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## The Einstein‐Maxwell Equations, Extremal Kähler Metrics, and Seiberg‐Witten Theory

*Claude LeBrun*

### in The Many Facets of Geometry: A Tribute to Nigel Hitchin

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0003
- Subject:
- Mathematics, Geometry / Topology

Nigel Hitchin has played a key role in the exploration of four-dimensional Riemannian geometry, and in particular has made foundational contributions to the theory of self-dual manifolds, ... More

## Gravitation

*Reinhold A. Bertlmann*

### in Anomalies in Quantum Field Theory

- Published in print:
- 2000
- Published Online:
- February 2010
- ISBN:
- 9780198507628
- eISBN:
- 9780191706400
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507628.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on gravitation. It first reviews some basic concepts of the Riemannian geometry and establishes notation. It then discusses the gravitational action, specifically the fermionic ... 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

## Relativity, Gravitation and Cosmology: A Basic Introduction

*Ta-Pei Cheng*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, the principle of equivalence, Riemannian ... More

## Geometry

*Olivier Darrigol*

### in Physics and Necessity: Rationalist Pursuits from the Cartesian Past to the Quantum Present

- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780198712886
- eISBN:
- 9780191781360
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198712886.003.0004
- Subject:
- Physics, History of Physics

This chapter deals with the foundations of geometry, with emphasis on Helmholtz’s approach to this problem in the late 1860s. It is first recalled how the ancient belief in the apodictic truth of the ... More

## Geodesics

*S. G. Rajeev*

### in Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199670857
- eISBN:
- 9780191775154
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199670857.003.0008
- Subject:
- Physics, Atomic, Laser, and Optical Physics

Riemannian geometry deals with distances of infinitesimally close points. A curve of minimal length is a geodesic. This variational principle happens to have the same structure as that of mechanics, ... More

## Riemann Surfaces

*Simon Donaldson*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.001.0001
- Subject:
- Mathematics, Geometry / Topology, Analysis

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an ... More

## Curvature and Instability

*S. G. Rajeev*

### in Fluid Mechanics: A Geometrical Point of View

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805021
- eISBN:
- 9780191843136
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805021.003.0011
- Subject:
- Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

The Euler equations of a rigid body can be understood as the geodesic equations for a metric on the rotation group. A rapid introduction to the Riemannian geometry of Lie groups (following Milnor) is ... More

## Towards a general theory of relativity

*Ta-Pei Cheng*

### in Einstein's Physics: Atoms, Quanta, and Relativity - Derived, Explained, and Appraised

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199669912
- eISBN:
- 9780191744488
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199669912.003.0012
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

In general relativity (GR) physics laws are not changed under general coordinate transformations. Among Einstein’s motivation for GR was his wish to have a deeper understanding of the empirically ... More

## Jet Isomorphism

*Charles Fefferman and C. Robin Graham*

### in The Ambient Metric (AM-178)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0008
- Subject:
- Mathematics, Geometry / Topology

A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered ... More

## General Relativity as a Geometric Theory of Gravity

*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/9780199693405.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

The mathematics of a curved manifold is Riemannian geometry. This chapter first presents some basic elements of Gaussian coordinates and the metric tensor. From the requirement that a curve have the ... More

## Basic notions

*Piotr T. Chruściel*

### in Geometry of Black Holes

- Published in print:
- 2020
- Published Online:
- December 2020
- ISBN:
- 9780198855415
- eISBN:
- 9780191889233
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198855415.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The aim of this chapter is to set the ground for the remainder of this work. We present our conventions and notations in Section 1.1. We review some basic facts about the topology of manifolds in ... More

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