Search Results (17)

We present a comparative exposition of Riemannian and Lorentzian geometries, organized through successive levels of increasing structure. Beginning with scalar products on vector spaces, we proceed through smooth manifolds, left-invariant metrics on Lie groups, homogeneous spaces and contact metric geometry. At each level, we emphasize which notions and results survive the transition from positive-definite to Lorentzian signature, which ones fail, and the mechanisms responsible for these differences. Recurring themes include: the emergence of causal structures, the loss of diagonalizability of self-adjoint operators, the failure of the equivalence between metric and geodesic completeness, and the topological obstruction to the existence of Lorentzian metrics on compact manifolds. Particular attention is devoted to the classification of three-dimensional homogeneous spaces, including Bianchi–Cartan–Vranceanu spaces and their Lorentzian counterparts, and to the contact metric setting, where an additional geometric structure restores a correspondence between the Riemannian and Lorentzian theories. Full article

The Hubble tension is usually formulated as a disagreement between two determinations of a single scalar parameter, $H_0$, within an exactly isotropic FLRW model. We develop a quantitative framework in which the tension is instead treated as a consistency test of the scalar FLRW compression of cosmological data in a homogeneous but anisotropically expanding Bianchi type I background. Beyond synthesizing established results on Bianchi I kinematics, null geodesics and optical propagation, the original contribution is a worked weak-shear, axisymmetric calculation that maps a specified shear history into a low-redshift luminosity-distance quadrupole. The calculation explicitly separates the direction-dependent redshift–affine-parameter mapping from the Jacobi-focusing contribution and then propagates the resulting distance quadrupole through an analytic polar-cap toy window. For freely decaying shear we obtain $A_D\left(z\right)=-B_{H0}+(2q_0-1)B_{H0}z/2+(5-q_0-18q_0^2+6j_0)B_{H0}{z}^2/12+\mathcal{O}(z^3,B_{H0}^2)$, where $B_{H0}=(H_{\Vert 0}-H_{\perp 0})/H_0$ and $j_0$ is the mean jerk parameter. A representative BBN limit, ${\mathrm{\Omega }}_{\sigma 0}\lesssim {10}^{-23}$, implies $|B_{H0}|\lesssim 9.5\times {10}^{-12}$ and a distance-modulus quadrupole below approximately $2.4\times {10}^{-11}$ mag at $z=0.15$. The early-Universe bound used in this comparison is adopted from previous work and is not itself a new result of the present analysis; the novelty is its propagation through the derived direction-dependent redshift and Sachs–Jacobi mapping into quantitative limits on the luminosity-distance quadrupole and on the catalog-window bias of an isotropic $H_0$ fit. By contrast, even a maximally aligned one-percent directional shift requires ${\mathrm{\Omega }}_{\sigma 0}\simeq 2.5\times {10}^{-5}$, while a shift comparable with the Planck 2018–SH0ES 2022 benchmark separation requires ${\mathrm{\Omega }}_{\sigma 0}\simeq 1.8\times {10}^{-3}$. Thus the minimal shear-only model cannot resolve the tension, although the framework supplies a falsifiable program for testing sustained late-time anisotropy with supernovae, BAO, distance-ladder measurements and future standard sirens. 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

Non-solid infill generation in Non-Planar Additive Manufacturing (NPAM) is still an open problem. This is due to mathematical complexities from curvature distortion, as well as bridging limitations inherent in some NPAM processes. Providing solutions to this problem may result in significant energy, build cycle time, and cost savings. In this context, the goal of this paper is to define a workflow for the generation of non-solid infill paths with quasi-uniform density within the layer. This was performed by defining the build geometry through an axisymmetric embedded map methodology, and the infill points were distributed via a geodesic repulsion energy-based algorithm. In addition to these core algorithms, several numeric optimizations were implemented to reduce runtime. The algorithm has been tested on several build platform geometries and slice polygons. The results were satisfactory, achieving a homogeneous kernel density distribution for all cases and reductions in geodesic distance standard deviations of around 70%. A first iteration of a path planning algorithm was also implemented to showcase the intended final results. This methodology is to be combined with other Design for Non-Planar Additive Manufacturing techniques to enable applications in the biomedical field, automotive and aerospace industry, or rapid mold manufacturing. Full article

We found statistical evidence for a mismatch between the (global) spatial curvature parameter K in the geodesic equation for incoming photons and the corresponding parameter in the Friedmann equation that determines the time evolution of the background spacetime and its perturbations. The mismatch, hereafter referred to as ‘curvature slip’, was especially evident when the SH0ES prior of the current expansion rate was assumed. This result is based on joint analyses of cosmic microwave background (CMB) observations with the PLANCK satellite (P18), the first year results of the Dark Energy Survey (DES), baryonic oscillation (BAO) data, and at a lower level of significance, the Pantheon SNIa (SN) catalog as well. For example, the betting odds against the null hypothesis were greater than ${10}^7$:1, 1400:1 and 1000:1 when P18+SH0ES, P18+DES+SH0ES and P18+BAO+SH0ES were considered, respectively. Datasets involving SNIa weakened this curvature slip considerably. Notably, even when the SH0ES prior was not imposed, the betting odds for the rejection of the null hypothesis were 70:1 and 160:1 in cases where P18+DES and P18+BAO were considered. When the SH0ES prior was imposed, the global fit of the modified model (that allows for a nonvanishing ‘curvature slip’) strongly outperformed that of $\Lambda$CDM, being manifested by significant deviance information criterion (DIC) gains ranging between 7 and 23, depending on the dataset combination considered. Even in comparison with K$\Lambda$CDM, the proposed model resulted in significant, albeit smaller, DIC gains when SN data were excluded. Our finding could possibly be interpreted as an inherent inconsistency between the (idealized) maximally symmetric nature of the FRW metric and the dynamical evolution of the GR-based homogeneous and isotropic $\Lambda$CDM models. As such, this implies that there is apparent tension between the metric curvature and the curvature-like term in the time evolution of the redshift. Full article

In this paper, we study some homogeneity properties of a semi-direct extension of the Heisenberg group, known in literature as the hyperbolic oscillator (or Boidol) group, equipped with the left-invariant metrics corresponding to the ones of the oscillator group. We identify the naturally reductive case by the existence of the corresponding special homogeneous structures. For the cases where these special homogeneous structures do not exist, we exhibit a complete description of the homogeneous geodesics. Full article

Methods for controlling electromagnetic fields in materials are presented that mitigate effects such as electrostatic discharge and electromagnetic/radio frequency interference. The first method determines the effective response of composite materials using a d-dimensional effective medium theory. The material consists of inhomogeneous two-layer inclusions with hyperspherical geometry. Non-integer dimensions represent fractal limits. The material medium is composed of a low hypervolume fraction of inclusions that are randomly distributed inside it. The effective response of the dielectric function is obtained using a virial expansion of the Maxwell–Garnett theory. The other method uses the transformation medium theory and involves the transformation of the material’s permittivity and permeability tensors so that the material exhibits a predefined effective response. By selecting appropriate transformations, a homogeneous material medium is transformed into an inhomogeneous version, forcing the electromagnetic fields to propagate along geodesic paths. These geodesics determine the behaviour of the fields inside the material. As a result, the material can be made to exhibit similar physical characteristics as those of a material composed of hyperspherical inclusions. The theoretical analysis presented is further studied and validated via the use of full-wave numerical simulations of Maxwell’s equations. Full article

On the basis of a general action principle, we revisit the scale invariant field equation using the cotensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor $\lambda$, which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler’s third law is a remarkable scale-invariant property. Full article

We investigate possible astronomical manifestations of space-time anisotropy. The homogeneous vacuum Kasner solution was chosen as a reference anisotropic cosmological model because there are no effects caused by inhomogeneity in this simple model with a constant degree of anisotropy. This anisotropy cannot become weak. The study of its geodesic structure made it possible to clarify the properties of this space-time. It showed that the degree of manifestation of anisotropy varies significantly depending on the travel time of the light from the observed object. For nearby objects, for which it does not exceed half the age of the universe, the manifestations of anisotropy are very small. Distant objects show more pronounced manifestations; for example, in the distribution of objects over the sky and over photometric distances. These effects for each of the individual objects decrease with time but, in general, the manifestations of anisotropy in the Kasner space-time remain constant due to the fact that new sources come from beyond the cosmological horizon. We analyze observable signatures of the Kasner-type anisotropy and compare it to observations. These effects were not found in astronomical observations, including the study of the CMB. We can assume that the Universe has always been isotropic or almost isotropic since the recombination era. This does not exclude the possibility of its significant anisotropy at the moment of the Big Bang followed by rapid isotropization during the inflationary epoch. Full article

A translation curve in a homogeneous space is a curve such that for a given unit vector at the origin, translation of this vector is tangent to the curve in its every point. Translation curves coincide with geodesics in most Thurston spaces, but not in twisted product Thurston spaces. Moreover, translation curves often seem more intuitive and simpler than geodesics. In this paper, we determine translation curves in ${\mathrm{Sol}}_1^4$ space. Their curvature properties are discussed and translation spheres are presented. Finally, characterization of geodesics in ${\mathrm{Sol}}_1^4$ space is given. Full article

Most quantum gravity theories quantize space-time on the order of Planck length (${\ell }_p^{}$). Some of these theories, such as loop quantum gravity (LQG), predict that this discreetness could be manifested through Lorentz invariance violations (LIV) over travelling particles at astronomical length distances. However, reports on LIV are controversial, and space discreetness could still be compatible with Lorentz invariance. Here, it is tested whether space quantization on the order of Planck length could still be compatible with Lorentz invariance through the application of a covariant geometric uncertainty principle (GeUP) as a constraint over geodesics in FRW geometries. Space-time line elements compatible with the uncertainty principle are calculated for a homogeneous, isotropic expanding Universe represented by the Friedmann–Lemaitre–Robertson–Walker solution to General Relativity (FLRW or FRW metric). A generic expression for the quadratic proper space-time line element is derived, proportional to Planck length-squared, and dependent on two contributions. The first is associated to the energy–time uncertainty, and the second depends on the Hubble function. The results are in agreement with space-time quantization on the expected length orders, according to quantum gravity theories, and within experimental constraints on putative LIV. Full article

Clustering-based methods of polarimetric synthetic aperture radar (PolSAR) image superpixel generation are popular due to their feasibility and parameter controllability. However, these methods pay more attention to improving boundary adherence and are usually time-consuming to generate satisfactory superpixels. To address this issue, a novel cross-iteration strategy is proposed to integrate various advantages of different distances with higher computational efficiency for the first time. Therefore, the revised Wishart distance (RWD), which has better boundary adherence but is time-consuming, is first integrated with the geodesic distance (GD), which has higher efficiency and more regular shape, to form a comprehensive similarity measure via the cross-iteration strategy. This similarity measure is then utilized alternately in the local clustering process according to the difference between two consecutive ratios of the current number of unstable pixels to the total number of unstable pixels, to achieve a lower computational burden and competitive accuracy for superpixel generation. Furthermore, hexagonal initialization is adopted to further reduce the complexity of searching pixels for relabelling in the local regions. Extensive experiments conducted on the AIRSAR, RADARSAT-2 and simulated data sets demonstrate that the proposed method exhibits higher computational efficiency and a more regular shape, resulting in a smooth representation of land cover in homogeneous regions and better-preserved details in heterogeneous regions. Full article

H-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that, in the homogeneous nearly Kähler ${\mathbb{S}}^3\times {\mathbb{S}}^3$, also H-umbilical Lagrangian submanifolds are automatically totally geodesic. Full article

The space $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form. Full article

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was positively answered in previous papers. However, the result is not optimal. In the present paper, this result is refined and the existence of at least two homogeneous geodesics in any homogeneous Finsler manifold is proved. In a previous paper, examples of Randers metrics which admit just two homogeneous geodesics were constructed, which shows that the present result is the best possible. Full article