## Links as Sasakian Manifolds

*Charles P. Boyer and Krzysztof Galicki*

### in Sasakian Geometry

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0010
- Subject:
- Mathematics, Geometry / Topology

This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The ... More

## Harmonic morphisms from compact 3-manifolds

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0010
- Subject:
- Mathematics, Pure Mathematics

This chapter shows that a non-constant harmonic morphism with compact fibres from a 3-manifold endows it with the structure of a Seifert fibre space — a 3-manifold with a certain type of ... More

## Curvature considerations

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0011
- Subject:
- Mathematics, Pure Mathematics

The curvature of the domain on various combinations of horizontal and vertical vectors is calculated for a horizontally conformal submersion and for a (submersive) harmonic morphism. It is related to ... More

## Fundamental properties of harmonic morphisms

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0004
- Subject:
- Mathematics, Pure Mathematics

This chapter characterizes harmonic morphisms between Riemannian manifolds and discusses their basic properties. The first non-constant term of the Taylor series of a harmonic morphism at a point ... More

## Mini-twistor theory on three-dimensional space forms

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0006
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses harmonic morphisms from 3-manifolds to surfaces, especially those of constant curvature. First, the local behaviour is considered, showing that the foliation given by its ... More

## Riemann Surfaces, Coverings, and Hypergeometric Functions

*Paula Tretkoff*

### in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and ... More

## Lorentz Geometry

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0001
- Subject:
- Mathematics, Applied Mathematics

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local ... More

## THE 3+1 FORMALISM

*Miguel Alcubierre*

### in Introduction to 3+1 Numerical Relativity

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

There are several different approaches to the problem of separating the Einstein field equations in a way that allows us to give certain initial data, and from there obtain the subsequent evolution ... More

## Time warps and bent space

*Louis A. Girifalco*

### in The Universal Force: Gravity - Creator of Worlds

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228966
- eISBN:
- 9780191711183
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228966.003.0015
- Subject:
- Physics, History of Physics

General relativity is the extension of the ideas of relativity to accelerated motion. It starts with the premise that the laws of physics should be the same for all, whatever the state of motion. It ... More

## Differential Geometry: Bundles, Connections, Metrics and Curvature

*Clifford Henry Taubes*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.001.0001
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

Bundles, connections, metrics, and curvature are the ‘lingua franca’ of modern differential geometry and theoretical physics. Many of the tools used in differential topology are introduced and the ... More

## Curvature on a Principal Bundle

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0017
- Subject:
- Mathematics, Educational Mathematics

This chapter examines curvature on a principal bundle. The curvature of a connection on a principal G-bundle is a g-valued 2-form that measures, in some sense, the deviation of the connection from ... More

## Grouping and Early Visual Processing in Avian Vision

*Robert G. Cook and Carl Erick Hagmann*

### in How Animals See the World: Comparative Behavior, Biology, and Evolution of Vision

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780195334654
- eISBN:
- 9780199933167
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334654.003.0004
- Subject:
- Psychology, Cognitive Neuroscience, Cognitive Psychology

This chapter examines early visual processing in birds revealed by the analysis of multi-element textured arrays. Pigeons, and presumably most other species of birds, share many common properties ... More

## Space

*Bede Rundle*

### in Time, Space, and Metaphysics

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199575114
- eISBN:
- 9780191722349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575114.003.0010
- Subject:
- Philosophy, Metaphysics/Epistemology

Absolutism with respect to space is subject to much the same lines of criticism as is absolutism with respect to time, and while with space we also have Kant's arguments concerning left and right ... More

## Brightness and Contrast

*Nigel Daw*

### in How Vision Works: The Physiological Mechanisms Behind What We See

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199751617
- eISBN:
- 9780199932375
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199751617.003.0003
- Subject:
- Psychology, Cognitive Neuroscience, Cognitive Psychology

The lightness of an object is determined in relation to the objects around it. The overall illumination of the scene is ignored, and so are shadows falling across the object, and curvature of the ... More

## Metric description of a curved space

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. ... More

## The three body problem

*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/9780199670857.003.0011
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter derives the consequences of scale invariance in the three body problem. Jacobi co-ordinates are introduced. The orbits are geodesics in a certain metric. The three body problem with ... More

## Tensors in general relativity

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0013
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Differentiation of tensor components in a curved space must be handled with extra care. By adding another term (related to Christoffel symbols) to the ordinary derivative operator, we can form a ... More

## The Ambient Metric (AM-178)

*Charles Fefferman and C. Robin Graham*

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

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient ... More

## Hertz’ s geometry of systems of points

*JESPER LÜTZEN*

### in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567370
- eISBN:
- 9780191717925
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567370.003.0013
- Subject:
- Physics, History of Physics

The relation between mechanics and geometry has changed dramatically over the last 350 years. The first physicist who used differential geometry in mechanics was Heinrich Hertz, and other physicists ... More

## Initialization and Extension (Theorems M6, M7, M8)

*Sergiu Klainerman and Jérémie Szeftel*

### in Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations: (AMS-210)

- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691212425
- eISBN:
- 9780691218526
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691212425.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter focuses on the proof for Theorem M6 concerning initialization, Theorem M7 concerning extension, and Theorem M8 concerning the improvement of higher order weighted energies. It first ... More