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.

scholarly journals Covariant derivatives on Kähler C-spaces

1996 ◽  
Vol 143 ◽  
pp. 31-57
Author(s):  
Koji Tojo
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivatives ◽  
Civita Connection ◽  
Levi Civita Connection

Let (M, g) be a Kähler C-space. R and ∇ denote the curvature tensor and the Levi-Civita connection of (M, g), respectively.In [6], Takagi have proved that there exists an integer n such that

Riemannian manifolds

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Covariant Derivative ◽  
Einstein Equations ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Hilbert Action

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to the properties of the curvature tensor. It states without proof the intrinsic versions of the properties of the Riemann–Christoffel tensor of a covariant derivative already given in Chapter 2. This chapter then performs the same derivation as in Chapter 4 by obtaining the Einstein equations of general relativity by varying the Hilbert action. However, this will be done in the intrinsic manner, using the tools developed in the present and the preceding chapters.

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.

On the uniqueness of the quadratic action Lagrangian in general relativity

1970 ◽  
Vol 48 (16) ◽  
pp. 1924-1932
Author(s):  
Borut Gogala
Keyword(s):  
General Relativity ◽  
Linear Combination ◽  
Curvature Tensor ◽  
Lagrangian Density ◽  
Computer Language ◽  
Riemann Curvature ◽  
Field Equations ◽  
Riemann Curvature Tensor ◽  
Quadratic Action ◽  
Uniqueness Proof

A procedure, suitable for translation into computer language and analogous to the uniqueness proof of Einstein's field equations in vierbein formulation, is developed for finding the most general quadratic action Lagrangian density in general relativity that satisfies the requirement of covariance and of invariance under coordinate dependent frame reorientation. It is found that this Lagrangian is identical with the most general linear combination that can be made up out of the three scalars formed by bilinear combinations of the Riemann curvature tensor.

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