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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


Circle Actions

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/9780691191751.003.0020
Subject:
Mathematics, Educational Mathematics

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler ... More


Vector-Valued Forms

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/9780691191751.003.0014
Subject:
Mathematics, Educational Mathematics

This chapter studies vector-valued forms. Ordinary differential forms have values in the field of real numbers. This chapter allows differential forms to take values in a vector space. When the ... More


Integration of Equivariant Forms

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/9780691191751.003.0028
Subject:
Mathematics, Educational Mathematics

This chapter illustrates integration of equivariant forms. An equivariant differential form is an element of the Cartan model. For a circle action on a manifold M, it is a polynomial in u with ... More


Basic Forms

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/9780691191751.003.0012
Subject:
Mathematics, Educational Mathematics

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic ... More


Introductory Lectures on Equivariant Cohomology: (AMS-204)

Loring W. Tu

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.001.0001
Subject:
Mathematics, Educational Mathematics

Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into ... More


The Lie Derivative and Interior Multiplication

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/9780691191751.003.0010
Subject:
Mathematics, Educational Mathematics

This chapter reviews two operations on differential forms, the Lie derivative and interior multiplication. These are necessary to the definition of invariant forms, horizontal forms, and basic forms ... More


Introduction to Kähler Manifolds

Eduardo Cattani and Phillip Griffiths

Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, Lê Dũng Tráng, Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)

in Hodge Theory (MN-49)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161341
eISBN:
9781400851478
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161341.003.0001
Subject:
Mathematics, Geometry / Topology

This chapter provides an introduction to the basic results on the topology of compact Kähler manifolds that underlie and motivate Hodge theory. This chapter consists of five sections which ... More


Calculus on surfaces

Simon Donaldson

in Riemann Surfaces

Published in print:
2011
Published Online:
December 2013
ISBN:
9780198526391
eISBN:
9780191774874
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526391.003.0005
Subject:
Mathematics, Geometry / Topology, Analysis

This chapter develops the theory of differential forms on smooth surfaces and Riemann surfaces. The analysis begins with smooth surfaces, covering cotangent spaces and 1-forms; 2-forms and ... More


The Cartan Model in General

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/9780691191751.003.0021
Subject:
Mathematics, Educational Mathematics

This chapter looks at the Cartan model. Specifically, it generalizes the Cartan model from a circle action to a connected Lie group action. The chapter assumes the Lie group to be connected, because ... 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


Manifolds and tensors

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/9780198822158.003.0004
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, ... More


The Electromagnetic Field

Laurent Baulieu, John Iliopoulos, and Roland Sénéor

in From Classical to Quantum Fields

Published in print:
2017
Published Online:
May 2017
ISBN:
9780198788393
eISBN:
9780191830310
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198788393.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The electromagnetic field. A brief review of Maxwell’s equations with a discussion of their invariance properties, both relativistic and gauge invariance. The formalism of Green’s functions is ... More


Differential Forms

Moataz H. Emam

in Covariant Physics: From Classical Mechanics to General Relativity and Beyond

Published in print:
2021
Published Online:
May 2021
ISBN:
9780198864899
eISBN:
9780191897313
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198864899.003.0009
Subject:
Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

In this chapter we present the modern theory of differential forms and see how it applies to the classical fields studied in the previous chapter. We apply the theory to Maxwell fields as well as to ... More


Covariant Physics: From Classical Mechanics to General Relativity and Beyond

Moataz H. Emam

Published in print:
2021
Published Online:
May 2021
ISBN:
9780198864899
eISBN:
9780191897313
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198864899.001.0001
Subject:
Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

This book is an introduction to the modern methods of the general theory of relativity, tensor calculus, space time geometry, the classical theory of fields, and a variety of theoretical physics ... More


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