Curvature

2021 ◽  
pp. 189-212
Author(s):  
Andrew M. Steane
Keyword(s):  
Curvature Tensor ◽  
Torsion Tensor ◽  
Parallel Transport ◽  
Lie Derivative ◽  
Riemannian Curvature ◽  
Riemann Curvature Tensor ◽  
Geodesic Deviation ◽  
Minkowski Metric ◽  
Covariant Derivatives ◽  
Global Properties

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.

scholarly journals On The Formulation of Bianchi Identity from Action Principle

2021 ◽  
Author(s):  
Shiladittya Debnath
Keyword(s):  
Curvature Tensor ◽  
Basic Property ◽  
Covariant Derivative ◽  
Bianchi Identity ◽  
Action Principle ◽  
Lie Derivative ◽  
General Covariance ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor

Abstract 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’.

scholarly journals Geometry of Nonholonomic Kenmotsu Manifolds

2021 ◽  
pp. 84-87
Author(s):  
A.V. Bukusheva
Keyword(s):  
Curvature Tensor ◽  
Einstein Manifold ◽  
Lie Derivative ◽  
Riemann Curvature Tensor ◽  
Intrinsic Geometry ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Schouten Tensor ◽  
Kenmotsu Manifolds ◽  
Formal Properties

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.

scholarly journals CONSTRAINING SPACETIME TORSION WITH THE MOON, MERCURY AND LAGEOS

2013 ◽  
Vol 53 (A) ◽  
pp. 817-820
Author(s):  
Riccardo March ◽  
Giovanni Bellettini ◽  
Roberto Tauraso ◽  
Simone Dell’Agnello
Keyword(s):  
General Relativity ◽  
Solar System ◽  
Curvature Tensor ◽  
Torsion Tensor ◽  
Test Body ◽  
The Moon ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor

We consider an extension of Einstein General Relativity where, beside the Riemann curvature tensor, we suppose the presence of a torsion tensor. Using a parametrized theory based on symmetry arguments, we report on some results concerning the constraints that can be put on torsion parameters by studying the orbits of a test body in the solar system.

scholarly journals Consistent curvature approximation on Riemannian shape spaces

2021 ◽  
Author(s):  
Alexander Effland ◽  
Behrend Heeren ◽  
Martin Rumpf ◽  
Benedikt Wirth
Keyword(s):  
Curvature Tensor ◽  
Vector Bundles ◽  
Parallel Transport ◽  
Time Discretization ◽  
Second Order ◽  
Shape Spaces ◽  
First Order ◽  
Covariant Derivatives ◽  
Low Dimensional ◽  
Curvature Approximation

Abstract We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end we extend the variational time discretization of geodesic calculus presented in Rumpf & Wirth (2015, Variational time discretization of geodesic calculus. IMA J. Numer. Anal., 35, 1011–1046), which just requires an approximation of the squared Riemannian distance that is typically easy to compute. First we obtain first-order discrete covariant derivatives via Schild’s ladder-type discretization of parallel transport. Second-order discrete covariant derivatives are then computed as nested first-order discrete covariant derivatives. These finally give rise to an approximation of the curvature tensor. First- and second-order consistency are proven for the approximations of the covariant derivative and the curvature tensor. The findings are experimentally validated on two-dimensional surfaces embedded in ${\mathbb{R}}^3$. Furthermore, as a proof of concept, the method is applied to a space of parametrized curves as well as to a space of shell surfaces, and discrete sectional curvature confusion matrices are computed on low-dimensional vector bundles.

Derivatives and curvature

2019 ◽  
pp. 37-51
Author(s):  
Steven Carlip
Keyword(s):  
General Relativity ◽  
Curvature Tensor ◽  
Differential Forms ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Tensor Calculus ◽  
Covariant Derivatives ◽  
Civita Connection ◽  
Physical Picture ◽  
Levi Civita Connection

This chapter develops tensor calculus: integration on manifolds, Cartan calculus for differential forms, connections and covariant derivatives, and the Levi-Civita connection used in general relativity. It then introduces the Riemann curvature tensor in several different ways, including the most directly physical picture of the curvature as a measure of the convergence of neighboring geodesics. The chapter concludes with a discussion of Cartan’s beautiful formulation of the connection and curvature in the language of differential forms.

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

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe 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.

CIRCULAR ORBITS AROUND SCHWARZSCHILD–AdS SPACETIME

2004 ◽  
Vol 19 (36) ◽  
pp. 2683-2695 ◽  
Author(s):  
A. M. DE M. CARVALHO ◽  
CLAUDIO FURTADO ◽  
FERNANDO MORAES
Keyword(s):  
Band Structure ◽  
Parallel Transport ◽  
Circular Orbits ◽  
Global Properties ◽  
Ads Spacetime

In this paper we discuss parallel transport of vectors and spinors around circular orbits in Schwarzschild–AdS spacetime. We study the global properties of this spacetime via loop variables or holonomy. A set of paths in this background is considered. We demonstrate that for some special radii there appears the so-called quantized band structure of holonomy invariance. This analysis is also extended to parallel transport of a spinor in this spacetime.

scholarly journals On semisymmetric connection

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1179-1184
Author(s):  
Ana Velimirovic ◽  
Milan Zlatanovic
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
The Other ◽  
Linear Combinations ◽  
Covariant Derivatives ◽  
Linearly Independent ◽  
Curvature Tensors

Using the non-symmetry of a connection, it is possible to introduce four types of covariant derivatives. Based on these derivatives, several types of Ricci?s identities and twelve curvature tensors are obtained. Five of them are linearly independent but the other curvature tensors can be expressed as linear combinations of these five linearly independent curvature tensors and the curvature tensor of the corresponding associated symmetric space. The semisymmetric connection is defined and the properties of two of the five independent curvature tensors are analyzed. In the same manner, the properties for three others curvature tensors may be derived.

scholarly journals On B- Covariant Derivative of First Order for Some Tensors in different Spaces

2021 ◽  
Vol 2 (2) ◽  
pp. 30-37
Author(s):  
Alaa A. Abdallah ◽  
A. A. Navlekar ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivative ◽  
Torsion Tensor ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Order ◽  
Recurrence Property ◽  
Necessary And Sufficient ◽  
The Relationship

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

Sign in / Sign up

Export Citation Format

Share Document