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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 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


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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