Search Results (80)

This article presents results on four-dimensional CR submanifolds of the homogeneous nearly Kähler product manifold ${\mathbb{S}}^3\times {\mathbb{S}}^3$. In the research of CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$, the most important role in the classification is played by the action of the almost product structure P. Here, the investigation of the action of the almost product structure on the tangent bundle of four-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ is extended. Classifications are obtained for certain types of submanifolds whose almost complex distribution is almost product invariant, such as the class characterized by a special type of angle functions, as well as those whose tangent bundle is almost product invariant. The previously mentioned classes of four-dimensional CR submanifolds lead to the classification of those submanifolds that are locally usual product manifolds of Lagrangian submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ and curves. Full article

The notion of 2-conformal vector fields on pseudo-Riemannian manifolds, which arose naturally in the study of hyperbolic solitons, is introduced by Fasihi-Ramandi, De, and Shamkhali. In this paper, we study invariant 2-conformal vector fields on four-dimensional non-reductive pseudo-Riemannian homogeneous manifolds $G/H$. Consequently, the complete classification of such vector fields is achieved, together with the necessary and sufficient conditions for their existence. The results are then applied to Lorentzian and neutral signatures, where 2-conformal vector fields provide an effective criterion for detecting 2-conformal equivalences in geometries with limited algebraic symmetries. Full article

Classical homogenization theory treats critical exponents as universal quantities depending only on spatial dimension, but recent evidence shows that this assumption fails for continuum composites once the mechanism of randomness generation is taken into account. We synthesize three complementary frameworks—structural approximation, structural sums, and self-similar renormalization—to develop a unified geometric theory of criticality in random composites. Dilute-regime expansions for the effective conductivity and shear modulus are expressed in terms of structural sums whose ensemble statistics depend sensitively on the randomness protocol. To bridge the dilute and critical regimes, we employ self-similar factor approximants, iterated-root approximants, additive approximants, and renormalization schemes based on minimal-difference and minimal-sensitivity conditions, combined with Borel summation. For maximally disordered protocols $\mathcal{P}\left(\tau \right)$, the conductivity index s and the elasticity index S fall within comparable numerical ranges, indicating a shared geometric origin and spectral response to the continuous breaking of translational symmetry. A regular periodic arrangement of inclusions ($\tau =0$) possesses full discrete translational symmetry; as a stochastic protocol $\mathcal{P}\left(\tau \right)$ is applied ($\tau$ increases), this symmetry is gradually degraded until statistical chaos is reached. For instance, the parameter $\tau$ can be considered as a time of stirring. During this evolution, the system traverses a continuous spectrum of critical indices, $s=s\left[\mathcal{P}\right(\tau \left)\right],$ which encodes the geometric and topological memory of the initial ordered state. It is established that the classical “universality” of percolation corresponds to a fixed point $\tau$ within a broader manifold of protocol-dependent critical behaviors. The framework developed here provides a coherent basis for inverse design, diagnostics, and classification of random composites by their disorder history, offering a geometric alternative to the universality paradigm. Full article

The burgeoning scale of Pre-trained Large Models (PLMs) has intensified the demand for efficient inference without compromising performance, while existing large model collaborative frameworks have shown promise, they often suffer from functional redundancy among experts and limited robustness in complex cross-domain scenarios. In this paper, we propose Tri-gate Routing for Inference via Decoupled Efficient Network Technologies (TRIDENT), a highly efficient and robust heterogeneous collaborative inference framework. TRIDENT leverages the complementary inductive biases of MLP (for statistical patterns) and KAN (for symbolic logic) to maximize reasoning potential with minimal parametric overhead. To address feature homogenization in traditional distillation, we introduce Orthogonal Feature Decoupling Distillation, utilizing an orthogonality loss ${\mathcal{L}}_{orth}$ for functional decoupling and a reconstruction loss ${\mathcal{L}}_{recon}$ to anchor decoupled features to the PLM knowledge manifold. During inference, a Dual-Threshold Arbiter effectively detects expert hallucinations by integrating individual confidence ${\tau }_{con}$ and heterogeneous consistency ${\tau }_{agree}$. Extensive experiments on CIFAR-100-LT, XNLI, and GSM8K demonstrate that TRIDENT significantly reduces the Invocation Rate (IR) of PLMs while maintaining high accuracy. Our findings reveal a distinct Pareto optimal balance and validate the spontaneous division of labor between heterogeneous experts. By transcending the limitations of single-architecture systems, TRIDENT provides a robust and interpretable pathway for efficient collaborative intelligence. Full article

This manuscript presents a comprehensive taxonomy of Riemann solitons within the framework of a spacetime manifold endowed with a metric exhibiting both spatial homogeneity and rotational characteristics. Furthermore, we undertake an analysis to determine the geometric nature of these solitons by establishing their correspondence to Killing vector fields, Ricci collineation vector fields, and gradient vector fields. Full article

We present a comprehensive general relativistic analysis of the Lemaître–Tolman–Bondi (LTB) metric, incorporating a cosmological constant $\mathrm{\Lambda }$ and a central pointlike mass $M_{\mathrm{d}}$ at the geometric origin. Within this framework, $M_{\mathrm{d}}$ is identified as the material source of dark matter in cosmology, yielding a scale-dependent total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ characterized by an $L^{-3}$ decay of its dark component ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$. We demonstrate that the Hubble and $S_8$ tensions are not independent anomalies but interconnected consequences of spacetime inhomogeneity. These discrepancies arise from a combination of physical and methodological factors: the probing of radial gradients at different characteristic scales and the subsequent interpretation of these data through a global FLRW template. This approach, compounded by the practice of isotropic sky averaging, masks the underlying LTB geometry and converts the physical variation of the manifold into the observed cosmological tensions. Our framework provides a self-consistent geometric explanation for current anomalies while preserving the Copernican principle, identifying the crisis in cosmology as arising from the application of homogeneous models to a manifold characterized by radial gradients and scale-dependent dynamics, where the observer and probes reside within the same inhomogeneous regime. Full article

Deep learning (DL), a hierarchical feature extraction method, has garnered increasing attention in the remote sensing community. Recently, self-supervised learning (SSL) methods in DL have gained wide recognition due to their ability to mitigate the dependence on both the quantity and quality of samples. This advantage is particularly significant when dealing with limited labeled samples in hyperspectral images (HSIs). However, conventional SSL methods face two main challenges. They struggle to construct self-supervised signals based on the unique characteristics of HSI. Moreover, they fail to design network optimization strategies that leverage the intrinsic manifold geometry within HSI. To tackle these issues, we propose a novel self-supervised learning method termed Manifold Geometry-Leveraged Self-supervised Learning (MSSL) for HSI classification. The approach employs a two-stage training strategy. In the initial pre-training stage, it develops self-supervised signals that exploit spatial homogeneity and spectral coherence properties of HSI. Furthermore, it introduces a manifold geometry-guided loss function that enhances feature discrimination by increasing intra-class compactness and inter-class separation. The second stage is a fine-tuning phase utilizing a small set of labeled samples. This stage optimizes the pre-trained model, enabling effective feature extraction from hyperspectral data for classification tasks. Experiments conducted on real-world HSI datasets demonstrate that MSSL achieves superior classification performance compared to several relevant state-of-the-art methods. Full article

In Kähler geometry, Calabi extremal metrics serves as a class of more available special metrics than Kähler metrics with constant scalar curvatures, as a generalization of Kähler Einstein metrics. In recent years, Maxwell–Einstein metrics (or conformally Kähler Einstein–Maxwell metrics) appeared as another alternative choice for Calabi extremal metrics. It turns out that some similar metrics defined by Futaki and Ono have similar roles in the Kähler geometry. In this paper, we prove that for some completions of certain line bundles, there is at least one k-generalized Maxwell–Einstein metric defined by Futaki and Ono conformally related to a metric in any given Kähler class for any integer $3\le k\le 13$. Full article

We investigate the longstanding question of whether compact Levi-flat hypersurfaces exist in the complex projective plane ${\mathbb{CP}}^2$. While the nonexistence of closed real-analytic Levi-flat hypersurfaces in ${\mathbb{CP}}^n$ for $n>2$ is well known, the case $n=2$ remains open. By combining techniques from the classification of homogeneous CR-manifolds with projective foliation geometry, we prove that no homogeneous Levi-flat hypersurfaces exist in ${\mathbb{CP}}^2$, thus partially resolving the problem under natural symmetry assumptions. Full article

In recommender systems research, the data sparsity problem has driven the development of hybrid recommendation algorithms integrating multimodal information and the application of graph neural networks (GNNs). However, conventional GNNs relying on homogeneous Euclidean embeddings fail to effectively model the non-Euclidean geometric manifold structures prevalent in real-world scenarios, consequently constraining the representation capacity for heterogeneous interaction patterns and compromising recommendation accuracy. As a consequence, the representation capability for heterogeneous interaction patterns is restricted, thereby affecting the overall representational power and recommendation accuracy of the models. In this paper, we propose a hyperbolic graph neural network model with contrastive learning for rating–review recommendation, implementing a dual-graph construction strategy. First, it constructs a review-aware graph to integrate rich semantic information from reviews, thus enhancing the recommendation system’s context awareness. Second, it builds a user–item interaction graph to capture user preferences and item characteristics. The hyperbolic graph neural network architecture enables joint learning of high-order features from these two graphs, effectively avoiding the embedding distortion problem commonly associated with high-order feature learning. Furthermore, through contrastive learning in hyperbolic space, the model effectively leverages review information and user–item interaction data to enhance recommendation system performance. Experimental results demonstrate that the proposed algorithm achieves excellent performance on multiple real-world datasets, significantly improving recommendation accuracy. 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 article presents a novel mathematical formalism for advanced manifold–metric pairs, enhancing the frameworks of geometry and topology. We construct various D-dimensional manifolds and their associated metric spaces using functional methods, with a focus on integrating concepts from mathematical physics, field theory, topology, algebra, probability, and statistics. Our methodology employs rigorous mathematical construction proofs and logical foundations to develop generalized manifold–metric pairs, including homogeneous and isotropic expanding manifolds, as well as probabilistic and entropic variants. Key results include the establishment of metrizability for topological manifolds via the Urysohn Metrization Theorem, the formulation of higher-rank tensor metrics, and the exploration of complex and quaternionic codomains with applications to cosmological models like the expanding spacetime. By combining spacetime generalized sets with information-theoretic and probabilistic approaches, we achieve a unified framework that advances the understanding of manifold–metric interactions and their physical implications. Full article

In this paper, we consider six homogeneous manifolds $GL(n,\mathbb{R})/O(n,\mathbb{R})$, $SL(n,\mathbb{R})/SO(n,\mathbb{R})$, $Sp(2n,\mathbb{R})/U\left(n\right),$$(GL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/O(n,\mathbb{R})$, $(SL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/SO(n,\mathbb{R}),$$(Sp(2n,\mathbb{R})⋉H_{\mathbb{R}}^{(n,m)})/(U\left(n\right)\times S(m,\mathbb{R})).$ They are homogeneous manifolds which are important geometrically and number theoretically. These first three spaces are well-known symmetric spaces and the other three are not symmetric spaces. It is well known that the algebra of invariant differential operators on a symmetric space is commutative. The algebras of invariant differential operators on these three non-symmetric spaces are not commutative and have complicated generators. We discuss invariant differential operators on these non-symmetric spaces and provide natural but difficult problems about invariant theory. Full article

We present a study of relic gravitational waves based on a foliated gauge field theory defined over a spacetime endowed with a noncommutative algebraic–geometric structure. As an ontological extension of general relativity—concerning manifolds, metrics, and fiber bundles—the conventional space and time coordinates, typically treated as classical numbers, are replaced by complementary quantum dual fields. Within this framework, consistent with the Bekenstein criterion and the Hawking–Hertog multiverse conception, singularities merge into a helix-like cosmic scale factor that encodes the topological transition between the contraction and expansion phases of the universe analytically continued into the complex plane. This scale factor captures the essence of an intricate topological quantum-leap transition between two phases of the branching universe: a contraction phase preceding the now-surpassed conventional concept of a primordial singularity and a subsequent expansion phase, whose transition region is characterized by a Riemannian topological foliated structure. The present linearized formulation, based on a slight gravitational field perturbation, also reveals a high sensitivity of relic gravitational wave amplitudes to the primordial matter and energy content during the universe’s phase transition. It further predicts stochastic homogeneous distributions of gravitational wave intensities arising from the interplay of short- and long-spacetime effects within the non-commutative algebraic framework. These results align with the anticipated future observations of relic gravitational waves, expected to pervade the universe as a stochastic, homogeneous background. Full article