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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


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


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


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


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


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


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


Classical And Quantum Gravity. Riemannian Manifolds And Tensors

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/9780198509233.003.0022
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter has two purposes; to present the few elements of differential geometry which are required in different places in this volume and to provide, for completeness, a short introduction to the ... 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


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