Search Results (68)

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

This editorial presents 24 research articles published in the Special Issue entitled Geometry of Manifolds and Applications of the MDPI Mathematics journal, which covers a wide range of topics from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in many branches of theoretical and applied mathematical studies, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as complex space forms, metallic Riemannian space forms, Hessian manifolds of constant Hessian curvature; optimal inequalities for submanifolds, such as generalized Wintgen inequality, inequalities involving $\delta$-invariants; homogeneous spaces and Poisson–Lie groups; the geometry of biharmonic maps; solitons (Ricci solitons, Yamabe solitons, Einstein solitons) in different geometries such as contact and paracontact geometry, complex and metallic Riemannian geometry, statistical and Weyl geometry; perfect fluid spacetimes [...] Full article

This paper is concerned with a class of fractional p-Laplacian systems with critical homogeneous nonlinearities. Under proper conditions, the existence and multiplicity results of nontrivial solutions are obtained by variational methods. To some extent, our results improve and supplement some existing relevant results. Full article

In this paper, we aim to solve the issue of pattern formation mechanisms in a spatiotemporally discrete activator–inhibitor model that incorporates self- and cross-diffusions. We seek to identify the conditions that lead to the emergence of complex patterns and to elucidate the principles governing the dynamic behaviors that result in these patterns. We first construct a corresponding coupled map lattice (CML) model based on the continuous activator–inhibitor reaction–diffusion system. In the reaction stage, we examine the existence, uniqueness, and stability of the homogeneous stationary state and specify the parametric conditions for realizing these properties. Furthermore, by applying the center manifold theorem, we perform a flip bifurcation analysis and confirm that the model is capable of undergoing flip bifurcation. In the diffusion stage, we focus on the analysis of Turing bifurcation and determine the parameter conditions for the emergence of Turing instability. Through numerical simulations, we validate and explain the results of our theoretical analysis. Our study reveals various Turing instability mechanisms by coupling Turing and flip bifurcations, which include pure-self-diffusion-Turing instability, pure-cross-diffusion-Turing instability, flip-self-diffusion-Turing instability, flip-cross-diffusion-Turing instability, and chaos-self-diffusion-Turing instability mechanisms. Under different mechanisms, we illustrate the corresponding Turing patterns and discover a rich variety of pattern types such as labyrinthine, mosaic, alternating mosaic, colorful mottled grid patterns with winding and twisted bands, and patterns with dense patches and twisted bands nested together. Our research provides a theoretical framework and numerical support for understanding the complex dynamical behaviors and pattern formations in activator–inhibitor models with self- and cross-diffusions. Full article

Let $\lambda \left(t\right)$ be the first eigenvalue of the operator $-∆+a{R}^b$ on locally three-dimensional homogeneous manifolds along the backward Ricci flow, where $a,b$ are real constants and R is the scalar curvature. In this paper, we study the properties of $\lambda \left(t\right)$ on Bianchi classes. We begin by deriving an evolution equation for the quantity $\lambda \left(t\right)$ on three-dimensional homogeneous manifolds in the context of the backward Ricci flow. Utilizing this equation, we subsequently establish a monotonic quantity that is contingent upon $\lambda \left(t\right)$. Additionally, we present both upper and lower bounds for $\lambda \left(t\right)$ within the framework of Bianchi classes. Full article

The outlet flow rates and changes in behaviors of five outlet ports where water and air–water (two-phase) mixtures pass horizontally in a manifold pipe system were investigated experimentally. The effects of different air-flow rates, vacuumed from the atmosphere with a Venturi device in the system, on the outlet flow rates and diameters of the manifold port outlets were compared by measuring the outlet jet lengths. The system performance provided homogeneity of manifold port outlet flows and was tested. As a result, it was observed that homogeneous jet lengths were obtained in both single and two-phase low main manifold flows and equal outlet port diameters. When the main manifold flow rate V is 1.5–2 m/s, the system is stable and produces high jet lengths. The manifold pipe systems used in the experimental setup provide suitable working conditions for d/D = 0.433. The system does not show a smooth flow pattern with Venturi devices for d/D < 0.433. The low flow rates in this study’s tests are key. They are vital for designing micro irrigation systems. This depends on the critical d/D ratio of the system. Full article

This article offers a fresh exegesis of Heidegger’s philosophy of art, focusing on his conceptualization of artwork as the reproduction of the thing’s general essence. Grounding the analysis in Heidegger’s revisit of Kant’s Transcendental Aesthetic, this study explores Heidegger’s interpretation of a thing as a “composed homogeneity” that reveals inherent determinations of temporality and spatiality in the self-presence of beings as a phenomenon grasped within finite human cognition. This is inextricably linked to the ecstatic temporality of Dasein, elucidating a cyclical human–thing dynamic integral to Heidegger’s ontology. Going deeper, I draw parallels between Kant’s “supersensible” realm and Heidegger’s “earth”, revealing a teleological (ethical) design manifested in art that captures the dual essence of Nature—using Kantian terminology, its purposiveness and contrapurposiveness—intersecting with Heidegger’s notion of the counter-essence of ἀλήθεια in relation to freedom. Finally, I show how the manifold aesthetic metamorphoses of this existential scheme within the existentiell ordinariness through nonradiant φαίνεσθαι, such as equipmentality, emerge as the everyday incarnation of this design. Full article

We embed an object with a singular horizon structure, reminiscent of (but fundamentally different from, except in a limiting case) a black hole event horizon, in an expanding, spherically symmetric, homogeneous, Universe that has a positive cosmological constant. Conformal representation is discussed. There is a temperature/pressure singularity and a corresponding scalar curvature singularity at the horizon. The expanding singular horizon ultimately bounds the entire spacetime manifold. It is is preceded by an expanding light front, which separates the spacetime affected by the singularity from that which is not yet affected. An appropriately located observer in front of the light front can have a Hubble–Lemaître constant that is consistent with that currently observed. Full article

In relativity, the Newtonian concepts of velocity and acceleration are observer-dependent quantities that vary with the chosen frame of reference. It is well established that in the comoving frame, cosmic expansion is currently accelerating; however, in the rest frame, this expansion is actually decelerating. In this paper, we explore the implications of this distinction. The traditional measure of cosmic acceleration, denoted by q, is derived from the comoving frame and describes the acceleration of the scale factor a for a 3D space-like homogeneous sphere. We introduce a new parameter $q_E$ representing the acceleration experienced between observers within the light cone. By comparing $q_E$ to the traditional q using observational data from Type Ia supernovae (SN) and the radial clustering of galaxies and quasars (BAO)—including the latest results from DESI2024—our analysis demonstrates that $q_E$ aligns more closely with these data. The core argument of the paper is that $\Lambda$—regardless of its origin—creates an event horizon that divides the manifold into two causally disconnected regions analogous to conditions inside a black hole’s interior, thereby allowing for a rest-frame perspective $q_E$ in which cosmic expansion appears to be decelerating and the horizon acts like a friction term. Such a horizon suggests that the universe cannot maintain homogeneity outside. The observed cosmological constant $\Lambda$ can then be interpreted not as a driver of new dark energy or a modification of gravity but as a boundary term exerting an attractive force, akin to a rubber band, resisting further expansion and preventing event horizon crossings. This interpretation calls for a reconsideration of current cosmological models and the assumptions underlying them. Full article

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification $G^{\mathbf{C}}$ of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. Full article

The simulation of engine combustion processes, such as autoignition, an important process in the co-optimization of fuel-engine design, can be computationally expensive due to the large number of thermo-chemical scalars needed to describe the full chemical system. Yet, the inherent correlations between the different chemical species during oxidation can significantly reduce the complexity of representing this system. One strategy is to select a subset of representative species that accurately captures the combustion process at a fraction of the computational cost of the full system. In this study, we compare the performance of four different techniques to select these species. They include the two-step principal component analysis (PCA) approach, directed relation graphs (DRGs), the global pathway selection (GPS) approach, and the manifold-informed species selection method. A parametric study of the representative species selection is carried out on data from the simulation of homogeneous and perfectly stirred reactors by investigating seven cumulative variances and 47 different cut-off percentages for the two-step PCA, and 65 and 51 thresholds for the DRGs and GPS, respectively. Results show that these selection methods capture key important species that can accurately describe the chemical system and track each stage of oxidation. The two-step PCA is sensitive to the cumulative variance, and DRGs and GPS are sensitive to the choice of target variables. By selecting key representative species and reducing the number of thermo-chemical scalars, these three methods can be used to develop computationally efficient hybrid chemistry schemes. Full article

This study explores the potentiality of low/zero carbon fuels such as methanol, methane and hydrogen for motor applications to pursue the goal of energy security and environmental sustainability. An experimental investigation was performed on a spark ignition engine equipped with both a port fuel and a direct injection system. Liquid fuels were injected into the intake manifold to benefit from a homogeneous charge formation. Gaseous fuels were injected in direct mode to enhance the efficiency and prevent abnormal combustion. Tests were realized at a fixed indicated mean effective pressure and at three different engine speeds. The experimental results highlighted the reduction of CO and CO₂ emissions for the alternative fuels to an extent depending on their properties. Methanol exhibited high THC and low NO_x emissions compared to gasoline. Methane and, even more so, hydrogen, allowed for a reduction in THC emissions. With regard to the impact of gaseous fuels on the NO_x emissions, this was strongly related to the operating conditions. A surprising result concerns the particle emissions that were affected not only by the fuel characteristics and the engine test point but also by the lubricating oil. The oil contribution was particularly evident for hydrogen fuel, which showed high particle emissions, although they did not contain carbon atoms. Full article