Nonzero Curvature Research Articles

Some classifications of 2-dimensional self-shrinkers

It is well-known that self-shrinkers play an important role in the study of mean curvature flows. In this paper, we develop new techniques to study the rigidity of self-shrinkers. We prove that any self-shrinker X:M→R3 with nonzero constant Gauss curvature is the round sphere S2(2). Moreover, we prove that any flat self-shrinker X:M→R3 is a plane R2, a cylinder S1(1)×R, a generalized cylinder Γ×R, where Γ is an Abresch-Langer curve. At last, we show that any self-shrinker X:M→R3 with constant mean curvature is a plane R2, a cylinder S1(1)×R or the round sphere S2(2).

Multi-stability of the hexagonal origami hypar based on group theory and symmetry breaking

The origami hyperbolic paraboloid, or the hypar, is widely known for its characteristic non-zero Gaussian curvature and multi-stable states. Previous investigations have mainly considered two particular cases of the origami hypar patterns, namely the square and the circular hypars. As a representative example of general polygonal hypar patterns, the hexagonal origami hypar displays more desirable energy properties and subtler multi-stable configurations, which is however, rarely studied. In this paper, we investigate the multi-stability of the hexagonal origami hypar by combining a group-theoretic approach, a symmetry-breaking method, and a bar-and-hinge structural model, to simplify the kinematic analysis of this highly symmetric structure . Notably, the kinematic path of the hexagonal origami hypar is divided into three bifurcation branches by symmetry breaking. Each branch corresponds to two symmetric stable states according to the equilibrium loading and potential energy simulated using the bar-and-hinge model. The non-rigid deformation of the hexagonal origami hypar is mainly controlled by the folding of creases and the bending of facets. Moreover, the energy barrier among the stable states becomes increasingly stronger with higher symmetry orders, thicker sheets of material, and longer creases. This work provides a new strategy for analyzing multi-stable origami structures with high symmetry orders, which can be useed in the design and development of novel adaptive or deployable engineering structures.

Testing spatial curvature and anisotropic expansion on top of the ΛCDM model

We explore the possible advantages of extending the $\Lambda$CDM model by more realistic backgrounds compared to its spatially flat RW spacetime assumption, while preserving the underpinning physics; in particular, by simultaneously allowing non-zero spatial curvature and anisotropic expansion on top of it, viz., the An-$o\Lambda$CDM model. This is to test whether the latest data support spatial flatness and/or isotropic expansion, and, if not, to explore the roles of spatial curvature and expansion anisotropy (due to its stiff fluid-like behavior) in addressing some of the cosmological tensions. We first present the theoretical background and explicit mathematical construction of An-$o\Lambda$CDM; combining the simplest anisotropic generalizations of the RW spacetime, viz., the Bianchi type I, V, and IX spacetimes. Then we constrain this model and its particular cases, viz., An-$\Lambda$CDM, $o\Lambda$CDM, and $\Lambda$CDM, by using the data sets from different probes, viz., Planck CMB(+Lens), BAO, SnIa Pantheon, and CC data, and discuss the results. Ultimately, we conclude that, within the setup under consideration, (i) the data confirm the spatial flatness and isotropic expansion, though a very small amount of present-day expansion anisotropy cannot be excluded, e.g., $\Omega_{\sigma0}\lesssim10^{-18}$ (95\% C.L.) for An-$\Lambda$CDM from CMB+Lens, (ii) the introduction of spatial curvature or anisotropic expansion, or both, on top $\Lambda$CDM does not offer a possible relaxation to the $H_0$ tension, and (iii) the introduction of anisotropic expansion neither affects the closed space prediction from CMB(+Lens) nor does it improve the drastically reduced value of $H_0$ led by the closed space. We discuss why it is important and indispensable to maintain the geometric generalization work program, especially in models that offer solutions to cosmological tensions. [abridged]

Singular Behavior of the Dark Universe under the Effect of Thermal Radiation in Curved Spacetime

We consider the late-time accelerated universe in the Friedmann–Robertson–Walker (FRW) spacetime with a nonzero curvature, and investigate cosmological models when the cosmic fluid is taken to be inhomogeneous and viscous (bulk viscous), coupled to dark matter. We consider the influence from thermal effects caused by Hawking radiation on the formation of singularities of various classified types, within a finite time. It is shown that under the influence of Hawking radiation, the time of formulation of a singularity and the nature of the singularity itself can change. It is also shown that by jointly taking into account radiation, viscosity, and space curvature, one can obtain a singularity-free universe. The symmetry properties of this kind of theory lie in the assumption about spatial isotropy. The spatial isotropy is also reflected in our use of a bulk instead of a shear viscosity.

On concurrent vector fields on Riemannian manifolds

<abstract><p>It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.</p></abstract>

Black Holes have Intrinsic Scalar Curvature

The scalar curvature [Formula: see text] is invariant under isometric symmetries (distance invariance) associated with metric spaces. Gravitational Riemannian manifolds are metric spaces. For Minkowski Space, the distance invariant is [Formula: see text], where [Formula: see text], [Formula: see text] are arbitrary 4-vectors. Thus the isometry symmetry associated with Minkowski Space is the Poincaré Group. The Standard Model Lagrangian density [Formula: see text] is also invariant under the Poincaré Group, so for Minkowski Space, the scalar curvature and the Standard Model Lagrangian density are proportional to each other. We show that this proportionality extends to general gravitational Riemannian manifolds, not just for Minkowski Space. This predicts that Black Holes have non-zero scalar curvatures [Formula: see text]. For Schwarzschild Black Holes, [Formula: see text] is predicted to be [Formula: see text], where [Formula: see text] is the Schwarzschild radius. The existence of [Formula: see text] means that Black Holes cannot evaporate.

Basic classes of timelike general rotational surfaces in the four-dimensional Minkowski space

In the present paper, we consider timelike general rotational surfaces in the Minkowski 4- space which are analogous to the general rotational surfaces in the Euclidean 4-space introduced by C. Moore. We study two types of such surfaces (with timelike and spacelike meridian curve, respectively) and describe analytically some of their basic geometric classes: flat timelike general rotational surfaces, timelike general rotational surfaces with flat normal connection, and timelike general rotational surfaces with non-zero constant mean curvature. We give explicitly all minimal timelike general rotational surfaces and all timelike general rotational surfaces with parallel normalized mean curvature vector field.

Self-similar cosmological solutions in symmetric teleparallel theory: Friedmann-Lemaître-Robertson-Walker spacetimes

The existence of self-similar solutions is discussed in symmetric teleparallel $f(Q)$ theory for a Friedmann-Lema\^{\i}tre-Robertson-Walker background geometry with zero and nonzero spatial curvature. For the four distinct families of connections that describe the specific cosmology in symmetric teleparallel gravity, the functional form of $f(Q)$ is reconstructed. Finally, to see if the analogy with GR holds, we discuss the relation of the self-similar solutions with the asymptotic behavior of more general $f(Q)$ functions.

Dynamical system analysis of Myrzakulov gravity

We perform a dynamical system analysis of Myrzakulov or F(R, T) gravity, which is a subclass of affinely connected metric theories, where ones uses a specific but non-special connection, which allows for non-zero curvature and torsion simultaneously. We consider two classes of models, we extract the critical points, and we examine their stability properties alongside their physical features. In the Class 1 models, which possess {\Lambda}CDM cosmology as a limit, we find the sequence of matter and dark energy eras, and we show that the Universe will result in a dark-energy dominated critical point for which dark energy behaves like a cosmological constant. Concerning the dark-energy equation-of-state parameter we find that it lies in the quintessence or phantom regime, according to the value of the model parameter. For the Class 2 models, we again find the dark-energy dominated, de Sitter late-time attractor, although the scenario does not possess {\Lambda}CDM cosmology as a limit. The cosmological behavior is richer, and the dark-energy sector can be quintessence-like, phantom-like, or experience the phantom-divide crossing during the evolution.

Tilting nondispersive bands in an empty microcavity

Recently, microcavities with anisotropic materials were shown to be able to create bands with non-zero local Berry curvature. The anisotropic refractive index of the cavity layer is believed to be critical in opening an energy gap at the tilted Dirac points. In this work, we show that the anticrossing between a cavity mode and a Bragg mode can also be realized within an empty microcavity without any birefringent materials in the cavity layer. Nondispersive bands are observed within the energy gap due to the particular refractive index distribution of the sample. The intrinsic TE-TM splitting and XY splitting of DBR mirrors induce the squeezing of the cavity modes in momentum space, so that the nondispersive bands are tilted and spin-dependent. Our results pave the way to investigate interesting physical phenomena of photonic modes close to or in the nondispersive bands without anisotropic cavity layers.

Intrinsic anomalous spin Hall effect

Charge-spin interconversion in magnetic materials is investigated by using the first-principles calculations. In addition to the conventional spin Hall effect (SHE) that requires mutual orthogonality of the charge current, spin-flow direction, and spin polarization, the recently proposed anomalous SHE (ASHE) is confirmed in Mn2Au and WTe2. The interaction of the order parameter with conduction electrons leads to sizeable non-zero spin Berry curvatures that give rise to anomalous spin Hall conductivity (ASHC). Our calculations show that the ASHE is intrinsic and originates from the order-parameter-controlled spin-orbit interaction, which generates an extra anomalous effective field. A useful relationship among the order parameter, the spin Berry curvature, and the ASHC is revealed. Our findings provide a new avenue for generating and detecting arbitrary types of spin currents.

A STUDY ON MANIFOLD KAEHLERIAIN SPACE

This work delves into the space-time theory of the 4-dimensional Kaehler manifold. Since the isotropic pressure, energy density, and energy momentum tensor all vanish in a perfect fluid Kaehler space-time manifold, we have established that this space-time manifold is an Einstein manifold and studied the Einstein equation with a cosmological constant in it. Finally, we demonstrated that, on a conformally flat, perfectly fluid Kaehler space-time manifold, the velocity vector field is infinitesimally spatially isotropic. Ideal fluid dilution to the point where only Ricci and minimal symmetry hold. We have proven that Kaehler space-time manifolds have either 0 scalar curvature or a connection between the respective rho and alpha vector fields via g(rho,alpha) = 4. To conclude, we have proved that a Kaehler space-time manifold cannot have both perfect fluidity and non-zero scalar curvature (weak Ricci symmetry).

Effect of crease curvature on the bistability of the origami waterbomb base

Largely due to its geometry-endowed bistability, the origami waterbomb base offers wide-ranging engineering potential. Here, we explore how nonzero crease curvature, which leads to panel bending, enhances the tunability of this structure. To reveal the influence of crease curvature on the deployed geometry and the mechanical response of the octagonal curved-crease waterbomb base, we combine physical experiments with a parametric numerical study. The crease curvature ranges from zero (i.e. the well-known straight-crease waterbomb base) to the maximum curvature possible within the allotted boundary. In addition, we perform finite element analysis (FEA), which incorporates crease plasticity and also allows us to examine the effects of crease stiffness and sheet thickness on mechanical behavior. Our results show that increasing crease curvature reduces the folded height and vertex range-of-motion, but raises the critical load for snap-through instability, without necessarily increasing the switching energy. Potential applications for these tunable structures range from energy harvesting mechanical metamaterials to architectural-scale adaptive shading devices, and deployable space-based solar power systems.

Spacelike minimal surfaces which are graphs in [formula omitted

In this work we study a system of two second order nonlinear partial differential equations corresponding to the equations of minimal spacelike graphs in the four-dimensional Minkownski space R14. We obtain a set of entire solutions of these equations, adapting the method of Karl Kommerell used for the analogous problem in R4. In our technique we replace ‘‘graphs of holomorphic functions” by ‘‘graphs of generalized harmonic curves” and then we obtain a class of minimal spacelike surfaces in R14 which are also entire graphs with non-zero Gauss curvature. We then provide two versions of the Bernstein Theorem corresponding for the two types of graphs which can occur in R14. Several explicit examples are given.

Modeling curvature-resisting material surfaces with isogeometric analysis

Improved understanding of solid surface energy and its role in the overall mechanical properties is of great interest due to the emerging manufacturing techniques of nanostructures, coatings, and synthetic/biological bilayer–polymer hybrids. Continuum numerical modeling of surface stresses efficiently incorporates a zero-thickness membrane bonded to a bulk, intrinsically accounting for surface tension and surface elasticity. Compressive surface stresses are not possible in a purely membrane formulation, ignoring the surface flexural resistance. The extension of material surfaces to account for flexural resistance, i.e., the Steigmann–Ogden model, requires spatial derivatives of second order, posing significant challenges to standard discretization techniques. Hence, the effect of surface curvature resistance on the overall mechanical behavior of complex geometries remains elusive. Here, we develop a three-dimensional computational formulation of curvature-dependent surface energetics at finite strains using surface-enriched isogeometric analysis. Coupled with a hyperelastic bulk, bending-resistance of material surfaces furnishes a new physical length scale, i.e., the elastobending length. We quantify the effect of elastobending deformations for several numerical examples involving soft materials with thin coatings and liquid-shell surfaces, capturing budding-like behavior observed at cell membranes. Our results demonstrate a stiffer overall mechanical behavior when material surfaces resist bending deformations and illustrate how curvature effects lead to complex budding deformations at non-zero initial curvature states. The proposed methodology provides a robust computational foundation to help improve our understanding and mechanical characterization of soft solids, nanostructures, and biological membranes at small scales.

Discrete constant mean curvature surfaces on general graphs

The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.

Active membrane recycling induced morphology changes in vesicles

Membranes of organelles in the intracellular trafficking pathway continuously undergo recycling through fission and fusion processes. The effect of these recycling processes on the large-scale morphology of organelles is not well understood. Using a dynamically triangulated surface model, we developed a membrane morphology simulator that allows for membrane trafficking, and analyzed the steady state shape of vesicles subjected to such active remodeling. We study a two-component vesicle composed of 1) active species which can have nonzero spontaneous curvature and participate in the recycling and 2) inactive species which do not participate in the recycling. We obtain a plethora of steady state morphologies as a function of the activity rate, spontaneous curvature, and the strength of interaction between species. We observe that morphology changes, as a function of rate of activity, are diametrically opposite for the two signs of the spontaneous curvature, but only have a weak effect on its magnitude. The interplay between the in-plane diffusion, the activity rate, and the spontaneous curvature are shown to determine the vesicle morphology at the steady state. It is shown that the spontaneous curvature and activity inhibits the formation of clusters of active species on the surface. We carry out linear stability analysis of a continuum model and show that the spherical shape of a vesicle is indeed unstable when subjected to active membrane recycling above a certain activity rate.

Finite inflation in curved space

We investigate the effects of non-zero spatial curvature on cosmic inflation in the light of cosmic microwave background (CMB) anisotropy measurements from the Planck 2018 legacy release and from the 2015 observing season of BICEP2 and the Keck Array. Even a small percentage of non-zero curvature today would significantly limit the total number of e-folds of the scale factor during inflation, rendering just-enough inflation scenarios with a kinetically dominated or fast-roll stage prior to slow-roll inflation more likely. Finite inflation leads to oscillations and a cutoff towards large scales in the primordial power spectrum and curvature pushes them into the CMB observable window. Using nested sampling, we carry out Bayesian parameter estimations and model comparisons taking into account constraints from reheating and horizon considerations. We confirm the preference of CMB data for closed universes with Bayesian odds of over $100:1$ and with a posterior on the curvature density parameter of $\Omega_{K,0}=-0.051\pm0.017$ for a curvature extension of LCDM and $\Omega_{K,0}=-0.031\pm0.014$ for Starobinsky inflation. Model comparisons of various inflation models give similar results as for flat universes with the Starobinsky model outperforming most other models.

Does relativistic cosmology software handle emergent volume evolution?

Several software packages for relativistic cosmological simulations that do not fully implement the Einstein equation have recently been developed. Two of the free-licensed ones are inhomog and gevolution. A key question is whether globally emergent volume evolution that is faster than that of a Friedmannian reference model results from the averaged effects of structure formation. Checking that emergent volume evolution is correctly modelled by the packages is thus needed. We numerically replace the software’s default random realisation of initial seed fluctuations by a fluctuation of spatially constant amplitude in a simulation’s initial conditions. The average volume evolution of the perturbed model should follow that of a Friedmannian expansion history that corresponds to the original Friedmannian reference solution modified by the insertion of the spatially constant perturbation. We derive the equations that convert from the perturbed reference solution to the effective solution. We find that inhomog allows emergent volume evolution correctly at first order through to the current epoch. For initial conditions with a resolution of N = 1283 particles and an initial non-zero extrinsic curvature invariant I i = 0.001, inhomog matches an exact Friedmannian solution to −0.0058% (Einstein–de Sitter, EdS) or −0.0033% (ΛCDM). We find that gevolution models the decaying mode to fair accuracy, and excludes the growing mode by construction. For N = 1283 and an initial scalar potential Φ = 0.001, gevolution is accurate for the decaying mode to 0.012% (EdS) or 0.013% (ΛCDM). We conclude that this special case of an exact non-linear solution for a perturbed Friedmannian model provides a robust calibration for relativistic cosmological simulations.

F(T,B) gravity in a Friedmann–Lemaître–Robertson–Walker universe with nonzero spatial curvature

We investigate exact solutions and the asymptotic dynamics for the Friedmann–Lemaître–Robertson–Walker universe with nonzero spatial curvature in the fourth‐order modified teleparallel gravitational theory known as theory. We show that the field equations admit a minisuperspace description, and they can reproduce any exact form of the scale factor. Moreover, we calculate the equilibrium points and analyze their stability. We show that Milne and Milne‐like solutions are supported, and the de Sitter universe is provided. To complete our analysis, we use Poincaré variables to investigate the dynamics at infinity.