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Covariant derivatives and metrics

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0015
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines relations between covariant derivatives and metrics. It covers metric compatible covariant derivatives; torsion free covariant derivatives on T*M; the Levi-Civita ... More


Riemannian Geometry

Valeri P. Frolov and Andrei Zelnikov

in Introduction to Black Hole Physics

Published in print:
2011
Published Online:
January 2012
ISBN:
9780199692293
eISBN:
9780191731860
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199692293.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

The key notions of the differential and Riemannian geometry necessary for understanding the General Relativity are introduced here. We provide the reader with the necessary tools for study the ... More


Tensor Formalism for General Relativity

Ta-Pei Cheng

in A College Course on Relativity and Cosmology

Published in print:
2015
Published Online:
August 2015
ISBN:
9780199693405
eISBN:
9780191803130
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199693405.003.0011
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter introduces the basic tensor formalism needed for a proper formulation of general relativity. In a curved space, one must work with the covariant derivative, which is a combination of the ... More


Covariant derivatives, connections and curvature

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0012
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines the notion of the curvature of a covariant derivative or connection. It begins by describing two notions involving differentiation of differential forms and vector fields that ... More


Curved Space

Nicholas Manton and Nicholas Mee

in The Physical World: An Inspirational Tour of Fundamental Physics

Published in print:
2017
Published Online:
July 2017
ISBN:
9780198795933
eISBN:
9780191837111
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198795933.003.0006
Subject:
Physics, Condensed Matter Physics / Materials

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that ... More


The covariant derivative and the curvature

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0063
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter first considers the tangent spaces of a non-connected manifold, in which the tangent t at the set of points p in the manifold is an element of the tangent space at p. Afterward, the ... More


Some elements of general relativity

Rodolfo Gambini and Jorge Pullin

in A First Course in Loop Quantum Gravity

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199590759
eISBN:
9780191774980
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199590759.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter introduces the idea that the gravitational field is described as a deformation of space-time. It briefly discusses general coordinates, and the definitions of vectors and tensors ... More


Derivatives and curvature

Steven Carlip

in General Relativity: A Concise Introduction

Published in print:
2019
Published Online:
March 2019
ISBN:
9780198822158
eISBN:
9780191861215
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198822158.003.0005
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

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 ... More


Curvilinear coordinates

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter presents a discussion on curvilinear coordinates in line with the introduction on Cartesian coordinates in Chapter 1. First, the chapter introduces a new system C of curvilinear ... More


Spacetime symmetries

Michael Kachelriess

in Quantum Fields: From the Hubble to the Planck Scale

Published in print:
2017
Published Online:
February 2018
ISBN:
9780198802877
eISBN:
9780191841330
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802877.003.0006
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the ... More


Differential geometry

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0004
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant ... More


Two-Dimensional Models and Bosonization Method

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0032
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 31 discussed the generic O(N) non-linear σ-model. We have noticed that the abelian case N = 2 is special because the RG β-function vanishes in two dimensions. The corresponding O(2) invariant ... More


Riemannian manifolds

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0064
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

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 ... More


ε‎-Invariance

Ercüment H. Ortaçgil

in An Alternative Approach to Lie Groups and Geometric Structures

Published in print:
2018
Published Online:
September 2018
ISBN:
9780198821656
eISBN:
9780191860959
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821656.003.0006
Subject:
Mathematics, Pure Mathematics

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.


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