Differential Geometry Research Articles

Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems

In this paper we present several new contributions to the oscillation theory of linear differential equations, in particular of linear Hamiltonian systems, when the traditional Legendre condition is absent. Following our recent work (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 2023), we introduce the multiplicity of a generalized right focal point and derive the corresponding local Sturmian separation theorem. We also examine the relation between the existence of finitely many generalized right focal points, or in the special case the nonexistence of generalized right focal points, with the Legendre condition. As the main tools we use new notions of the minimal multiplicities at a given point and the dual comparative index — an object from matrix analysis or differential geometry (Maslov index theory). Furthermore, we study local limit properties of the dual comparative index and the comparative index and apply them for deriving new oscillation results phrased in terms of the generalized right and left focal points. The investigation of the interplay between generalized right and left focal points leads to conditions characterizing the situation, when in the local Sturmian separation theorem the corresponding multiplicities attain the minimal possible value. This also provides a generalization of the concepts of the right and left proper focal point defined by Kratz (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 2007) to the setting, which does not impose the Legendre condition. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm–Liouville differential equations of arbitrary even order.

A Comprehensive Review of Golden Riemannian Manifolds

In differential geometry, the concept of golden structure represents a compelling area with wide-ranging applications. The exploration of golden Riemannian manifolds was initiated by C. E. Hretcanu and M. Crasmareanu in 2008, following the principles of the golden structure. Subsequently, numerous researchers have contributed significant insights with respect to golden Riemannian manifolds. The purpose of this paper is to provide a comprehensive survey of research on golden Riemannian manifolds conducted over the past decade.

Quantum geometrical properties of topological materials

The momentum space of topological insulators and topological superconductors is equipped with a quantum metric defined from the overlap of neighboring valence band states or quasihole states. We investigate the quantum geometrical properties of these materials within the framework of Dirac models and differential geometry. Their momentum space is found to be always a maximally symmetric space with a constant Ricci scalar, and the vacuum Einstein equation is satisfied in 3D with a finite cosmological constant. For linear Dirac models, several geometrical properties are found to be independent of the band gap, including a peculiar straight line geodesic, constant volume of the curved momentum space, and the exponential decay form of the nonlocal topological marker, indicating the peculiar yet universal quantum geometrical properties of these models.

A Heuristic Philosophical Discourse on Various Applications of Abstract Differential Geometry in Quantum Gravity Research

A Heuristic Philosophical Discourse on Various Applications of Abstract Differential Geometry in Quantum Gravity Research

On the differential geometry of smooth ruled surfaces in 4‐space

On the differential geometry of smooth ruled surfaces in 4‐space

A Geometric View of Seismic Wavefields: Implications for Imaging Dense Clusters of Events

Summary Imaging dense clusters of seismicity is crucial to many problems in seismology: to delineate complex systems of faults, provide constraints on the causes of volcanic and cryogenic swarms, and to shed light on possible means to prevent damaging induced seismicity in mining, geothermal and oil and gas extraction activities. Current imaging methods rely upon high-resolution relative location techniques, commonly requiring arrival-time picks for seismic phases. This paper examines an alternative approach, based upon concepts drawn from differential geometry, that images directly from waveform data. It relies upon the common assumption of spatial continuity of seismic wavefield observations, which implies that a differentiable map exists between the source region to be imaged and waveform observations considered as elements of a vector space. The map creates an image of event clusters on a Riemannian manifold embedded in that vector space. The image can be visualized by projecting the observations into a tangent space of the manifold and is a distorted rendering of cluster geometry. However, the distortion can be predicted and removed if a model for wavefield propagation is available. This visualization approach is applicable to clusters of uniform events with highly similar waveforms, such as are commonly acquired with correlation detectors or other pattern matching techniques. To assess its performance, it is applied to the closely-related reciprocal problem of imaging the (known) geometry of an array from observations by the array of several regional events. Differences between the original problem and its reciprocal analog are noted and controlled for in the analysis. Chief among the differences is the necessity for aligning the waveforms in the original problem, which, to maintain consistency with the original problem, is solved in the reciprocal problem by a generalization of the VanDecar-Crosson algorithm. The VanDecar-Crosson algorithm exhibits a bias, shown through an analysis of the situation when the observed wavefields are adequately modeled as plane waves. In that circumstance, the bias can be predicted and removed. In a test using a portion of a large-N array, this imaging approach is shown to successfully reconstruct the array geometry. The method is applicable directly to infinitesimal array apertures, but is extended to a larger aperture by partitioning the image into local, effectively infinitesimal overlapping subsets. These are inverted, then assembled into a global picture of the array geometry using constraints provided by the overlapped regions. Although demonstrated in a reciprocal array context, the method appears viable for imaging clusters of events with highly similar source mechanisms and time histories.

Some remarks on the history of Ricci’s absolute differential calculus

The article offers a general account of the genesis of the absolute differential calculus (ADC), paying special attention to its links with the history of differential geometry. In relatively recent times, several historians have described the development of the ADC as a direct outgrowth either of the theory of algebraic and differential invariants or as a product of analytical investigations, thus minimizing the role of Riemann’s geometry in the process leading to its discovery. Our principal aim consists in challenging this historiographical tenet and analyzing the intimate connection between the development of Riemannian geometry and the birth of tensor calculus.

Deep learning as Ricci flow.

Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton's Ricci flow-a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of 'global Ricci network flow' that can be used to assess a DNN's ability to disentangle complex data geometries to solve classification problems. By training more than 1500 DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

Impact of Black Swan Events on Ethereum blockchain ERC20 token transaction networks

The Ethereum blockchain and its ERC20 token standard have revolutionized the landscape of digital assets and decentralized applications. ERC20 tokens are programmable and interoperable tokens, enabling various applications and token economies. Transaction graphs, representing the flow of the value between wallets within the Ethereum network, have played a crucial role in understanding the system’s dynamics, such as token transfers and the behavior of traders. Here, we explore the evolution of daily transaction graphs of ERC20 token transactions, which sheds light on the trader’s behavior during the Black Swan Events – 2018 crypto crash and the COVID-19 pandemic. By using the tools from network science and differential geometry, we analyze 0.98 billion of ERC20 token transaction data from November 2015 to January 2023. Our analysis reveals an increase in diverse interaction among the traders and a greater adoption of ERC20 tokens in a maturing Ethereum ERC20 financial ecosystem after the Crypto Crash 2018 and the COVID-19 pandemic. Before the crash and the COVID-19 pandemic, most traders interacted with other traders in an isolated or restricted manner, with each trader focusing solely on either buying or selling activities. However, after the crash and during the pandemic, most traders diversely interacted among themselves by participating in both buying and selling activities. In addition, we observe no significant negative impact of the COVID-19 pandemic on user behavior in the financial ecosystem.

Applications of Cantor Set to Fractal Geometry

Fractal geometry is a subfield of mathematics that allows us to explain many of the complexities in nature. Considering this remarkable feature of fractal geometry, this study examines the Cantor set, which is one of the most basic examples of fractal geometry. First of all for the Cantor set, which is one of the basic example and important structure of it. Firstly, generalization of Cantor set in on ${\mathbb{R}}$, ${\mathbb{R}}^2$ and ${\mathbb{R}^3}$ are taken into consideration. Then the given structures are examined over curve and surface theory. This approach enables to given a relationship between fractal geometry and differential geometry. Finally, some examples are established.

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion tensor imaging (DTI) is a powerful neuroimaging technique that provides valuable insights into the microstructure and connectivity of the brain. By measuring the diffusion of water molecules along neuronal fibers, DTI allows the visualization and study of intricate networks of neural pathways.DTI is a noise-sensitive method, where a low signal-to-noise ratio (SNR) results in significant errors in the estimated tensor field. Tensor field regularization is an effective solution for noise reduction.Diffusion tensors are represented by symmetric positive-definite (SPD) matrices. The space of SPD matrices may be viewed as a Riemannian manifold after defining a suitable metric on its tangent bundle. The Log-Cholesky metric is a recently developed concept with advantages over previously defined Riemannian metrics, such as the affine-invariant and Log-Euclidean metrics. The utility of the Log-Cholesky metric for tensor field regularization and noise reduction has not been investigated in detail.This manuscript provides a quantitative investigation of the impact of Log-Cholesky filtering on noise reduction in DTI. It also provides sufficient details of the linear algebra and abstract differential geometry concepts necessary to implement this technique as a simple and effective solution to filtering diffusion tensor fields.

Optimized continuous dynamical decoupling via differential geometry and machine learning

Optimized continuous dynamical decoupling via differential geometry and machine learning

Engineering flexible machine learning systems by traversing functionally invariant paths

Contemporary machine learning algorithms train artificial neural networks by setting network weights to a single optimized configuration through gradient descent on task-specific training data. The resulting networks can achieve human-level performance on natural language processing, image analysis and agent-based tasks, but lack the flexibility and robustness characteristic of human intelligence. Here we introduce a differential geometry framework—functionally invariant paths—that provides flexible and continuous adaptation of trained neural networks so that secondary tasks can be achieved beyond the main machine learning goal, including increased network sparsification and adversarial robustness. We formulate the weight space of a neural network as a curved Riemannian manifold equipped with a metric tensor whose spectrum defines low-rank subspaces in weight space that accommodate network adaptation without loss of prior knowledge. We formalize adaptation as movement along a geodesic path in weight space while searching for networks that accommodate secondary objectives. With modest computational resources, the functionally invariant path algorithm achieves performance comparable with or exceeding state-of-the-art methods including low-rank adaptation on continual learning, sparsification and adversarial robustness tasks for large language models (bidirectional encoder representations from transformers), vision transformers (ViT and DeIT) and convolutional neural networks.

Noncommutative differential geometry on crossed product algebras

We provide a differential structure on arbitrary cleft extensions B:=AcoH⊆A for an H-comodule algebra A. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra B#σH from the data of a bicovariant calculus on the structure Hopf algebra H and a calculus on the base algebra B, which is compatible with the 2-cocycle and measure of the crossed product. The result is a quantum principal bundle with canonical strong connection and we describe the induced bimodule covariant derivatives on associated bundles of the crossed product. All results specialize to trivial extensions and smash product algebras B#H and we give a characterization of the smash product calculus in terms of the differentials of the cleaving map j:H→A and the inclusion B↪A. The construction is exemplified for pointed Hopf algebras. In particular, the case of Radford Hopf algebras H(r,n,q) is spelled out in detail.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

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.

Simplified discrete model for axisymmetric dielectric elastomer membranes with robotic applications

Soft robots utilizing inflatable dielectric membranes can realize intricate functionalities through the application of non-mechanical fields. However, given the current limitations in simulations, including low computational efficiency and difficulty in dealing with complex external interactions, the design and control of such soft robots often require trial and error. Thus, a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical model is developed for analyzing the highly nonlinear mechanics in axisymmetric inflatable dielectric membranes. The model captures the intricate dynamics of these membranes under both inflationary pressure and electrical stimulation. Comprehensive validations using hyperelastic benchmarks demonstrate the model’s accuracy and reliability. Additionally, the focus on the electro-mechanical coupling elucidates critical insights into the membrane’s behavior under varying internal pressures and electrical loads. The research further translates these findings into innovative soft robotic applications, including a spherical soft actuator, a soft circular fluid pump, and a soft toroidal gripper, where the snap-through of electroelastic membrane plays a crucial role. Our analyses reveal that the functional ranges of soft robots are amplified by the snap-through of an electroelastic membrane upon electrical stimuli. This study underscores the potential of DDG-based simulations to advance the understanding of the nonlinear mechanics of electroelastic membranes and guide the design of electroelastic actuators in soft robotics applications.

Information geometry and parameter sensitivity of non-Hermitian Hamiltonians

Information geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the application of information geometry in the realm of non-Hermitian quantum systems, focusing on the Fisher-Rao metric as a measure of parameter sensitivity. We approximate the Lindblad master equation for non-Hermitian Hamiltonians to analyze the temporal evolution of the quantum geometric metric. Utilizing the quantum spin Ising model with an imaginary magnetic field as an exemplar, we investigate the energy spectrum and geometric metric evolution within PT-symmetry Hamiltonians. We demonstrate that the detrimental effects of dissipation can be counteracted by introducing a control Hamiltonian, leading to improved accuracy in parameter estimation. Our work provides insights into the role of quantum control in mitigating dissipative impacts and enhancing the precision of quantum metrological tasks.

Universal mechanical modeling of fiber bundle under torsion: An experimental and numerical study

AbstractIn this paper, a universal mechanical model for the torsional behavior of fiber bundle was established, grounded in the principles of spatial geometry and continuum mechanics, incorporating differential geometry formulas. The model was validated through self‐designed experimental equipment and finite element simulations using Ansys Workbench, showing strong agreement between predicted torsional fracture behavior and experimental results. The study further examined the contributions of fiber bending, torsion, and tension to the overall torque, revealing that torsion plays the dominant role. Additionally, the effect of filament count on the torsional performance of fiber bundles was analyzed, demonstrating that while the number of fibers significantly influences stress distribution, its impact on overall torque is minimal.Highlights A universal mechanical model for fiber bundle torsional is proposed. High consistency between simulation and experimental results. Examined filament count effect on torsional strength. More filaments reduce stress concentration, improving performance.

Study on contact characteristics of double-disk arc-contact lapping for cylindrical rollers based on influence coefficient method

The double-disk arc-contact lapping (DDACL) method is a novel for precision machining of the rolling surface of cylindrical roller. The contact characteristics of DDACL process can affect the lapping quality of the bearing roller. However, there is a limited amount of study on the contact theory of DDACL. Therefore, the objective of this research is to create a semi-analytical modeling approach that can effectively and accurately forecast the contact characteristics of DDACL. In the initial step, the spatial coordinate transformation was employed to establish accurate mathematical models that represents the surfaces of the cylindrical roller and arc groove. Subsequently, the potential contact area was determined and subdivided into discrete elements based on the differential geometry principles. The accurate flexibility of the potential contact area is achieved by utilizing the interpolation theory and considering both the global and local deformations of the roller and groove. By employing the influence coefficient method and the force-deformation coordination relation, the deformation and contact force can be calculated. Finally, the accuracy and efficiency of this calculation is confirmed through a comparison between the finite element method (FEM) and the proposed method.

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.