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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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