Gabriele, Franco Milicchio, Stefan Shlager, Ian Dryden, and Paolo Piras. "Geodesics in the TPS Space." Mathematics 10, no. 9 (2022): 1562. http://dx.doi.org/10.3390/math10091562.

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In shape analysis, the interpolation of shapes’ trajectories is often performed by means of geodesics in an appropriate Riemannian Shape Space. Over the past several decades, different metrics and shape spaces have been proposed, including Kendall shape space, LDDMM based approaches, and elastic contour, among others. Once a Riemannian space is chosen, geodesics and parallel transports can be used to build splines or piecewise geodesics paths. In a recent paper, we introduced a new Riemannian shape space named TPS Space based on the Thin Plate Spline interpolant and characterized by an appropr

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Ahdid, Rachid, Khaddouj Taifi, Mohamed Fakir, Said Safi, and Bouzid Manaut. "Two-Dimensional Face Recognition Methods Comparing with a Riemannian Analysis of Iso-Geodesic Curves." Journal of Electronic Commerce in Organizations 13, no. 3 (2015): 15–35. http://dx.doi.org/10.4018/jeco.2015070102.

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In this paper, the authors performed a comparative study of two-dimensional face recognition methods. This study was based on existing methods (PCA, LDA, 2DPCA, 2DLDA, SVM...) and 2D face surface analysis using a Riemannian geometry. The last system uses the representation of the image at gray level as a 2D surface in a 3D space where the third coordinate represent the intensity values of the pixels. The authors' approach is to represent the human face as a collection of closed curves, called facial curves, and apply tools from the analysis of the shape of curves using the Riemannian geometry.

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Costa, André, Vincent Grandjean, and Maria Michalska. "Global Lipschitz geometry of conic singular sub-manifolds with applications to algebraic sets." Documenta Mathematica 29, no. 6 (2024): 1341–66. http://dx.doi.org/10.4171/dm/975.

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We prove that a connected globally conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: its outer and inner metric space structures are equivalent. Moreover, we show that generic \mathbb{K} -analytic germs as well as generic affine algebraic sets in \mathbb{K}^{n} , where \mathbb{K}=\mathbb{C} or \mathbb{R} , are globally conic singular sub-manifolds. Consequently, a generic \mathbb{K} -analytic germ or a generic algebraic subset of \mathbb{K}^{n} is Lipschitz Normally Embedded.

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Dipierro, Serena, Zu Gao, and Enrico Valdinoci. "Global gradient estimates for nonlinear parabolic operators." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 21. http://dx.doi.org/10.1051/cocv/2021016.

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We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also th

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Angers, Jean-Francois, and Peter T. Kim. "Symmetry and Bayesian Function Estimation1." Calcutta Statistical Association Bulletin 56, no. 1-4 (2005): 57–80. http://dx.doi.org/10.1177/0008068320050504.

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Summary This paper develops Bayesian function estimation on compact Riemannian manifolds. The approach is to combine Bayesian methods along with aspects of spectral geometry associated with the Laplace-Beltrami operator on Riemannian manifolds. Although frequentist nonparametric function estimation in Euclidean space abound, to date, no attempt has been made with respect to Bayesian function estimation on a general Riemannian manifold. The Bayesian approach to function estimation is very natural for manifolds because one can elicit very specific prior information on the possible symmetries in

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Rýparová, Lenka, Irena Hinterleitner, Sergey Stepanov, and Irina Tsyganok. "Infinitesimal Transformations of Riemannian Manifolds—The Geometric Dynamics Point of View." Mathematics 11, no. 5 (2023): 1114. http://dx.doi.org/10.3390/math11051114.

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In the present paper, we study the geometry of infinitesimal conformal, affine, projective, and harmonic transformations of complete Riemannian manifolds using the concepts of geometric dynamics and the methods of the modern version of the Bochner technique.

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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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Noakes, Lyle, and Tomasz Popiel. "Geometry for robot path planning." Robotica 25, no. 6 (2007): 691–701. http://dx.doi.org/10.1017/s0263574707003669.

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SUMMARYThere have been many interesting recent results in the area of geometrical methods for path planning in robotics. So it seems very timely to attempt a description of mathematical developments surrounding very elementary engineering tasks. Even with such limited scope, there is too much to cover in detail. Inevitably, our knowledge and personal preferences have a lot to do with what is emphasised, included, or left out.Part I is introductory, elementary in tone, and important for understanding the need for geometrical methods in path planning. Part II describes the results on geometrical

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Vasilakis, Nikolaos, Christos Chorianopoulos, and Elias N. Zois. "A Riemannian Dichotomizer Approach on Symmetric Positive Definite Manifolds for Offline, Writer-Independent Signature Verification." Applied Sciences 15, no. 13 (2025): 7015. https://doi.org/10.3390/app15137015.

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Automated handwritten signature verification continues to pose significant challenges. A common approach for developing writer-independent signature verifiers involves the use of a dichotomizer, a function that generates a dissimilarity vector with the differences between similar and dissimilar pairs of signature descriptors as components. The Dichotomy Transform was applied within a Euclidean or vector space context, where vectored representations of handwritten signatures were embedded in and conformed to Euclidean geometry. Recent advances in computer vision indicate that image representati

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van der Schaft, Arjan, and Bernhard Maschke. "Geometry of Thermodynamic Processes." Entropy 20, no. 12 (2018): 925. http://dx.doi.org/10.3390/e20120925.

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Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, whi

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Sakai, Hiroyuki, and Hideaki Iiduka. "Hybrid Riemannian conjugate gradient methods with global convergence properties." Computational Optimization and Applications 77, no. 3 (2020): 811–30. http://dx.doi.org/10.1007/s10589-020-00224-9.

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Liu, Xi, Zhengming Ma, and Guo Niu. "Mixed Region Covariance Discriminative Learning for Image Classification on Riemannian Manifolds." Mathematical Problems in Engineering 2019 (February 28, 2019): 1–11. http://dx.doi.org/10.1155/2019/1261398.

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Covariance matrices, known as symmetric positive definite (SPD) matrices, are usually regarded as points lying on Riemannian manifolds. We describe a new covariance descriptor, which could improve the discriminative learning ability of region covariance descriptor by taking into account the mean of feature vectors. Due to the specific geometry of Riemannian manifolds, classical learning methods cannot be directly used on it. In this paper, we propose a subspace projection framework for the classification task on Riemannian manifolds and give the mathematical derivation for it. It is different

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Aazami, Amir Babak, and Charles M. Melby-Thompson. "On the principal Ricci curvatures of a Riemannian 3-manifold." Advances in Geometry 19, no. 2 (2019): 251–62. http://dx.doi.org/10.1515/advgeom-2018-0020.

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Abstract We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a posit

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Noakes, Lyle. "A Global algorithm for geodesics." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 65, no. 1 (1998): 37–50. http://dx.doi.org/10.1017/s1446788700039380.

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AbstractThe problem of finding a george joinning given points x0, x1in a connected complete Riemannian manifold requires much more effort than determining a geodesic from initial data. Boundary value problems of this type are sometimes solved using shooting methods, which work best when good initial guesses are available expectually when x0, x1are nearby. Galerkin methods have their drawbacks too. The situation is much more difficult with general variational problems, which is why we focus on the Riemannian case.Our global algorithm is very simple to implement, and works well in practice, with

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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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Gzyl, H., and F. Nielsen. "Geometry of the probability simplex and its connection to the maximum entropy method." Journal of Applied Mathematics, Statistics and Informatics 16, no. 1 (2020): 25–35. http://dx.doi.org/10.2478/jamsi-2020-0003.

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AbstractThe use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities.Fo

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Tian, Jian-Sheng, Wei Wang, Fei Xue, and Pei-Yong Cong. "Boundary Stabilization of the Wave Equation with Time-Varying and Nonlinear Feedback." Mathematical Problems in Engineering 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/176583.

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We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying and nonlinear term. By the Riemannian geometry methods and a suitable assumption of nonlinearity and the time-varying term, we obtain the uniform decay of the energy of the system.

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Gong, Bei, and Xiaopeng Zhao. "Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/728760.

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We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.

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Kiosak, V., L. Kusik, and V. Isaiev. "Geodesic Ricci-symmetric pseudo-Riemannian spaces." Proceedings of the International Geometry Center 15, no. 2 (2022): 109–19. http://dx.doi.org/10.15673/tmgc.v15i2.2224.

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We introduced special pseudo-Riemannian spaces, called geodesic A-symmetric spaces, into consideration. It is proven that there are no geodesic symmetric spaces and no geodesic Ricci symmetric spaces, which differ from spaces of constant curvature and Einstein spaces respectively. The research is carried out locally, by tensor methods, without any limitations imposed on a metric and a sign.

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