Constant Positive Curvature Research Articles

Lorentzian G-manifolds of constant positive curvature

We study the orbits arising from isometric actions of connected Lie groups on Lorentzian manifolds with constant positive curvature.

On quasicomplete k$k$‐surfaces in 3‐dimensional space‐forms

AbstractIn the study of immersed surfaces of constant positive extrinsic curvature in space‐forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such surfaces under the hypotheses of quasicompleteness. In particular, we show that, for , the only quasicomplete immersed surfaces of constant extrinsic curvature equal to in the 3‐dimensional space‐form of constant sectional curvature equal to are the geodesic spheres. Together with earlier work of the author, this completes the classification of quasicomplete immersed surfaces of constant positive extrinsic curvature in 3‐dimensional space‐forms.

Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.

Sustaining Quasi De-Sitter Inflation with Bulk Viscosity

The de-Sitter spacetime is a maximally symmetric Lorentzian manifold with constant positive scalar curvature that plays a fundamental role in modern cosmology. Here, we investigate bulk-viscosity-assisted quasi de-Sitter inflation, that is the period of accelerated expansion in the early universe during which −H˙≪H2, with H(t) being the Hubble expansion rate. We do so in the framework of a causal theory of relativistic hydrodynamics, which takes into account non-equilibrium effects associated with bulk viscosity, which may have been present as the early universe underwent an accelerated expansion. In this framework, the existence of a quasi de-Sitter universe emerges as a natural consequence of the presence of bulk viscosity, without requiring introducing additional scalar fields. As a result, the equation of state, determined by numerically solving the generalized momentum-conservation equation involving bulk viscosity pressure turns out to be time dependent. The transition timescale characterising its departure from an exact de-Sitter phase is intricately related to the magnitude of the bulk viscosity. We examine the properties of the new equation of state, as well as the transition timescale in the presence of bulk viscosity pressure. In addition, we construct a fluid description of inflation and demonstrate that, in the context of the causal formalism, it is equivalent to the scalar field theory of inflation. Our analysis also shows that the slow-roll conditions are realised in the bulk-viscosity-supported model of inflation. Finally, we examine the viability of our model by computing the inflationary observables, namely the spectral index and the tensor-to-scalar ratio of the curvature perturbations, and compare them with a number of different observations, finding good agreement in most cases.

Exploring elliptic geometry in neural networks : The concept of elliptic neural networks (ENN)

The exploration of non-Euclidean geometries in the context of neural network architectures presents a novel avenue for enhancing the processing of complex data structures. This paper introduces the concept of Elliptic Neural Networks (ENN), a framework that integrates the intrinsic properties of elliptic geometry into the fabric of neural computation. Characterized by constant positive curvature and closed geodesic paths, elliptic geometry provides a unique perspective for representing data, especially advantageous for configurations exhibiting cyclic and dense hierarchical interconnections. We begin by delineating the fundamental aspects of elliptic geometry, focusing on its impact on conventional notions of distance and global structural interpretation within data sets. This foundational understanding paves the way for our proposed mathematical schema for ENNs. Within this framework, we articulate the methodologies for embedding data in elliptic spaces and adapt neural network operations by encompassing neuron activation, signal propagation, and learning paradigms to conform to the topological idiosyncrasies of elliptic geometry. In this paper addressing the implementation of ENNs, we discuss the computational challenges and the development of novel algorithms requisite for navigating the elliptic geometric landscape. The prospective applications of ENNs are extensive and diverse, ranging from the enhancement of cyclic pattern recognition in time-series analysis to more sophisticated representations in graph-based data and natural language processing tasks. This theoretical exposition aims to set the groundwork for subsequent empirical research to validate the proposed model and assess its practical performance relative to conventional neural network architectures. By harnessing the distinctive characteristics of elliptic geometry, ENNs mark a significant stride towards enriching the arsenal of machine learning methodologies, potentially leading to more versatile and efficacious computational tools in data analysis and artificial intelligence.

Graphs with Constant Positive Ricci Curvature

Graphs with Constant Positive Ricci Curvature

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in Sm(c)×R

Let Sm(c) denote a sphere with a positive constant curvature c and Mn(n≥3) be an n-dimensional compact pseudo-umbilical submanifold in a Riemannian product space Sm(c)×R with a nonzero parallel mean curvature vector (PMC), where R is a Euclidean line. In this paper, we prove a sequence of pinching theorems with respect to the Ricci, sectional and scalar curvatures of Mn, which allow us to generalize some classical curvature pinching results in spheres.

Some results on four-manifolds with nonnegative biorthogonal curvature

Let (M,g) be a 4-dimensional compact oriented Riemannian manifold with nonnegative biorthogonal curvature and W− be the anti-self-dual component of the Weyl curvature tensor W. If M has constant scalar curvature and |W−| of M is constant, or if M has harmonic Weyl tensor and detW− or |W−| of M is constant, then we give two classification theorems for M. As two applications, a 4-dimensional compact oriented Einstein manifold with nonnegative sectional curvature whose detW− or |W−| is constant is isometric to one of S4, CP2, S2×S2 or a flat manifold; a 4-dimensional compact oriented self-dual Riemannian manifold with positive sectional curvature and constant scalar curvature either is conformal to S4, CP2, or is diffeomorphic to CP2♯CP2, CP2♯CP2♯CP2.

Cosmic dynamics of isotropic models with inhomogeneous EoS: A dynamical system perspective

We examine the homogeneous and isotropic models sourced by the fluid having an inhomogenous equation of state (EoS) using qualitative methods. The flat Friedmann–Lemaitre–Robertson–Walker (FLRW) model explains the accelerating universe expansion of the present era having past of decelerating era dominated by radiation and matter. The open FLRW model incorporates the radiation, matter and dark energy dominated era in order with the spatial curvature playing role just before the accelerated universe evolution. The dominating behaviors of radiation, matter and spatial curvature affect the evolution trajectories of the deceleration and effective EoS parameter, which are obtained by the numerical solutions of autonomous system. The model parameters affect the nature of dark energy and therefore the EoS parameter may also lie in the quintessence or phantom regions. The closed FLRW model having constant positive curvature may yield the bouncing and turnaround universe evolution with dark energy-dominated phase at late-times. The application of statefinder diagnostic in the cosmological dynamical systems highlights that the universe may approach the [Formula: see text] cold dark matter ([Formula: see text]CDM) universe as a particular case in the future.

Manifolds with harmonic Weyl tensor and nonnegative curvature operator of the second kind

We show that a compact manifold with harmonic Weyl tensor and nonnegative curvature operator of the second kind is globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. Moreover, we obtain that a compact manifold with harmonic curvature and nonnegative curvature operator of the second kind is isometric to a space of constant curvature. These improve Kashiwada's result that Riemannian manifolds with harmonic curvature tensor for which the curvature operator of the second kind is nonnegative and positive are locally symmetric spaces and constant curvature spaces, respectively.

Fitting Type Ia Supernova Data to a Cosmological Model Based on Einstein–Newcomb–De Sitter Space

Einstein–Newcomb–de Sitter (ENdS) space is de Sitter’s modification of spherical space used by Einstein in his first cosmological model paper published in 1917. The modification by de Sitter incorporated the topological identification of antipodal points in space previously proposed by Newcomb in 1877. De Sitter showed that space topologically modified in this way (called elliptical or projective space) satisfies Einstein’s field equations. De Sitter also found that in a space with constant positive curvature, spectral lines of remote galaxies would be red-shifted (called the de Sitter effect). However, de Sitter’s formulae relating distances to red shifts do not satisfy observational data. The likely reason for this mismatch is that de Sitter mainly focused on space curvature and ignored the identification of antipodal points. Herein, we demonstrate that it is this particular feature that allows an almost perfect fit of the ENdS-based cosmological model to observational data. We use 1701 sources from the type Ia supernovae data sample called Pantheon+, which was previously used to fit the ΛCDM model. ΛCDM and ENdS diverge in their predictions for red shifts exceeding z∼2.3. Since there are no available type Ia supernovae (SNe) data for higher red shifts, both models can be validated by using an additional sample of 193 gamma-ray bursts (GRBs) spanning red shifts up to z∼8. This validation shows that the minimum χ2 for the SNe+GRBs sample is about 2.7% smaller for the ENdS space model than for the ΛCDM model.

Killing vector fields on Riemannian and Lorentzian 3‐manifolds

AbstractWe give a complete local classification of all Riemannian 3‐manifolds admitting a nonvanishing Killing vector field T. We then extend this classification to timelike Killing vector fields on Lorentzian 3‐manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature S of g and the function , where Ric is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of T. Our classification generalizes that of Sasakian structures, which is the special case when . We also give necessary, and separately, sufficient conditions, both expressed in terms of , for g to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that T has unit length and the coordinates derived in our classification are globally defined on , we give conditions under which S completely determines when the metric will be geodesically complete. In the event that the 3‐manifold M is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.

Elliptic KdV potentials and conical metrics of positive constant curvature, I

Elliptic KdV potentials and conical metrics of positive constant curvature, I

A priori bounds and existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with log-convex measure

<abstract><p>In the present paper, we prove the existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $ (M, g) $ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $ L_p $ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.</p></abstract>

A family of special case sequential warped-product manifolds

<abstract><p>We derive the general formulas for a special configuration of the sequential warped-product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a generic diagonal conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $ n_1 $-dimensional and $ n_2 $-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $ (n_1-1) $-dimensional group of transformations to the case of positive constant Ricci curvature ($ \lambda > 0 $).</p></abstract>

On Momentum Operators Given by Killing Vectors Whose Integral Curves Are Geodesics

The paper considers momentum operators on intrinsically curved manifolds. Given that momentum operators are Killing vector fields whose integral curves are geodesics, the corresponding manifold is flat or of the compact type with positive constant sectional curvature and dimensions equal to 1, 3, or 7. Explicit representations of momentum operators and the associated Casimir element are discussed for the 3-sphere S3. It is verified that the structural constants of the underlying Lie algebra are proportional to 2 ℏ/R, where R is the curvature radius of S3 and ℏ is the reduced Planck’s constant. This results in a countable energy and momentum spectrum of freely moving particles in S3. The maximal resolution of the possible momenta is given by the de Broglie wave length, λR=πR, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates, and the associated commutator relations of position and momentum are established.

Motion of charged particles in spacetimes with magnetic fields of spherical and hyperbolic symmetry

The motion of charged particles in spacetimes containing a submanifold of constant positive or negative curvature is considered, with the electromagnetic tensor proportional to the volume two-form form of the submanifold. In the positive curvature case, this describes spherically symmetric spacetimes with a magnetic monopole, while in the negative curvature case, it is a hyperbolic spacetime with magnetic field uniform along hyperbolic surfaces. Constants of motion are found by considering Poisson brackets defined on a phase space with gauge-covariant momenta. In the spherically-symmetric case, we find a correspondence between the trajectories on the Poincar\'{e} cone with equatorial geodesics in a conical defect spacetime. In the hyperbolic case, the analogue of the Poincar\'{e} cone is defined as a surface in an auxiliary Minkowski spacetime. Explicit examples are solved for the Minkowski, $\mathrm{AdS}_4\times S^2$, and the hyperbolic AdS-Reissner--Nordstr\"{o}m spacetimes.

Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces

We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions.

Volume minimising unit vector fields on three dimensional space forms of positive curvature

We prove that on a closed 3-manifold of constant positive curvature the unit vector fields of minimum volume are exactly the Hopf vector fields.

Opening nodes in the DPW method: Co-planar case

We combine the DPW method and opening nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.