Riemann Tensor Research Articles

On the geometry of the Mus-Cheeger-Gromoll metric

Let (M, g) be an n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G
 and called the Mus-Cheeger-Gromoll metric on the tangent bundle TM. We calculate its Levi-Civita connection and Riemannian curvature tensor. Also, we study the geometry of (TM, G).

Conformally-rescaled Schwarzschild metrics do not predict flat galaxy rotation curves

For conformally invariant gravity theories defined on Riemannian spacetime and having the Schwarzschild–de-Sitter (SdS) metric as a solution in the Einstein gauge, we consider whether one may conformally rescale this solution to obtain flat rotation curves, such as those observed in galaxies, without the need for dark matter. Contrary to recent claims in the literature, we show that if one works in terms of quantities that can be physically measured, then in any conformal frame the trajectories followed by ‘ordinary’ matter particles are merely the timelike geodesics of the SdS metric, as one might expect. This resolves the apparent frame dependence of physical predictions and unambiguously yields rotation curves with no flat region. We also show that attempts to model rising rotation curves by fitting the coefficient of the quadratic term in the SdS metric individually for each galaxy are precluded, since this coefficient is most naturally interpreted as proportional to a global cosmological constant. We further extend our analysis beyond static, spherically-symmetric systems to show that the invariance of particle dynamics to the choice of conformal frame holds for arbitary metrics, again as expected. Moreover, we show that this conclusion remains valid for conformally invariant gravity theories defined on more general Weyl–Cartan spacetimes, which include Weyl, Riemann–Cartan and Riemannian spacetimes as special cases.

On the toal curvature of closed curves in Riemannian manifolds

On the toal curvature of closed curves in Riemannian manifolds

Almost C(α)-Manifold on M-Projective Curvature Tensor

In this article, the behavior of the C(α)-manifold satisfying the conditions R(X,Y)W∗ = 0,W∗ (X,Y)R = 0,W∗ (X,Y)Z˜ = 0, W∗ (X,Y)S = 0 and W∗ (X,Y)C˜ = 0 on the M−projective curvature tensor is investigated. The C(α)−Manifold is characterized according to these states of the curvature tensor. Here, W∗ ,R,S,Z˜ and C˜ are M−projective, Riemann, Ricci, concircular and quasiconformal curvature tensors

Quantum Riemannian geometry of quantum projective spaces

We study the quantum Riemannian geometry of quantum projective spaces of any dimension. In particular, we compute the Riemann and Ricci tensors using previously introduced quantum metrics and quantum Levi-Civita connections. We show that the Riemann tensor is a bimodule map and derive various consequences of this fact. We prove that the Ricci tensor is proportional to the quantum metric, giving a quantum analogue of the Einstein condition, and compute the corresponding scalar curvature.

Notes on the second-order tangent bundles with the deformed Sasaki metric

The paper deals with the second-order tangent bundle T2MT2M with the deformed Sasaki metric ¯gg¯ over an n−dimensional Riemannian manifold gover an n−dimensional Riemannian manifold (M,g)(M,g). We calculate all Riemannian curvature tensor fields of the deformed Sasaki metric ¯gg¯ and search Einstein property of and search Einstein property of T2MT2M. Also the weakly symmetry properties of the deformed Sasaki metric are presented.

Main Curvatures Identities on Lightlike Hypersurfaces of Statistical Manifolds and Their Characterizations

In this study, some identities involving the Riemannian curvature invariants are presented on lightlike hypersurfaces of a statistical manifold in the Lorentzian settings. Several inequalities characterizing lightlike hypersurfaces are obtained. These inequalities are also investigated on lightlike hypersurfaces of Lorentzian statistical space forms.

Curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime

The curvature scalar invariants of the Riemann tensor are important in general relativity because they allow a manifestly coordinate invariant characterisation of certain geometrical properties of spacetimes such as, among others, curvature singularities, gravitomagnetism. We calculate explicit analytic expressions for the set of Zakhary–McIntosh curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime as well as for the Kerr–Newman–(anti-)de Sitter black hole. These black hole metrics belong to the most general type D solution of the Einstein–Maxwell equations with a cosmological constant. Explicit analytic expressions for the Euler–Poincare density invariant, which is relevant for the computation of the Euler–Poincare characteristic χ(M), and the Kretschmann scalar are also provided for both cases. We perform a detailed plotting of the curvature invariants that reveal a rich structure of the spacetime geometry surrounding the singularity of a rotating, electrically charged and accelerating black hole. These graphs also help us in an exact mathematical way to explore the interior of these black holes. Our explicit closed form expressions show that the above gravitational backgrounds possess a non-trivial Hirzebruch signature density. Possible physical applications of this property for the electromagnetic duality anomaly in curved spacetimes that can spoil helicity conservation are briefly discussed.

Spacetime curvature from ultrarapid measurements of quantum fields

We write the curvature of spacetime in terms of the excitation probability of particle detectors ultra-rapidly coupled to a quantum field. More precisely, we provide an expansion for the excitation probability of a smeared UDW detector delta-coupled to a real scalar quantum field in a curved background. Using a short distance expansion for the Wightman function, we express the excitation probability of a detector as the transition probability in Minkowski spacetime plus correction terms written as a function of the curvature tensors and the detector size. Comparing the excitation probability in curved spacetimes with its flat analog, we are able to express the components of the Ricci and Riemann curvature tensors as a function of physically measurable excitation probabilities of different shaped detectors.

Gravitational wave memory and the wave equation

Gravitational wave memory and its electromagnetic analog are shown to be straightforward consequences of the wave equation. From Maxwell’s equations one can derive a wave equation for the electric field, while from the Bianchi identity one can derive a wave equation for the Riemann tensor in linearized gravity. Memory in both cases is derived from the structure of the source of those wave equations.

An Accelerated First-Order Method for Non-convex Optimization on Manifolds

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

Covariant Evolution of Gravitoelectromagnetism

The long-range gravitational terms associated with tidal forces, frame-dragging effects, and gravitational waves are described by the Weyl conformal tensor, the traceless part of the Riemann curvature that is not locally affected by the matter field. The Ricci and Bianchi identities provide a set of dynamical and kinematic equations governing the matter coupling and evolution of the electric and magnetic parts of the Weyl tensor, so-called gravitoelectric and gravitomagnetic fields. A detailed analysis of the Weyl gravitoelectromagnetic fields can be conducted using a number of algebraic and differential identities prescribed by the 1+3 covariant formalism. In this review, we consider the dynamical constraints and propagation equations of the gravitoelectric/-magnetic fields and covariantly debate their analytic properties. We discuss the special conditions under which gravitational waves can propagate, the inconsistency of a Newtonian-like model without gravitomagnetism, the nonlinear generalization to multi-fluid models with different matter species, as well as observational effects caused by the Weyl fields via the kinematic quantities. The 1+3 tetrad and 1+1+2 semi-covariant methods, which can equally be used for gravitoelectromagnetism, are briefly explained, along with their correspondence with the covariant formulations.

F(G,TαβTαβ) theory and complex cosmological structures

The basic objective of this investigation is to explore the impact of a novel gravitational modification, specifically, the $f(\mathcal{G}, \mathbf{T}^2)$ (where $\mathbf{T}^2 \equiv T_{\alpha\beta}T^{\alpha\beta}$, $T^{\alpha\beta}$ denotes the stress-energy tensor) model of gravitation, upon the complexity of time-dependent dissipative as well as non-dissipative spherically symmetric celestial structures. To find the complexity factor $(\mathbb{C}_{\mathbf{F}})$ from the generic version of the structural variables, we performed Herrera's scheme for the orthogonal cracking of Riemann tensor. In this endeavor, we are mainly concerned with the issue of relativistic gravitational collapse of the dynamically relativistic spheres fulfilling the presumption of minimal $\mathbb{C}_{\mathbf{F}}$. The incorporation of a less restrictive condition termed as quasi-homologous $(\mathbb{Q}_{\mathbf{H}})$ condition together with the zero $\mathbb{C}_{\mathbf{F}}$, allows us to formulate a range of exact solutions for a particular choice of $f(\mathcal{G}, \mathbf{T}^2)$ model. We find that some of the given exact solutions relax the Darmois junction conditions and describe thin shells by satisfying the Israel conditions, while some exhibit voids by fulfilling the Darmois constraints on both boundary surfaces. Eventually, few expected applications of the provided solutions in the era of modern cosmology are debated.

Sprays of weakly isotropic curvature

Spray geometry and Finsler geometry are related to each other because every Finsler metric on a manifold induces a spray on it. In the past decades, some sprays of scalar curvature and isotropic curvature are studied to tell the relations with Finsler metrics of scalar flag curvature and isotropic flag curvature. It is a natural way to study more new sprays with special curvature to find more interesting properties of sprays and their relations with Finsler metrics. In this paper, we introduce a new notion of curvature named weakly isotropic curvature which is more general than isotropic curvature for sprays. The relationship between this new curvature and the existed curvatures (Riemann curvature, χ-curvature and H-curvature) is revealed. Then we prove this curvature notion is a natural extension of weakly isotropic flag curvature for Finsler metrics. Two new groups of sprays of weakly isotropic curvature not induced by any Finsler metric are constructed based on our result.

Fractional Order Riemann Curvature Tensor in Differential Geometry

This study discussed some interesting aspects and features of fractional curvature in the differential manifold. In particular, Riemannian fractional curvature tensor, Livi-Civita fractional connection and Bianchi fractional identity are presented.

Laplacian on fuzzy de Sitter space

We study details of geometry of noncommutative de Sitter space: we determine the Riemann and Ricci curvature tensors, the energy and the Laplacian. We find, in particular, that fuzzy de Sitter space is an Einstein space, R ab = −3ζη ab . The Laplacian, defined in the noncommutative frame formalism, is not Hermitian and gives nonunitary evolution. When symmetrically ordered, it has the usual quadratic form Δ = Π a Π a (when acting on functions in representation space, ): we find its eigenstates and discuss its spectrum. This result is a first step in a study of the scalar field Laplacian, Δ = [Π a , [Π a , ]], and its propagator.

Curvature conditions for spatial isotropy

In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalized) Robertson-Walker space-time is important. In particular, it is a requirement for the development of initial data to reproduce or approximate the standard cosmological model. Usually these conditions involve the Einstein field equations, which change if one considers alternative theories of gravity or if the coupling matter fields change. Therefore, the derivation of conditions which do not depend on the field equations is an advantage. In this work we present a geometric derivation of such a condition. We require the existence of a unit vector field to distinguish at each point of space two (non-equal) sectional curvatures. This is equivalent for the Riemann tensor to adopt a specific form. Our geometrical approach yields a local isometry between the space and a Robertson-Walker space of the same dimension, curvature and metric tensor sign (the dimension of the largest subspace on which the metric tensor is negative definite). Remarkably, if the space is simply-connected, the isometry is global. Our result generalizes to a class of spaces of non-constant curvature the theorem that spaces of the same constant curvature, dimension and metric tensor sign must be locally isometric. Because we do not make any assumptions regarding field equations, matter fields or metric tensor sign, one can readily use this result to study cosmological models within alternative theories of gravity or with different matter fields.

Complexity of a dynamical dissipative cylindrical system in non-minimally coupled theory

This paper aims to formulate certain scalar factors associated with the matter variables for a self-gravitating non-static cylindrical geometry by considering a standard model R+ζQ of f(R,T,Q) gravity, where Q=RφϑTφϑ and ζ is the arbitrary coupling parameter. We split the Riemann tensor orthogonally to calculate four scalars and deduce YTF as a complexity factor for the fluid configuration. This scalar incorporates the influence of the inhomogeneous energy density, heat flux and pressure anisotropy along with correction terms for the modified gravity. We discuss the dynamics of the cylinder by considering the two simplest modes of structural evolution. We then take YTF=0 with a homologous condition to determine the solution for the dissipative as well as non-dissipative scenarios. Finally, we discuss the criterion under which the complexity-free condition shows stable behavior throughout the evolution. It is concluded that the complex functional of this theory results in a more complex structure.

Dynamical complexity and the gravitational collapse of compact stellar objects

We investigate the dynamics of the gravitational collapse of a compact object via a complexity factor scalar which arises from the orthogonal splitting of the Riemann tensor. This scalar has the property of vanishing for systems which are isotropic in pressure and homogeneous in the energy density. In this way, the complexity factor can give further details of the progression of inhomogeneity as the collapse proceeds. Furthermore, we show that complexity may be used in comparing models and justifying their physical viability. Thus, it could become an integral part of the physical analysis of relativistic collapse in addition to energy conditions analysis, (in)stability, and recently investigated force dynamics.

Einstein Field Equation, Recursion Operators, Noether and Master Symmetries in Conformable Poisson Manifolds

We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute the corresponding deformed recursion operator. Besides, using the Hamiltonian–Jacobi separability, we construct recursion operators for Hamiltonian vector fields in conformable Poisson–Schwarzschild and Friedmann–Lemaître–Robertson–Walker (FLRW) manifolds, and derive the related constants of motion, Christoffel symbols, components of Riemann and Ricci tensors, Ricci constant and components of Einstein tensor. We highlight the existence of a hierarchy of bi-Hamiltonian structures in both the manifolds, and compute a family of recursion operators and master symmetries generating the constants of motion.