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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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