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## BUNDLES

*Andrew Ranicki*

### in Algebraic and Geometric Surgery

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter brings together the basic properties of fibre bundles, fibrations, and vector bundles required in surgery theory. The tangent and normal bundles of a manifold are introduced. The ... More

## Some Aspects of the Theory of Higgs Pairs

*S. Ramanan*

### 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/9780199534920.003.0006
- Subject:
- Mathematics, Geometry / Topology

This chapter presents an exposition of certain aspects of the theory of Higgs pairs (also called ‘Higgs bundles’ by some authors). Topics covered include the moduli of vector bundles, Hecke ... More

## Introduction to vector bundles

*Clifford Henry Taubes*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0003
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines vector bundles. It begins with a standard definition of vector bundles, and this is followed by some first examples of vector bundles. It then discusses the tangent bundle with ... More

## Principal bundles

*Clifford Henry Taubes*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0010
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines principal bundles, which is defined as the Lie group analog of a vector bundle. It covers principal bundles constructed from vector bundles; examples of Lie group quotients; ... More

## Vector bundles with ℂn as fiber

*Clifford Henry Taubes*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0006
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

Just as there are vector spaces over ℂ, there are vector bundles whose fibres can be consistently viewed as ℂn for some n. This chapter first defines these objects and then provides number of ... More

## Linear symplectic geometry

*Dusa McDuff and Dietmar Salamon*

### in Introduction to Symplectic Topology

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.003.0003
- Subject:
- Mathematics, Analysis, Geometry / Topology

The second chapter introduces the basic concepts of symplectic topology in the linear algebra setting, such as symplectic vector spaces, the linear symplectic group, Lagrangian subspaces, and the ... More

## Curvature polynomials and characteristic classes

*Clifford Henry Taubes*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0014
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter explains how the curvatures of connections can be used to construct De Rham cohomology classes that distinguish isomorphism classes of vector bundles and principal bundles. These classes ... More

## Covariant derivatives and connections

*Clifford Henry Taubes*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0011
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines the related notions of covariant derivative and connection. It covers the space of covariant derivatives. It also gives a relatively straightforward construction of a covariant ... More

## Metrics on vector bundles

*Clifford Henry Taubes*

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

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0007
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter discusses the following: metrics and transition functions for real vector bundles; metrics and transition functions for complex vector bundles; metrics, algebra and maps; and a metric on ... More

## The Topology of a Group Action

*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/9780691191751.003.0025
- Subject:
- Mathematics, Educational Mathematics

This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces ... More

## Localization Formulas

*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/9780691191751.003.0030
- Subject:
- Mathematics, Educational Mathematics

This chapter highlights localization formulas. The equivariant localization formula for a torus action expresses the integral of an equivariantly closed form as a finite sum over the fixed point set. ... More

##
The hypoelliptic Laplacian on *X* = *G/K*

*Jean-Michel Bismut*

### in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691151298
- eISBN:
- 9781400840571
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151298.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter constructs the hypoelliptic Laplacian ℒbX > 0 acting on the total space of a vector bundle TX ⊕ N ≃ g over the symmetric space X = G/K. The operator ℒbX is obtained using general ... More

## Shtukas with one leg III

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0014
- Subject:
- Mathematics, Geometry / Topology

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following ... More

##
Vector bundles and *G*-torsors on the relative Fargues-Fontaine curve

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0022
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses vector bundles and G-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important ... More

## Shtukas with one leg II

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0013
- Subject:
- Mathematics, Geometry / Topology

This chapter offers a second lecture on one-legged shtukas. It shows that a shtuka over Spa Cb, a priori defined over Y[0,INFINITY) = Spa Ainf REVERSE SOLIDUS {xk, xL}, actually extends to Y = Spa ... More

## Klein Geometries

*Ercüment H. Ortaçgil*

### in An Alternative Approach to Lie Groups and Geometric Structures

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198821656
- eISBN:
- 9780191860959
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821656.003.0017
- Subject:
- Mathematics, Pure Mathematics

Up to now, the discussion has been mainly concerned with Lie groups and their curved analogs, namely, parallelizable manifolds and their curvatures. The problem is to generalize this construction to ... More

## Rost’s Chain Lemma

*Christian Haesemeyer and Charles A. Weibel*

### in The Norm Residue Theorem in Motivic Cohomology

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780691191041
- eISBN:
- 9780691189635
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191041.003.0009
- Subject:
- Mathematics, Geometry / Topology

This chapter states and proves Rost's Chain Lemma. The proof (due to Markus Rost) does not use the inductive assumption that BL(n − 1) holds. Throughout this chapter, 𝓁 is a fixed prime, and 𝑘 is a ... More

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