Manifold Research Articles

From Ricci Soliton to Almost Contact Metric Structures

In this paper, we construct almost contact metric structures on a three-dimensional Riemannian manifold equipped with an almost Ricci soliton. Then, we give the techniques necessary to define the nature of such structures. Concrete examples are given.

Fast and Efficient Drone Path Planning Using Riemannian Manifold in Indoor Environment

This paper introduces an innovative dual-path planning algorithm rooted in a topological three-dimensional Riemannian manifold (T3DRM) to optimize drone navigation in complex environments. It seamlessly integrates strategies for both discrete and continuous obstacles, employing spherical navigation for the former and hyperbolic paths for the latter. Serving as a transformative tool, the T3DRM facilitates efficient path planning by transitioning between discrete and continuous domains. In uncertain environments with unpredictable obstacle positions, our methodology categorizes these positions as discrete or continuous based on their distribution patterns. Discrete obstacles exhibit random distributions, while continuous obstacles display symmetrical patterns with continuity. Leveraging topological metrics, the T3DRM efficiently classifies these patterns for effective path planning. The findings of this research demonstrate the efficiency of path planning based on classified obstacle positions, enabling swift and efficient drone navigation. This research introduces a pioneering application of a T3DRM, accelerating drone navigation in uncertain environments through a dual approach that simultaneously transforms navigation in primal and dual domains. By enabling spherical and hyperbolic navigation concurrently, the T3DRM offers a comprehensive solution to discrete and continuous path planning challenges. The proposed approach can be used for various indoor applications, especially for warehouse management, surveillance and security, navigation in complex structures, indoor farming, site inspection, healthcare facilities, etc.

Immersion and Invariance-Based Linear Tracking and Regulation Controller for Depth Position of an AUV

This paper addresses the tracking control challenge in the diving motion system of a specific class of autonomous underwater vehicles (AUVs) characterized by a torpedo-like shape. A decoupled and reduced-order three degrees-of-freedom linearized diving motion model is employed for depth position control. A control law is synthesized using the immersion and invariance (I&I) technique to achieve the control objectives. The primary aim is to attain tracking by immersing a stable, lower-order target (second-order) dynamic system into a three-dimensional manifold, upon which the closed-loop system evolves. We address the regulation problem as a specialized instance of the tracking problem, with the reference input set as a predetermined known depth that requires regulation. The efficacy of the proposed control law is evaluated through simulation studies involving various scenarios. Robustness tests are conducted to assess the control law’s performance under modeling uncertainties and underwater disturbances. The computer simulation employs an AUV named MAYA, utilizing experimentally validated diving motion parameters. A comparative analysis is performed between the proposed control law and other benchmark controllers to gauge its performance. Additionally, the effectiveness of the proposed control law is confirmed by validating its application to the nonlinear model of the diving motion system.

An Information-Geometric Formulation of Pattern Separation and Evaluation of Existing Indices.

Pattern separation is a computational process by which dissimilar neural patterns are generated from similar input patterns. We present an information-geometric formulation of pattern separation, where a pattern separator is modeled as a family of statistical distributions on a manifold. Such a manifold maps an input (i.e., coordinates) to a probability distribution that generates firing patterns. Pattern separation occurs when small coordinate changes result in large distances between samples from the corresponding distributions. Under this formulation, we implement a two-neuron system whose probability law forms a three-dimensional manifold with mutually orthogonal coordinates representing the neurons' marginal and correlational firing rates. We use this highly controlled system to examine the behavior of spike train similarity indices commonly used in pattern separation research. We find that all indices (except scaling factor) are sensitive to relative differences in marginal firing rates, but no index adequately captures differences in spike trains that result from altering the correlation in activity between the two neurons. That is, existing pattern separation metrics appear (A) sensitive to patterns that are encoded by different neurons but (B) insensitive to patterns that differ only in relative spike timing (e.g., synchrony between neurons in the ensemble).

PARA-SASAKIAN MANIFOLD ADMITTING RICCI-YAMABE SOLITONS WITH QUARTER SYMMETRIC METRIC CONNECTION

In the year 2019, Guler and Crasmareanu [6] conducted an investigation into another geometric flow known as the Ricci-Yamabe map. This map is nothing but a scalar combination of the Ricci and the Yamabe flow [7]. The primary objective of the current paper is to provide a characterization of a Ricci Yamabe soliton on a para-Sasakian manifold [17]. To commence, we prove that a para-Sasakian manifold admits a nearly quasi-Einstein manifold. Moreover, we discuss whether such a manifold is shrinking, expanding, or steady. Subsequently, we generalize these findings to Ricci-Yamabe solitons on para-Sasakian manifolds equipped with a quarter symmetric metric connection. Lastly, we furnish an illustration of a three-dimensional para-Sasakian manifold admitting a Ricci-Yamabe soliton which satisfies our results.

Conformal trajectories in three-dimensional space forms

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds [Formula: see text]. Given a conformal vector field [Formula: see text], a conformal trajectory of [Formula: see text] is a regular curve [Formula: see text] in [Formula: see text] satisfying [Formula: see text], for some fixed nonzero constant [Formula: see text]. In this paper, we study the conformal trajectories in the space forms [Formula: see text], [Formula: see text] and [Formula: see text]. For (non-Killing) conformal vector fields in [Formula: see text] (respectively, in [Formula: see text]), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space [Formula: see text], we obtain the same result for the radial vector field and characterizing all conformal trajectories.

A characterization of homogeneous three-dimensional CR manifolds

We characterize homogeneous three-dimensional CR manifolds, in particular Rossi spheres, as critical points of a certain energy functional that depends on the Webster curvature and torsion of the pseudohermitian structure.

The Translation Surfaces on Statistical Manifolds with Normal Distribution

In this paper, we will investigate translation surfaces on statistical manifolds. Statistical manifolds are mathematical structures that describe the geometric properties of statistical models. We will focus on minimal statistical translation surfaces and then classify statistical translation surfaces of null sectional curvature in three-dimensional hyperbolic statistical manifolds

Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum.

A parallel algorithm for three-dimensional numerical manifold elements generation

Numerical Manifold Method (NMM) has gained widespread application in engineering practice due to its capacity to effectively address both continuous and discontinuous problems in a unified framework. With the advancement of 3D-NMM, there remains a deficiency in ready-made preprocessing tools. Hence, the generation of 3D manifold elements (MEs) is the primary prerequisite for 3D-NMM. In this study, a parallel algorithm to generate the MEs is proposed. Firstly, a novel 3D block identification algorithm is developed to improve the modeling efficiency. Then, 3D joint networks are simulated using the geological statistical results. Thirdly, the mathematical cover system is constructed using uniformly regular hexahedral meshes. Subsequently, the physical domain is cut using the Boolean algorithm to generate the related manifold blocks (MBs) and the Union-Find algorithm is used to determine the relationship of MBs with mathematical patches (MPs) and physical patches (PPs). The proposed program incorporates the OpenMP parallel library for CPU parallelization to enhance its efficiency and the OpenGL library is used to build a real-time visualization program. Finally, the proposed program is validated using several representative examples, and the obtained results demonstrate its efficient capability in generating MEs.

Scenarios for the appearance of strange attractors in a model of three interacting microbubble contrast agents

We study nonlinear dynamics in a model of three interacting encapsulated gas bubbles in a liquid. The model is a system of three coupled nonlinear oscillators with an external periodic force. Such bubbles have numerous applications, for instance, they are used as contrast agents in ultrasound visualization. Certain types of bubbles oscillations may be beneficial or undesirable depending on a given application and, hence, the dependence of the regimes of bubbles oscillations on the control parameters is worth studying. We demonstrate that there is a wide variety of types of dynamics in the model by constructing a chart of dynamical regimes in the control parameters space. Here we focus on hyperchaotic attractors characterized by three positive Lyapunov exponents and strange attractors with one or two positive Lyapunov exponents possessing an additional zero Lyapunov exponent, which have not been observed previously in the context of bubbles oscillations. We also believe that we provide a first example of a hyperchaotic attractor with additional zero Lyapunov exponent. Furthermore, the mechanisms of the onset of these types of attractors are still not well studied. We identify two-parametric regions in the control parameter space where these hyperchaotic and chaotic attractors appear and study one-parametric routes leading to them. We associate the appearance of hyperchaotic attractors with three positive Lyapunov exponents with the inclusion of a periodic orbit with a three-dimensional unstable manifold, while the onset of chaotic oscillations with an additional zero Lyapunov exponent is connected to the partial synchronization of bubbles oscillations. We propose several underlying bifurcation mechanisms that explain the emergence of these regimes. We believe that these bifurcation scenarios are universal and can be observed in other systems of coupled oscillators.

Low speed impact simulation of explosive charge based on three-dimensional numerical manifold method

The present work addresses the challenge of evaluating the low speed impact response of polymer bonded explosive charges with three-dimensional numerical manifold method. Based on the cover-based contact theory, the contact algorithm is implemented by transforming position relationships between reference points a0 and entrance block E(A, B). Meanwhile, based on statistical micro-cracks, a Visco Scram mechanical model for the explosive charge is established. Using the classic Steven test, the influence of charge thickness, diameter, and warhead shape on the mechanical response are analyzed under low speed impact. The simulation results indicate that, increasing the thickness of the charge can effectively avoid internal damage of the explosive caused by projectile extrusion and holder confinement; Increasing the aspect ratio of the charge can alleviate the large deformation; Increasing the shape and size of the warhead can make the charge extrusion shallower, alleviating the concave shear damage of the charge. The simulation results verifies that the developed contact algorithm in this paper can effectively solve 3D impact problems, and provides guidance in the design of charge structures to some extent in the weapon system.

Parametric Expansions of an Algebraic Variety Near Its Singularities II

The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω, which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold Ω near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity.

Enhanced boiling heat transfer via microporous copper surface integration in a manifold microgap

Heat flux dissipation from electronic devices has increased with their miniaturization owing to the increasing performance demands and development of microfabrication technologies. Improving the heat transfer performance is crucial for enhancing heat dissipation. However, this often leads to an increase in pressure drop, which reduces energy efficiency. Microgap heat sink is a promising approach in this regard owing to its geometrical simplicity and facile fabrication while providing heat transfer performance comparable to that of microchannel heat sinks with substantially lower pressure drops. In this study, we develop a manifold microgap heat sink integrated with a porous copper surface that effectively enhances heat transfer while using significantly low pumping power. A three-dimensional liquid routing manifold is used to achieve better flow distribution and alleviate temperature non-uniformity while providing improved heat transfer by enabling jet impingement and mixing of the thermal boundary layer on the microgap surface with minimal additional pressure drop. Moreover, a microscale inverse opal structure is used to facilitate nucleate boiling on the microgap surface, which improves heat transfer while maintaining a relatively low flow resistance. A maximum heat flux of 322.8 W/cm2 was achieved at the mass flux of 472 kg/m2 s, and the corresponding pressure drop and maximum heater temperature were 0.6 kPa and 140 ℃, respectively. The coefficient of performance (COP) achieved herein was significantly higher than those reported in previous relevant studies, indicating that the proposed cooling technique can potentially be used for energy-efficient thermal management in electronics devices.

Investigations on a Riemannian Manifold with a Semi- Symmetric Non-Metric Connection and Gradient Solitons

This article carries out the investigation of a three-dimensional Riemannian manifold N3 endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on N3. It is established that a N3 with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of N3 with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either N3 is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold N3 with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and m-quasi Einstein solitons of gradient type, respectively.

Certain vector fields on f-Kenmotsu manifold with Schouten-van Kampen connection

The present study investigates various characteristics of conformal Ricci solitons with a Schouten-van Kampen connection. Characterizations are obtained when the potential vector field involves a torse-forming vector field. Moreover, applications related to submanifolds are also provided. Lastly, we provided an example of conformal Ricci solitons on a three-dimensional f-Kenmotsu manifold to validate our findings.

On Homeomorphisms of Three-Dimensional Manifolds with Pseudo-Anosov Attractors and Repellers

On Homeomorphisms of Three-Dimensional Manifolds with Pseudo-Anosov Attractors and Repellers

Stability of minimal surfaces in the sub-Riemannian manifold $\widetilde{E(2)}$

In the paper we study smooth oriented surfaces in the universal covering space of the group of orientation-preserving Euclidean plane isometries, which has a three-dimensional sub-Riemannian manifold structure. This structure is constructed as a restriction of the Euclidean metric on the group to some completely non-integrable left invariant distribution. The sub-Riemannian area of a surface is then defined as the integral of the length of its unit normal field projected orthogonally onto this distribution. We calculate the first variation formula of the sub-Riemannian surface area and derive the minimality criterion from it. Here we call a surface minimal if it is a critical point of the sub-Riemannian area functional under normal variations with compact support. We show that the minimality in this case is not equivalent to the vanishing of the sub-Riemannian mean curvature. We then prove that a Euclidean plane is minimal if and only if it is parallel or orthogonal to the $z$-axis (where the $z$-coordinate corresponds to the rotation angle of an isometry). Also we obtain the minimality condition for a graph and give examples of minimal graphs. The examples considered in the paper demonstrate, in particular, that the minimality of a surface in the Riemannian (in this case Euclidean) sense does not imply its sub-Riemannian minimality, and vice versa. Next, we consider the stability of minimal surfaces. For this purpose, we derive the second variation formula of the sub-Riemannian area and show with it that minimal Euclidean planes are stable. We introduce a class of surfaces for which the tangent planes are perpendicular to the planes of the sub-Riemannian structure, and call them vertical surfaces. In particular, for such surfaces the second variation formula is simplified significantly. Then we prove that complete connected vertical minimal surfaces are either Euclidean planes or helicoids and that helicoids are unstable. This implies a following Bernstein type result: a complete connected vertical minimal surface is stable if and only if it is a Euclidean plane orthogonal to the $z$-axis.

Chern-Simons action and the Carrollian Cotton tensors

In three-dimensional pseudo-Riemannian manifolds, the Cotton tensor arises as the variation of the gravitational Chern-Simons action with respect to the metric. It is Weyl-covariant, symmetric, traceless and covariantly conserved. Performing a reduction of the Cotton tensor with respect to Carrollian diffeomorphisms in a suitable frame, one discloses four sets of Cotton Carrollian relatives, which are conformal and obey Carrollian conservation equations. Each set of Carrollian Cotton tensors is alternatively obtained as the variation of a distinct Carroll-Chern-Simons action with respect to the degenerate metric and the clock form of a strong Carroll structure. The four Carroll-Chern-Simons actions emerge in the Carrollian reduction of the original Chern-Simons ascendant. They inherit its anomalous behaviour under diffeomorphisms and Weyl transformations. The extremums of these Carrollian actions are commented and illustrated.

On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes left{ left( mathbb {S}^2times mathbb {S}^1, g_circ -frac{d^2}{c^2}dt^2right) right} _{d,cin mathbb {N}^+text { coprime}}, with g_circ the round metric on mathbb {S}^2 and t the mathbb {S}^1-coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on STmathbb {S}^2cong L(2,1) under the natural projection L(2,1)rightarrow L(2c,1). We extend this computation to Ztimes mathbb {S}^1 for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.