Riemann Tensor Research Articles

On non‐Hopf Ricci‐pseudosymmetric hypersurfaces in CP2$\mathbb {C}P^{2}$ and CH2$\mathbb {C}H^{2}$

AbstractIn this paper, we study an open problem raised by Cecil and Ryan [Geometry of Hypersurfaces, Springer Monographs in Mathematics, p. 531] which asked whether there exist non‐Hopf Ricci‐pseudosymmetric hypersurfaces in and . As our main results, we first prove the nonexistence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in . Then, we prove the existence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in . Finally, applying the preceding results and sharpening Theorem 4.1 of Wang and Zhang [J. Geom. Phys. 181 (2022), 104648], we prove the nonexistence of non‐Hopf weakly Einstein hypersurfaces with constant norm of Riemannian curvature tensor in both and .

Geometric properties of almost pure metric plastic pseudo-Riemannian manifolds

This paper investigates the geometric and structural properties of almost plastic pseudo-Riemannian manifolds, with a specific focus on three-dimensional cases. We explore the interplay between an almost plastic structure and a pseudo-Riemannian metric, providing a comprehensive analysis of the conditions that define pure metric plastic P-Kählerian manifolds. In this context, the fundamental tensor field is symmetric and also represents another pure metric. A key finding is the necessary and sufficient condition for the integrability of the plastic structure by means of partial differential equations, which is characterized by a specific function depending solely on one variable. Furthermore, the necessary and sufficient condition for an almost pure metric plastic pseudo-Riemannian manifold to be pure metric plastic P-Kählerian is obtained. The study also reveals that the Riemannian curvature tensor vanishes under specific conditions, while the scalar curvature vanishes if a particular polynomial form is satisfied. The analysis extends to the properties of vector fields, identifying conditions under which they become Killing vector fields or form Ricci soliton structures. Additionally, we examine three-dimensional Walker manifolds, detailing the conditions for vanishing scalar curvature, Killing vector fields, and Ricci soliton structures. Our findings provide a detailed framework for understanding almost plastic manifolds, contributing to the broader field of differential geometry by clarifying the relationship between plastic structures and pseudo-Riemannian metrics.

Lie group geometry: Riemann and Ricci tensors and normal forms of Lie algebras

Lie group geometry: Riemann and Ricci tensors and normal forms of Lie algebras

Exploring Torus Black-Holes In (1+3)-Dimensions: A Novel Approach to Higher Genus Solution

A torus black-hole solution of the vacuum gravitational field equation of general relativity in (1 + 3)-dimensions is obtained. Starting with a metricansatz associated with the torus, our method is based on straight forward computations the usual geometric mathematical tools of the Christoffel symbols and the Riemann tensor. Specifically, after deriving such mathematical tools the field equations of general relativity are considered. The resultatning equations are properly combained to find the solution. Moreover, the novelty and potential implications of this solution emerges from the fact that is based on a coordinate transformation metric ansatz. This provides with broad implications and future research directions. In particular we argue that our formalism can properly be used for a search of higher genus black-hole solution.

On the non-flatness nature of noncommutative Minkowski spacetime and the singular behavior of probes

It is more than a century-old concept that the Minkowski spacetime is flat. From the pure geometric point of view, we explicitly address the issue of whether a noncommutative Minkowski spacetime is flat or not. In the framework of the twisted-diffeomorphism approach to noncommutative gravity with canonical type noncommutative (NC) coordinate structure, one important result that we get is that the NC Minkowski spacetime parametrized either with spherical polar coordinates or with parabolic coordinates has nontrivial NC corrections to Riemann curvature tensor, Ricci tensor and curvature scalars. Another crucial result is that the curvature scalars have singular behavior at certain points, and these singularities are not coordinate singularities. The nature of these singularities clearly points towards the idea of high-energetic probes turning into black holes. The absence of any such noncommutative corrections, and thus any such singularities, in the cases of Cartesian coordinates and cylindrical coordinates leads to the conclusion that high-energetic probes do not turn into black holes when the canonical NC structure is considered for these coordinates.

Curvature and Weitzenböck formula for spectral triples

AbstractUsing the Levi‐Civita connection on the noncommutative differential 1‐forms of a spectral triple , we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac spectral triples and derive a general Weitzenböck formula for them. We apply these tools to ‐deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under ‐deformation, whereas the connection Laplacian, Clifford representation of the curvature, and the scalar curvature are all invariant under deformation.

Complexity factor for a static self-gravitating sphere in Rastall–Rainbow gravity

We generalized Herrera’s definition of complexity factor for static spherically symmetric fluid distributions to Rastall–Rainbow theory of gravity. For this purpose, an energy-dependent equation of motion is employed in accordance with the principle of gravity’s rainbow. It is found that the complexity factor appears in the orthogonal splitting of the Riemann curvature tensor, and measures the deviation of the value of the active gravitational mass from the simplest system under the combined corrections of Rastall and rainbow. In the low-energy limit, all the results we have obtained reduce to the counterparts of general relativity when the non-conserved parameter is taken to be one. We also demonstrate how to build an anisotropic or isotropic star model using complexity approach. In particular, the vanishing complexity factor condition in Rastall–Rainbow gravity is exactly the same as that derived in general relativity. This fact may imply a deeper geometric foundation for the complexity factor.

Analytical solutions of planar geometry with zero complexity in extended gravity

In this paper, we discuss various analytical solutions of the planner geometry with anisotropic fluid as well as vanishing complexity factor condition in the framework of [Formula: see text] theory. The matching conditions are employed at the interior and exterior hypersurfaces. The connection between the matter parameters and the Weyl tensor has been constructed. For dissipative and non-dissipative structures, we examine four different conformal Killing vector possibilities, while some possibilities adhere to the [Formula: see text] condition along with the matching conditions, where [Formula: see text] indicates the trace free part of the electric component of the Riemann tensor. Several theoretical models are provided to show how a non-static system evolves.

Engineering flexible machine learning systems by traversing functionally invariant paths

Contemporary machine learning algorithms train artificial neural networks by setting network weights to a single optimized configuration through gradient descent on task-specific training data. The resulting networks can achieve human-level performance on natural language processing, image analysis and agent-based tasks, but lack the flexibility and robustness characteristic of human intelligence. Here we introduce a differential geometry framework—functionally invariant paths—that provides flexible and continuous adaptation of trained neural networks so that secondary tasks can be achieved beyond the main machine learning goal, including increased network sparsification and adversarial robustness. We formulate the weight space of a neural network as a curved Riemannian manifold equipped with a metric tensor whose spectrum defines low-rank subspaces in weight space that accommodate network adaptation without loss of prior knowledge. We formalize adaptation as movement along a geodesic path in weight space while searching for networks that accommodate secondary objectives. With modest computational resources, the functionally invariant path algorithm achieves performance comparable with or exceeding state-of-the-art methods including low-rank adaptation on continual learning, sparsification and adversarial robustness tasks for large language models (bidirectional encoder representations from transformers), vision transformers (ViT and DeIT) and convolutional neural networks.

Quantum extremal modular curvature: modular transport with islands

Modular Berry transport is a useful way to understand how geometric bulk information is encoded in the boundary CFT: the modular curvature is directly related to the bulk Riemann curvature. We extend this approach by studying modular transport in the presence of a non-trivial quantum extremal surface. Focusing on JT gravity on an AdS background coupled to a non-gravitating bath, we compute the modular curvature of an interval in the bath in the presence of an island: the Quantum Extremal Modular Curvature (QEMC). We highlight some important properties of the QEMC, most importantly that it is non-local in general. In an OPE limit, the QEMC becomes local and probes the bulk Riemann curvature in regions with an island. Our work gives a new approach to probe physics behind horizons.

Magnetic Curvature and Existence of a Closed Magnetic Geodesic on Low Energy Levels

Abstract To a Riemannian manifold $(M,g)$ endowed with a magnetic form $\sigma $ and its Lorentz operator $\Omega $ we associate an operator $M^{\Omega }$, called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric $g$ together with terms of perturbation due to the magnetic interaction of $\sigma $. From $M^{\Omega }$ we derive the magnetic sectional curvature $\textrm{Sec}^{\Omega }$ and the magnetic Ricci curvature $\textrm{Ric}^{\Omega }$ that generalize in arbitrary dimension the already known notion of magnetic curvature previously considered by several authors on surfaces. On closed manifolds, under the assumption of $\textrm{Ric}^{\Omega }$ being positive on an energy level below the Mañé critical value, with a Bonnet–Myers argument, we establish the existence of a contractible periodic orbit. In particular, when $\sigma $ is nowhere vanishing, this implies the existence of a contractible periodic orbit on every energy level close to zero. Finally, on closed oriented even dimensional manifolds, we discuss about the topological restrictions that appear when one requires $\textrm{Sec}^{\Omega }$ to be positive.

A phenomenological approach to the dark energy models in the Finsler–Randers framework

This study extends two phenomenological models of dark energy within the framework of Finsler–Randers space–time, accommodating anisotropies. The models consider the cosmological constant Λ in two scenarios: one where Λ is proportional to the second time derivative of the scale factor ä, and another where it varies with the matter-energy density ρ. Earlier, such Λ-decaying cosmologies were proposed to address long-standing cosmological constant problems. However, following the discovery of late-time cosmic acceleration, the focus shifted to modeling dark energy. Since Λ is widely viewed as the most significant and suitable candidate for driving cosmic acceleration, it is worthwhile to revisit the phenomenological approach in this context. This work uses this approach to find solutions for the scale factor a(t) and other geometrical and physical parameters. Additionally, we analyze the evolution of density parameters Ωm, ΩΛ, and Ωκ, representing matter, dark energy, and curvature, from early to late times. The phenomenological approach is employed to solve the field equations, with model parameters constrained using recent observational data, yielding ranges consistent with observations. The solutions converge to the ΛCDM cosmology at early and late times. The added complexity introduced by Finsler–Randers geometry enhances accuracy compared to analogous solutions in Riemannian space–time.

A note on field equations in generalized theories of gravity

In the work [Phys. Rev. D 84, 124 041 (2011)], to obtain a simple and economic formulation of field equations for generalised theories of gravity described by the Lagrangian −gLgαβ,Rμνρσ , the key equality ∂L/∂gμνRαβκω=2PμλρσRνλρσ was derived. In this note, it is demonstrated that such an equality can be directly derived from an off-shell Noether current associated with an arbitrary vector field. As byproducts, a generalized Bianchi identity related to the divergence for the expression of field equations, together with the Noether potential, is obtained. On the basis of the above, we further propose a systematic procedure to derive the equations of motion from the Noether current, and then this procedure is extended to more general higher-order gravities endowed with the Lagrangian encompassing additional terms of the covariant derivatives of the Riemann tensor. To our knowledge, both the detailed expressions for field equations and the Noether potential associated with such theories are first given at a general level. All the results reveal that using the Noether current to determine field equations establishes a straightforward connection between the symmetry of the Lagrangian and the equations of motion and such a remedy even can avoid calculating the derivative of the Lagrangian density with respect to the metric.

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The C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^3$$\\end{document} approach is an invariant formalism that utilizes the eigenvalues of the Riemann curvature tensor to match spacetimes across a specific matching surface. We apply this approach to match an anisotropic fluid with an exterior vacuum solution, including the case in which discontinuities appear on the matching surface. As a particular example, a class of analytic solutions, which describe the gravitational field of realistic neutron stars, is matched to the exterior Schwarzschild spacetime.

On Riemann Curvature of Spherically Symmetric Metrics

On Riemann Curvature of Spherically Symmetric Metrics

Strong Cosmic Censorship with bounded curvature

Abstract In this paper we propose a weaker version of Penrose’s much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextendability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in Lp. Lipschitz continuity is the threshold regularity for causal structures, and curvature bounds rule out infinite tidal accelerations, arguing for physical significance of this weaker SCC conjecture. The main result of this paper, under the assumption that no extensions exist with higher connection regularity W1,p loc, proves in the affirmative this SCC conjecture with bounded curvature for p sufficiently large, (p > 4 with uniform bounds, p > 2 without uniform bounds).

Scalar induced gravitational waves in chiral scalar–tensor theory of gravity

We study the scalar induced gravitational waves (SIGWs) from a chiral scalar–tensor theory of gravity. The parity-violating (PV) Lagrangian contains the Chern–Simons (CS) term and PV scalar–tensor terms, which are built of the quadratic Riemann tensor term and first-order derivatives of a scalar field. We consider SIGWs in two cases, in which the semi-analytic expression to calculate SIGWs can be obtained. Then, we calculate the fractional energy density of SIGWs with a monochromatic power spectrum for the curvature perturbation. We find that the SIGWs in chiral scalar–tensor gravity behave differently from those in GR before and after the peak frequency, which results in a large degree of circular polarization.

Geometric Amplitude Accompanying Local Responses: Spinor Phase Information from the Amplitudes of Spin-Polarized STM Measurements.

Solving the Hamiltonian of a system yields the energy dispersion and eigenstates. The geometric phase of the eigenstates generates many novel effects and potential applications. However, the geometric properties of the energy dispersion go unheeded. Here, we provide geometric insight into energy dispersion and introduce a geometric amplitude, namely, the geometric density of states (GDOS) determined by the Riemann curvature of the constant-energy contour. The geometric amplitude should accompany various local responses, which are generally formulated by the real-space Green's function. Under the stationary phase approximation, the GDOS simplifies the Green's function into its ultimate form. In particular, the amplitude factor embodies the spinor phase information of the eigenstates, favoring the extraction of the spin texture for topological surface states under an in-plane magnetic field through spin-polarized STM measurements. This work opens a new avenue for exploring the geometric properties of electronic structures and excavates the unexplored potential of spin-polarized STM measurements to probe the spinor phase information of eigenstates from their amplitudes.

Gravitons in a gravitational plane wave

Gravitational plane waves (when Ricci flat) belong to the VSI family. The achronym VSI stands for vanishing scalar invariants, meaning that all scalar invariants built out of Riemann tensor and its derivatives vanish, although the Riemann tensor itself does not. In the particular case of plane waves many interesting phenomena have been uncovered for strings propagating in this background. Here we comment on gravitons propagating in such a spacetime.

A Causal Net approach to Relativistic Quantum Mechanics and Gravitation

In this paper we discuss a causal network model to describe quantum mechanics and gravittation, where space–time is built solely from point events and connecting probabilities. A causal net provides a representation which combines the quantum complementary features of both partticles and waves as each vertex on the causal net represents a possible point event or particle observation. Simultaneity on the causal net is defined by hyperplanes of equivalent proper time. The causal net model when constructed in Minkowksi space–time is shown to lead to a formulation for relativistic quantum mechanics including the Dirac equation. In a curved Riemann space–time the causal paths of the causal net are geodesics and in the local Lorentz frame the net probabilities and the Dirac formalism are preserved. The variation of space–time density of events in the causal paths modifies the metric and provides a space–time curvature leading to the Hilbert action associated with general relativity. Consideration of the Schwarzchild metric shows that in classical limit the perturbation in the density of possible events is proportional to the Newtonian gravitational potential.