Search Results (326)

Manifold learning is a significant computer vision task used to describe high-dimensional visual data in lower-dimensional manifolds without sacrificing the intrinsic structural properties required for 3D reconstruction. Isomap, Locally Linear Embedding (LLE), Laplacian Eigenmaps, and t-SNE are helpful in data topology preservation but are typically indifferent to the intrinsic differential geometric characteristics of the manifolds, thus leading to deformation of spatial relations and reconstruction accuracy loss. This research proposes an Optimization of Manifold Learning using Differential Geometry Framework (OML-DGF) to overcome the drawbacks of current manifold learning techniques in 3D reconstruction. The framework employs intrinsic geometric properties—like curvature preservation, geodesic coherence, and local–global structure correspondence—to produce structurally correct and topologically consistent low-dimensional embeddings. The model utilizes a Riemannian metric-based neighborhood graph, approximations of geodesic distances with shortest path algorithms, and curvature-sensitive embedding from second-order derivatives in local tangent spaces. A curvature-regularized objective function is derived to steer the embedding toward facilitating improved geometric coherence. Principal Component Analysis (PCA) reduces initial dimensionality and modifies LLE with curvature weighting. Experiments on the ModelNet40 dataset show an impressive improvement in reconstruction quality, with accuracy gains of up to 17% and better structure preservation than traditional methods. These findings confirm the advantage of employing intrinsic geometry as an embedding to improve the accuracy of 3D reconstruction. The suggested approach is computationally light and scalable and can be utilized in real-time contexts such as robotic navigation, medical image diagnosis, digital heritage reconstruction, and augmented/virtual reality systems in which strong 3D modeling is a critical need. Full article

We investigate Yamabe solitons on Riemannian hypersurfaces induced by conformal vector fields in Riemannian and Lorentzian manifolds, with an emphasis on the tangential component. We show that these hypersurfaces are totally umbilical, and when the ambient manifold is Einstein, a rigidity condition emerges connecting the mean and scalar curvatures. Using this, we classify compact Yamabe solitons: each hypersurface is either totally geodesic or an extrinsic sphere. Additionally, we prove the non-existence of trivial Yamabe solitons on oriented hypersurfaces of higher dimension in Einstein manifolds. These results highlight the classification of compact hypersurfaces and rigidity phenomena in the ambient spaces, providing a clear understanding of the geometric structures associated with Yamabe solitons. Full article

On a compact n-dimensional Riemannian manifold without boundary $(M,g)$, it is well-known that the $L^2$-normalized Laplace eigenfunctions with semiclassical parameter h satisfy the universal $L^{\infty }$ growth bound of $O(h^{{\displaystyle \frac{1-n}{2}}})\mathrm{as}h\to 0^{+}.$ In the context of a quantum completely integrable system on M, which consists of n commuting self-adjoint pseudodifferential operators $P_1\left(h\right),\dots ,P_n\left(h\right)$, where $P_1\left(h\right)=-h^2{\mathrm{\Delta }}_g+V\left(x\right)$, Galkowski-Toth showed polynomial improvements over the standard $O(h^{{\displaystyle \frac{1-n}{2}}})$ bounds for typical points. Specifically, in the two-dimensional case, such an improved upper bound is $O(h^{-1/4})$. In this study, we aim to further enhance this bound to $O\left(\right|lnh{|}^{1/2})$ at the points where a strictly monotonic condition is satisfied. Full article

This article investigates fundamental inequalities within a golden-like statistical manifold (GLSM) equipped with a semi-symmetric metric connection (SSMC). We explore key geometric and analytical properties, including curvature relations and inequalities analogous to those in classical information geometry. The interplay between the golden-like structure and the SSMC yields new insights into the underlying differential geometric framework. Our results extend known inequalities in the statistical manifold (SM), providing a foundation for further studies in optimization and divergence theory within this generalized framework. Full article

The product manifold ${\mathbb{S}}^3\times {\mathbb{S}}^3$, which belongs to the homogenous six-dimensional nearly Kähler manifolds, admits two structures, the almost complex structure J and the almost product structure P. The investigation of embeddings of different classes of CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ was started some time ago by investigating three-dimensional CR submanifolds. It resulted that the almost product structure P is very important for the study of CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$, since submanifolds characterized by different actions of the almost product structure on base vector fields often appear as a result of the study of some specific types of CR submanifolds. Therefore, the investigation of four-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ is initiated in this article. The main result is the classification of four-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$, whose almost complex distribution ${\mathcal{D}}_1$ is almost product orthogonal on itself. First, it was proved that such submanifolds have a non-integrable almost complex distribution, and then it was proved that these submanifolds are locally product manifolds of curves and three-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ of the same type, and they were therefore constructed in this way. Full article

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples. Full article

This paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants of submanifolds. The methodology involves deriving generalized Chen-type and $\delta (2,2)$ curvature inequalities using curvature tensor analysis and dual affine connections. A concrete example is provided to verify the theoretical framework. The novelty of this work lies in extending classical curvature inequalities to a newly introduced statistical structure, thereby opening new perspectives in the study of geometric inequalities in information geometry and related mathematical physics contexts. Full article

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature. Full article

The subject of this study is almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds. The considerations are restricted to a special class of these manifolds, namely those of the Sasaki-like type, because of their geometric construction and the explicit expression of their classification tensor by the pair of B-metrics. Here, each of the two B-metrics is considered as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form. The soliton potential is chosen to be a conformal Killing vector field (in particular, concircular or concurrent) and then a generalization of the notion of conformality using contact conformal transformations of B-metrics. The resulting manifolds, equipped with the introduced almost solitons, are geometrically characterized. In the five-dimensional case, an explicit example on a Lie group depending on two real parameters is constructed, and the properties obtained in the theoretical part are confirmed. Full article

A brain–computer interface (BCI) provides a direct communication pathway between the human brain and external devices, enabling users to control them through thought. It records brain signals and classifies them into specific commands for external devices. Among various classifiers used in BCI, those directly classifying covariance matrices using Riemannian geometry find broad applications not only in BCI, but also in diverse fields such as computer vision, natural language processing, domain adaption, and remote sensing. However, the existing Riemannian-based methods exhibit limitations, including time-intensive computations, susceptibility to disturbances, and convergence challenges in scenarios involving high-dimensional matrices. In this paper, we tackle these issues by introducing the Bures–Wasserstein (BW) distance for covariance matrices analysis and demonstrating its advantages in BCI applications. Both theoretical and computational aspects of BW distance are investigated, along with algorithms for Fréchet Mean (or barycenter) estimation using BW distance. Extensive simulations are conducted to evaluate the effectiveness, efficiency, and robustness of the BW distance and barycenter. Additionally, by integrating BW barycenter into the Minimum Distance to Riemannian Mean classifier, we showcase its superior classification performance through evaluations on five real datasets. Full article

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 representations to the Riemannian Symmetric Positive Definite (SPD) manifolds outperform vector space representations. In offline signature verification, both writer-dependent and writer-independent systems have recently begun leveraging Riemannian frameworks in the space of SPD matrices, demonstrating notable success. This work introduces, for the first time in the signature verification literature, a Riemannian dichotomizer employing Riemannian dissimilarity vectors (RDVs). The proposed framework explores a number of local and global (or common pole) topologies, as well as simple serial and parallel fusion strategies for RDVs for constructing robust models. Experiments were conducted on five popular signature datasets of Western and Asian origin, using blind intra- and cross-lingual experimental protocols. The results indicate the discriminative capabilities of the proposed Riemannian dichotomizer framework, which can be compared to other state-of-the-art and computationally demanding architectures. Full article

We present a review of recent developments in cosmological models based on Finsler geometry, as well as geometric extensions of general relativity formulated within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend not only on position but also on an additional internal degree of freedom, typically represented by a vector field at each point of the spacetime manifold. We examine in detail the possibility that Finsler-type geometries can describe the physical properties of the gravitational interaction, as well as the cosmological dynamics. In particular, we present and review the implications of a particular implementation of Finsler geometry, based on the Barthel connection, and of the $(\alpha ,\beta )$ geometries, where $\alpha$ is a Riemannian metric, and $\beta$ is a one-form. For a specific construction of the deviation part $\beta$, in these classes of geometries, the Barthel connection coincides with the Levi–Civita connection of the associated Riemann metric. We review the properties of the gravitational field, and of the cosmological evolution in three types of geometries: the Barthel–Randers geometry, in which the Finsler metric function F is given by $F=\alpha +\beta$, in the Barthel–Kropina geometry, with $F={\alpha }^2/\beta$, and in the conformally transformed Barthel–Kropina geometry, respectively. After a brief presentation of the mathematical foundations of the Finslerian-type modified gravity theories, the generalized Friedmann equations in these geometries are written down by considering that the background Riemannian metric in the Randers and Kropina line elements is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equations are also presented, and they are interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. We investigate the cosmological properties of the Barthel–Randers and Barthel–Kropina cosmological models in detail. In these scenarios, the additional geometric terms arising from the Finslerian structure can be interpreted as an effective geometric dark energy component, capable of generating an effective cosmological constant. Several cosmological solutions—both analytical and numerical—are obtained and compared against observational datasets, including Cosmic Chronometers, Type Ia Supernovae, and Baryon Acoustic Oscillations, using a Markov Chain Monte Carlo (MCMC) analysis. A direct comparison with the standard $\Lambda$CDM model is also carried out. The results indicate that Finslerian cosmological models provide a satisfactory fit to the observational data, suggesting they represent a viable alternative to the standard cosmological model based on general relativity. Full article

This study aims to analyze electrocardiogram (ECG) data for the classification of five cardiac rhythms: sinus bradycardia (SB), sinus rhythm (SR), atrial fibrillation (AFIB), supraventricular tachycardia (SVT), and sinus tachycardia (ST). While SR is considered normal, the other four represent types of cardiac arrhythmias. A range of methods is utilized, including the supervised learning technique K-Nearest Neighbors (KNNs), combined with dimensionality reduction approaches such as Principal Component Analysis (PCA) and Uniform Manifold Approximation and Projection (UMAP), a modern method based in Riemannian topology. Additionally, logistic regression was applied using both maximum likelihood and Bayesian methods, with two distinct prior distributions: an informative normal prior and a non-informative Jeffreys prior. Performance was assessed using evaluation metrics such as positive predictive value (PPV), negative predictive value (NPV), specificity, sensitivity, accuracy, and F1-score. Ultimately, the UMAP-KNN method demonstrated the best overall performance. Full article

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated $\eta$-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article

Selecting input data points in the context of high-dimensional, nonlinear, and complex data in Riemannian space is challenging. While optimal experimental design theory is well-established in Euclidean space, its extension to Riemannian manifolds remains underexplored. Li and Del Castillo recently obtained new theoretical results on D-optimal and G-optimal designs on Riemannian manifolds. This paper follows their framework to investigate A-optimal designs on such manifolds. We prove an equivalence theorem for A-optimality under the manifold regularization model. Based on this result, a sequential algorithm for identifying A-optimal designs on manifold data is developed. Numerical studies using both synthetic and real datasets show the validity of the proposed method. Full article