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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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