Riemann Tensor Research Articles

Gravitational distortion on photon state at the vicinity of the Earth

As a photon propagates along a null geodesic, the space-time curvature around the geodesic distorts its wave function. We utilize the Fermi coordinates adapted to a general null geodesic and derive the equation for interaction between the Riemann tensor and the photon wave function. The equation is solved by being mapped to a time-dependent Schr\"odinger equation in ($2+1$) dimensions. The results show that as a Gaussian time-bin wave packet with a narrow bandwidth travels over a null geodesic, it gains an extra phase that is a function of the Riemann tensor evaluated and integrated over the propagation trajectory. This extra phase is calculated for communication between satellites around the Earth and is shown to be measurable by current technology.

Collapsing axial system and f(𝒢,T) gravity

The motive of this paper is to investigate the evolution of dissipative axially symmetry collapsing fluid in the presence of dark sources. For this purpose, we use the modified Gauss–Bonnet gravity as an exotic energy candidate. We formulate the dynamical variables and investigate the impacts of dark sources upon the heat dissipation and pressure anisotropy. We find scalar functions through orthogonal decomposition of Riemann tensor. The physical behavior of these scalars has been examined for matter as well as dark source configurations. Moreover, we develop a set of EEs related to the evolution of dynamical variables, heat transport equation, Weyl tensor and super-Poynting vector. These equations discuss collapse features like thermodynamics, density inhomogeneity and gravitational radiations in presences of exotic terms. It has been shown that the dark source terms have an effect on the thermodynamics of the system, evolution of kinematical variables and density inhomogeneity. The exotic terms can generate repulsive radiations which can induce cosmic expansion. Finally, the physical significance of modified Gauss–Bonnet gravity is explored in association to the stellar 4U1820-30.

SO(2)×SO(3)$SO(2)\times SO(3)$‐invariant Ricci solitons and ancient flows on S4$\mathbb {S}^4$

Consider the standard action of S O ( 2 ) × S O ( 3 ) $SO(2)\times SO(3)$ on R 5 = R 2 ⊕ R 3 $\mathbb {R}^5=\mathbb {R}^2\oplus \mathbb {R}^3$ . We establish the existence of a uniform constant C > 0 $\mathcal {C}>0$ so that any S O ( 2 ) × S O ( 3 ) $SO(2)\times SO(3)$ -invariant Ricci soliton on S 4 ⊂ R 5 $\mathbb {S}^4\subset \mathbb {R}^5$ with Einstein constant 1 must have Riemann curvature and volume bounded by C $\mathcal {C}$ , and injectivity radius bounded below by 1 C $\frac{1}{\mathcal {C}}$ . This observation, coupled with basic numerics, gives strong evidence to suggest that the only S O ( 2 ) × S O ( 3 ) $SO(2)\times SO(3)$ -invariant Ricci solitons on S 4 $\mathbb {S}^4$ are round. We also prove existence of the so-called ‘pancake’ ancient solution of the Ricci flow.

On Generalized Φ- Recurrent of Kenmotsu Type Manifolds

The present paper studies the generalized Φ- recurrent of Kenmotsu type manifolds. This is done to determine the components of the covariant derivative of the Riemannian curvature tensor. Moreover, the conditions which make Kenmotsu type manifolds to be locally symmetric or generalized Φ- recurrent have been established. It is also concluded that the locally symmetric of Kenmotsu type manifolds are generalized recurrent under suitable condition and vice versa. Furthermore, the study establishes the relationship between the Einstein manifolds and locally symmetric of Kenmotsu type manifolds.

Asymptotic behavior of null geodesics near future null infinity. II. Curvatures, photon surface, and dynamically transversely trapping surface

Bearing in mind our previous study on asymptotic behavior of null geodesics near future null infinity, we analyze the behavior of geometrical quantities such as a certain extrinsic curvature and Riemann tensor in the Bondi coordinates. In the sense of asymptotics, the condition for an $r$-constant hypersurface to be a photon surface is shown to be controlled by a key quantity that determines the fate of photons initially emitted in angular directions. As a consequence, in four dimensions, such a non-expanding photon surface can be realized even near future null infinity in the presence of enormous energy flux for a short period of time. By contrast, in higher-dimensional cases, no such a photon surface can exist. This result also implies that the dynamically transversely trapping surface, which is proposed as an extension of a photon surface, can have an arbitrarily large radius in four dimensions.

Riemannian Formulation of Pontryagin’s Maximum Principle for the Optimal Control of Robotic Manipulators

In this work, we consider robotic systems for which the mass tensor is identified to be the metric in a Riemannian manifold. Cost functional invariance is achieved by constructing it with the identified metric. Optimal control evolution is revealed in the form of a covariant second-order ordinary differential equation featuring the Riemann curvature tensor that constrains the control variable. In Pontryagin’s framework of the maximum principle, the cost functional has a direct impact on the system Hamiltonian. It is regarded as the performance index, and optimal control variables are affected by this fundamental choice. In the present context of cost functional invariance, we show that the adjoint variables are the first-order representation of the second-order control variable evolution equation. It is also shown that adding supplementary invariant terms to the cost functional does not modify the basic structure of the optimal control covariant evolution equation. Numerical trials show that the proposed invariant cost functionals, as compared to their non-invariant versions, lead to lower joint power consumption and narrower joint angular amplitudes during motion. With our formulation, the differential equations solver gains stability and operates dramatically faster when compared to examples where cost functional invariance is not considered.

On projective Riemann curvature

With a volume form on a manifold, every spray can be deformed to a projective spray. The Riemann curvature of a projective spray is called the projective Riemmann curvature. In this paper, we introduce the notion of projectively R-flat sprays/Finsler metrics. As an application, we investigate and characterize projectively R-flat Randers metrics. Moreover, we obtained a rigidity theorem on a closed Finsler manifold.

The proportionality principle for Osserman manifolds

We give the necessary and sufficient conditions for Jacobi operators that determine an algebraic curvature tensor. This motivates us to introduce the new concept of Jacobi-proportional Riemannian tensors, whose special case is the Rakić duality principle. We prove that all known Osserman tensors (both two-root Osserman and Clifford) are Jacobi-proportional. After the results given by Nikolayevsky, it is known that possible counterexamples of the Osserman conjecture can occur in dimension 16 only, while the reduced Jacobi operator has an eigenvalue of multiplicity 7 or 8. We prove that Jacobi-proportional Osserman tensors that do not satisfy the Osserman conjecture are 2-root with multiplicities 8 and 7, or 3-root with multiplicities 7, 7, and 1.

Structure scalars and an extension to LTB metric

The main goal of this paper is to present the Lemaître–Tolman–Bondi (LTB) spacetime for the radiating case by extending the study of [L. Herrera, A. Di Prisco, J. Ospino and J. Carot, Phys. Rev. D 82 (2010) 024021] in [Formula: see text] theory, where [Formula: see text] is the Ricci scalar and [Formula: see text] is the energy–momentum tensor trace. We will start with some fundamental equations, such as the kinematical variables, mass function introduced by Misner and Sharp, the Weyl tensor under the influence of the [Formula: see text] modified Bianchi identities. We have also considered the evolution of the modified constraint equation. Later on, we have analyzed LTB in terms of modified structure scalars emerging from the orthogonal splitting of the Riemann tensor, and the transport equation in [Formula: see text] gravity theory. We will also discuss a case in order to generalize LTB metric.

Submersion and Homogeneous Spray Geometry

We introduce the submersion between two spray structures and propose the submersion technique in spray geometry. Using this technique, as well as global invariant frames on a Lie group, we setup the general theoretical framework for homogeneous spray geometry. We define the spray vector field \(\eta \) and the connection operator N for a homogeneous spray manifold \((G/H,{\mathbf {G}})\) with a linear decomposition \({\mathfrak {g}}={\mathfrak {h}}+{\mathfrak {m}}\). These notions generalize their counter parts in homogeneous Finsler geometry. We prove the correspondence between \({\mathbf {G}}\) and \(\eta \) when the given decomposition is reductive, and that between geodesics on \((G/H,{\mathbf {G}})\) and integral curves of \(-\eta \). We find the ordinary differential equations on \({\mathfrak {m}}\) describing parallel translations on \((G/H,{\mathbf {G}})\), and we calculate the S-curvature and Riemann curvature of \((G/H,{\mathbf {G}})\), generalizing L. Huang’s curvature formulae for homogeneous Finsler manifolds.

Infinite-derivative linearized gravity in convolutional form

This article aims to transform the infinite-order Lagrangian density for ghost-free infinite-derivative linearized gravity into non-local form. To achieve it, we use the theory of generalized functions and the Fourier transform in the space of tempered distributions . We show that the non-local operator domain is not defined on the whole functional space but on a subset of it. Moreover, we prove that these functions and their derivatives are bounded in all and, consequently, the Riemann tensor is regular and the scalar curvature invariants do not present any spacetime singularity. Finally, we explore what conditions we need to satisfy so that the solutions of the linearized equations of motion exist in .

Categorical Smoothness of 4-Manifolds from Quantum Symmetries and the Information Loss Paradox.

In this paper, we focus on some aspects of the relation of spacetime and quantum mechanics and the study counterparts (in Set) of the categorical local symmetries of smooth 4-manifolds. In the set-theoretic limit, there emerge some exotic smoothness structures on (hence the Riemannian nonvanishing curvature), which fit well with the quantum mechanical lattice of projections on infinite-dimensional Hilbert spaces. The method we follow is formalization localized on the open covers of the spacetime manifold. We discuss our findings in the context of the information paradox assigned to evaporating black holes. A black hole can evaporate entirely, but the smoothness structure of spacetime will be altered and, in this way, the missing information about the initial states of matter forming the black hole will be encoded. Thus, the possible global geometric remnant of black holes in spacetime is recognized as exotic 4-smoothness. The full-fledged verification of this proposal will presumably be possible within the scope of future quantum gravity theory research.

The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

Several Riemannian metrics and families of Riemannian metrics were defined on the manifold of Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define families of metrics: the principle of deformed metrics. We relate the recently introduced family of alpha-Procrustes metrics to the general class of mean kernel metrics by providing a sufficient condition under which elements of the former belong to the latter. Secondly, we focus on the principle of balanced bilinear forms that we recently introduced. We give a new sufficient condition under which the balanced bilinear form is a metric. It allows us to introduce the Mixed-Euclidean (ME) metrics which generalize the Mixed-Power-Euclidean (MPE) metrics. We unveal their link with the (u,v)-divergences and the (α,β)-divergences of information geometry and we provide an explicit formula of the Riemann curvature tensor. We show that the sectional curvature of all ME metrics can take negative values and we show experimentally that the sectional curvature of all MPE metrics but the log-Euclidean, power-Euclidean and power-affine metrics can take positive values.

Modified Teleparallel Gravity induced by quantum fluctuations

In the semi-classical regime, quantum fluctuations embedded in a Riemannian spacetime can be effectively recast as classical back reactions and manifest themselves in the form of non-minimal couplings between matter and curvature. In this work, we exhibit that this semi-classical description can also be applied within the teleparallel formulation. In the teleparallel formulation, quantum fluctuations generically lead to non-minimal torsion-matter couplings. Due to the equivalence between the (classical) Einstein gravity in the Riemannian description and that in teleparallel description, some effective models which were constructed using Riemannian description can be reproduced completely using the teleparallel description. Besides, when the effective quantum correction term is proportional to the torsion scalar $T$, we obtain a subclass of novel $f(T,B,\mathcal{T})$ gravity, where $B$ is a boundary term, and $\mathcal{T}$ is the trace of the energy-momentum tensor. Next, we investigate the cosmological properties in this $f(T,B,\mathcal{T})$ theory by assuming that the matter Lagrangian is solely constructed by a dynamical scalar field. We exhibit some interesting cosmological solutions, such as those with decelerating expansion followed by a late-time accelerating phase. In addition, the non-minimal torsion-matter couplings induced by quantum corrections naturally lead to energy transfers between gravity and cosmological fluids in the universe.

Soft-collinear gravity beyond the leading power

We construct “soft-collinear gravity”, the effective field theory which describes the interaction of collinear and soft gravitons with matter (and themselves), to all orders in the soft-collinear power expansion. Despite the absence of collinear divergences in gravity at leading power, the construction exhibits remarkable similarities with soft-collinear effective theory of QCD (gauge fields). It reveals an emergent soft background gauge symmetry, which allows for a manifestly gauge-invariant representation of the interactions in terms of a soft covariant derivative, the soft Riemann tensor, and a covariant generalisation of the collinear light-cone gauge metric field. The gauge symmetries control both the unsuppressed collinear field components and the inherent inhomogeneity in λ of the invariant objects to all orders, resulting in a consistent expansion.

Towards a gravitational and supergravitational Elko theory

We describe a possible and alternative route to connect gravity with Elko theory. Our approach is based on the possibility to introduce a totally antisymmetric gauge field in the generalized Elko field equation, which is an alternative extension of the Dirac field equation. The corresponding totally antisymmetric field strength ([Formula: see text]-form) is then associated with Grassmann–Plücker coordinates and therefore with a decomposable [Formula: see text]-form. We show that such a totally antisymmetric field strength can be considered as the square root of the Riemann tensor associated to a homogeneous space. Motivated by this result we conjecture that a full connection with Riemann geometry and therefore with general relativity must be possible if such a field strength is not decomposable. We also show how a supergravity version could arise from a generalization of the above ideas.

Complexity of static sphere in energy–momentum squared gravity

In this paper, we study complexity of a static spherically symmetric object containing anisotropic fluid in the background of energy–momentum squared gravity (EMSG). We formulate the field equations and describe the energy content of the fluid via Misner–Sharp as well as Tolman mass definitions. In order to compute the complexity factor, the Riemann tensor is orthogonally split to produce structure scalars which incorporate the fundamental properties of the system. The scalar involving anisotropic pressure and inhomogeneous energy density is chosen as the complexity factor. We consider a minimally coupled model to formulate the condition for vanishing complexity which is employed to formulate two solutions. It is concluded that additional matter source terms appearing due to energy–momentum squared gravity increase the complexity of the system.

Complexity of dynamical cylindrical system in f(G, T) gravity

The purpose of this paper is to formulate a complexity factor for a non-static cylindrical structure in the background of [Formula: see text] gravity, where [Formula: see text] and [Formula: see text] represent the Gauss–Bonnet term and trace of the energy–momentum tensor, respectively. Different physical factors such as inhomogeneous energy density, anisotropic pressure, heat dissipation and modified terms are considered as candidates of complexity. In order to determine a complexity factor encompassing the essential aspects of the system, we apply the technique of orthogonal splitting to the Riemann tensor. We also study evolution of the cylinder through two modes: homologous and homogeneous. Further, we utilize the complexity-free and homologous conditions to examine the dissipative and non-dissipative self-gravitating system. Finally, the factors responsible for inducing complexity during the evolution system are inspected. We deduce that the modified terms in [Formula: see text] gravity make the system more complex.

Complexity for dynamical anisotropic sphere in [formula omitted] gravity

This paper is devoted to the formulation of a complexity factor for dynamical anisotropic sphere in the framework of f(G,T) gravity, where G is the Gauss–Bonnet invariant and T is the trace of energy–momentum tensor. Inhomogeneous energy density, anisotropic pressure, heat dissipation and modified terms create complexity within the self-gravitating system. We evaluate the structure scalars by orthogonal splitting of the Riemann tensor to evaluate a complexity factor which incorporates all the fundamental properties of the system. Moreover, we examine the dynamics of the sphere by assuming homologous mode as the simplest pattern of evolution. We also discuss dissipative as well as non-dissipative scenarios corresponding to homologous and complexity free conditions. Finally, we establish a criterion under which the complexity free condition remains stable throughout the process of evolution. We conclude that the presence of dark source terms of f(G,T) gravity increase the system’s complexity.

A local singularity analysis for the Ricci flow and its applications to Ricci flows with bounded scalar curvature

We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum’s result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Finally, we show some applications of the theory to Ricci flows with bounded scalar curvature.