Curvature

The mathematics of Riemannian curvature is presented. The Riemann curvature tensor and its role in parallel transport, in the metric, and in geodesic deviation are expounded at length. We begin by defining the curvature tensor and the torsion tensor by relating them to covariant derivatives. Then the local metric is obtained up to second order in terms of Minkowski metric and curvature tensor. Geometric issues such as the closure or non-closure of parallelograms are discussed. Next, the relation between curvature and parallel transport around a loop is derived. Then we proceed to geodesic deviation. The influence of global properties of the manifold on parallel transport is briefly expounded. The Lie derivative is then defined, and isometries of spacetime are discussed. Killing’s equation and properties of Killing vectors are obtained. Finally, the Weyl tensor (conformal tensor) is introduced.

R+RG gravity with maximal Noether symmetry

A Noether symmetric, 3rd order polynomial in the Riemann curvature tensor R αβμν extension of the General Relativity (GR) without cosmological constant (R+RG gravity) is suggested and discussed as a possible fundamental theory of gravity in 4-dimensional space-time with the geometric part of the Lagrangian to be L R + R G = − g 2 k R ( 1 + G G P ) . Here k = 8 π G N c 4 is the Einstein constant, g = det ( g μ ν ) , g μ ν - the metric tensor, GN - the Newton constant, c - the speed of light, R = R μ ν μ ν - the Ricci scalar, G = R 2 − 4 R μ ν R μ ν + R α β μ ν R α β μ ν - the Gauss-Bonnet topological invariant, and GP - a new constant of the gravitational self-interaction to model the cosmological bounce, inflation, accelerated expansion of the Universe, etc. The best fit to the Baryon Acoustic Oscillations data for the Hubble parameter H (z) at the redshifts z<2.36 leads to G P 1 / 4 = ( 0.557 ± 0.014 ) T p c − 1 with the mean square weighted deviation from the data about 3 times smaller than for the standard cosmological (ΛCDM) model. Due to the self-gravitating term ∼RG the respective Einstein equation in the R+RG gravity contains the additional (tachyonic in the past and now) scalar (spin = 0) graviton and the perfect geometric fluid tensor with pressure-and matter-like terms equal to the respective terms in the ΛCDM model at |z| 1. Some predictions of this R+RG gravity for the Universe are also done.

Generalized Lemaítre–Tolman–Bondi spacetime under the influence of electric charge and Palatini f(R) gravity

The objective of this article is to investigate the effects of electromagnetic field on the generalization of Lemaître–Tolman–Bondi (LTB) spacetime by keeping in view the Palatini f(R) gravity and dissipative dust fluid. For performing this analysis, we followed the strategy deployed by Herrera et al. (Phys Rev D 82(2):024021, 2010). We have explored the modified field equations along with kinematical quantities and mass function and constructed the evolution equations to study the dynamics of inhomogeneous universe along with Raychauduary and Ellis equations. We have developed the relation for Palatini f(R) scalar functions by splitting the Riemann curvature tensor orthogonally and associated them with metric coefficients using modified field equations. We have formulated these scalar functions for LTB and its generalized version, i.e., GLTB under the influence of charge. The properties of GLTB spacetime are consistent with those of the LTB geometry and the scalar functions found in both cases are comparable in the presence of charge and Palatini f(R) curvature terms. The symmetric properties of generalized LTB spacetime are also studied using streaming out and diffusion approximations.

Influence of f(G) gravity on the complexity of relativistic self-gravitating fluids

In this paper, we extend the notion of complexity for the case of nonstatic self-gravitating spherically symmetric structures within the background of modified Gauss–Bonnet gravity (i.e. [Formula: see text] gravity), where [Formula: see text] denotes the Gauss–Bonnet scalar term. In this regard, we have formulated the equations of gravity as well as the relations for the mass function for anisotropic matter configuration. The Riemann curvature tensor is broken down orthogonally through Bel’s procedure to compose some modified scalar functions and formulate the complexity factor with the help of one of the scalar functions. The CF (i.e. complexity factor) comprehends specific physical variables of the fluid configuration including energy density inhomogeneity and anisotropic pressure along with [Formula: see text] degrees of freedom. Moreover, the impact of the dark source terms of [Formula: see text] gravity on the system is analyzed which revealed that the complexity of the fluid configuration is increased due to the modified terms.

On The Formulation of Bianchi Identity from Action Principle

In this letter, we investigate the basic property of the Hilbert-Einstein action principle and its infinitesimal variation under suitable transformation of the metric tensor. We find that for the variation in action to be invariant, it must be a scalar so as to obey the principle of general covariance. From this invariant action principle, we eventually derive the Bianchi identity (where, both the 1st and 2nd forms are been dissolved) by using the Lie derivative and Palatini identity. Finally, from our derived Bianchi identity, splitting it into its components and performing cyclic summation over all the indices, we eventually can derive the covariant derivative of the Riemann curvature tensor. This very formulation was first introduced by S Weinberg in case of a collision less plasma and gravitating system. We derive the Bianchi identity from the action principle via this approach; and hence the name ‘Weinberg formulation of Bianchi identity’.

Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

We study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

Geometry of Nonholonomic Kenmotsu Manifolds

The concept of the intrinsic geometry of a nonholonomic Kenmotsu manifold M is introduced. It is understood as the set of those properties of the manifold that depend only on the framing of the D^ distribution D of the manifold M, on the parallel transformation of vectors belonging to the distribution D along curves tangent to this distribution. The invariants of the intrinsic geometry of the nonholonomic Kenmotsu manifold are: the Schouten curvature tensor; 1-form η generating the distribution D; the Lie derivative of the metric tensor g along the vector field ; Schouten — Wagner tensor field P, whose components in adapted coordinates are expressed using the equalities . It is proved that, as in the case of the Kenmotsu manifold, the Schouten — Wagner tensor of the manifold M vanishes. It follows that the Schouten tensor of a nonholonomic Kenmotsu manifold has the same formal properties as the Riemann curvature tensor. It is proved that the alternation of the Ricci — Schouten tensor coincides with the differential of the structural form. This property of the Ricci — Schouten tensor is used in the proof of the main result of the article: a nonholonomic Kenmotsu manifold cannot carry the structure of an η-Einstein manifold.

Curvature Identities for Generalized Kenmotsu Manifolds

In the present paper we obtained 2 identities, which are satisfied by Riemann curvature tensor of generalized Kenmotsu manifolds. There was obtained an analytic expression for third structure tensor or tensor of f-holomorphic sectional curvature of GK-manifold. We separated 2 classes of generalized Kenmotsu manifolds and collected their local characterization.

How to smooth a crinkled map of space–time: Uhlenbeck compactness for L ∞ connections and optimal regularity for general relativistic shock waves by the Reintjes–Temple equations

We present the authors’ new theory of the RT-equations (‘regularity transformation’ or ‘Reintjes–Temple’ equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem( Γ ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle T M of arbitrary manifolds M , including Lorentzian manifolds of general relativity.

A new formula for conserved charges of Lovelock gravity in AdS space–times and its generalization

Within the framework of the Lovelock gravity theory, we propose a new rank-four divergenceless tensor consisting of the Riemann curvature tensor and inheriting its algebraic symmetry characters. Such a tensor can be adopted to define conserved charges of the Lovelock gravity theory in asymptotically anti-de Sitter (AdS) space–times. Besides, inspired with the case of the Lovelock gravity, we put forward another general fourth-rank tensor in the context of an arbitrary diffeomorphism invariant theory of gravity described by the Lagrangian constructed out of the curvature tensor. On basis of the newly-constructed tensor, we further suggest a Komar-like formula for the conserved charges of this generic gravity theory.