Differential geometry [electronic resource] : bundles, connections, metrics and curvature / Clifford Henry Taubes

Author: Taubes, Clifford, 1954-
Published: Oxford : Oxford University Press, 2011.
Physical Description: 1 online resource (xiii, 298 pages) : illustrations

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Oxford scholarship online: ezaccess.libraries.psu.edu

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Series

Oxford graduate texts in mathematics ; 23.

Contents

Machine generated contents note: 1.1.Smooth manifolds -- 1.2.The inverse function theorem and implicit function theorem -- 1.3.Submanifolds of Rm -- 1.4.Submanifolds of manifolds -- 1.5.More constructions of manifolds -- 1.6.More smooth manifolds: The Grassmannians -- Appendix 1.1 How to prove the inverse function and implicit function theorems -- Appendix 1.2 Partitions of unity -- Additional reading -- 2.1.The general linear group -- 2.2.Lie groups -- 2.3.Examples of Lie groups -- 2.4.Some complex Lie groups -- 2.5.The groups SI(n; C), U(n) and SU(n) -- 2.6.Notation with regards to matrices and differentials -- Appendix 2.1 The transition functions for the Grassmannians -- Additional reading -- 3.1.The definition -- 3.2.The standard definition -- 3.3.The first examples of vector bundles -- 3.4.The tangent bundle -- 3.5.Tangent bundle examples -- 3.6.The cotangent bundle -- 3.7.Bundle homomorphisms -- 3.8.Sections of vector bundles -- 3.9.Sections of TM and T*M -- Additional reading -- 4.1.Subbundles -- 4.2.Quotient bundles -- 4.3.The dual bundle -- 4.4.Bundles of homomorphisms -- 4.5.Tensor product bundles -- 4.6.The direct sum -- 4.7.Tensor powers -- Additional reading -- 5.1.The pull-back construction -- 5.2.Pull-backs and Grassmannians -- 5.3.Pull-back of differential forms and push-forward of vector fields -- 5.4.Invariant forms and vector fields on Lie groups -- 5.5.The exponential map on a matrix group -- 5.6.The exponential map and right/left invariance on Gl(n; C) and its subgroups -- 5.7.Immersion, submersion and transversality -- Additional reading -- 6.1.Definitions -- 6.2.Comparing definitions -- 6.3.Examples: The complexification -- 6.4.Complex bundles over surfaces in R3 -- 6.5.The tangent bundle to a surface in R3 -- 6.6.Bundles over 4-dimensional submanifolds in 1185 -- 6.7.Complex bundles over 4-dimensional manifolds -- 6.8.Complex Grassmannians -- 6.9.The exterior product construction -- 6.10.Algebraic operations -- 6.11.Pull-back -- Additional reading -- 7.1.Metrics and transition functions for real vector bundles -- 7.2.Metrics and transition functions for complex vector bundles -- 7.3.Metrics, algebra and maps -- 7.4.Metrics on TM -- Additional reading -- 8.1.Riemannian metrics and distance -- 8.2.Length minimizing curves -- 8.3.The existence of geodesics -- 8.4.First examples -- 8.5.Geodesics on SO(n) -- 8.6.Geodesics on U(n) and SU(n) -- 8.7.Geodesics and matrix groups -- Appendix 8.1 The proof of the vector field theorem -- Additional reading -- 9.1.The maximal extension of a geodesic -- 9.2.The exponential map -- 9.3.Gaussian coordinates -- 9.4.The proof of the geodesic theorem -- Additional reading -- 10.1.The definition -- 10.2.A cocycle definition -- 10.3.Principal bundles constructed from vector bundles -- 10.4.Quotients of Lie groups by subgroups -- 10.5.Examples of Lie group quotients -- 10.6.Cocycle construction examples -- 10.7.Pull-backs of principal bundles -- 10.8.Reducible principal bundles -- 10.9.Associated vector bundles -- Appendix 10.1 Proof of Proposition 10.1 -- Additional reading -- 11.1.Covariant derivatives -- 11.2.The space of covariant derivatives -- 11.3.Another construction of covariant derivatives -- 11.4.Principal bundles and connections -- 11.5.Connections and covariant derivatives -- 11.6.Horizontal lifts -- 11.7.An application to the classification of principal G-bundles up to isomorphism -- 11.8.Connections, covariant derivatives and pull-back bundles -- Additional reading -- 12.1.Exterior derivative -- 12.2.Closed forms, exact forms, diffeomorphisms and De Rham cohomology -- 12.3.Lie derivative -- 12.4.Curvature and covariant derivatives -- 12.5.An example -- 12.6.Curvature and commutators -- 12.7.Connections and curvature -- 12.8.The horizontal subbundle revisited -- Additional reading -- 13.1.Flat connections -- 13.2.Flat connections on bundles over the circle -- 13.3.Foliations -- 13.4.Automorphisms of a principal bundle -- 13.5.The fundamental group -- 13.6.The flat connections on bundles over M -- 13.7.The universal covering space -- 13.8.Holonomy and curvature -- 13.9.Proof of the classification theorem for flat connections -- Appendix 13.1 Smoothing maps -- Appendix 13.2 The proof of the Frobenius theorem -- Additional reading -- 14.1.The Bianchi Identity -- 14.2.Characteristic forms -- 14.3.Characteristic classes: Part 1 -- 14.4.Characteristic classes: Part 2 -- 14.5.Characteristic classes for complex vector bundles and the Chern classes -- 14.6.Characteristic classes for real vector bundles and the Pontryagin classes -- 14.7.Examples of bundles with nonzero Chern classes -- 14.8.The degree of the map g->gm from SU(2) to itself -- 14.9.A Chern[–]Simons form -- Appendix 14.1 The ad-invariant functions on M(n; C) -- Appendix 14.2 Integration on manifolds -- Appendix 14.3 The degree of a map -- Additional reading -- 15.1.Metric compatible covariant derivatives -- 15.2.Torsion free covariant derivatives on T*M -- 15.3.The Levi-Civita connection/covariant derivative -- 15.4.A formula for the Levi-Civita connection -- 15.5.Covariantly constant sections -- 15.6.An example of the Levi-Civita connection -- 15.7.The curvature of the Levi-Civita connection -- Additional reading -- 16.1.Spherical metrics, flat metrics and hyperbolic metrics -- 16.2.The Schwarzchild metric -- 16.3.Curvature conditions -- 16.4.Manifolds of dimension 2: The Gauss[–]Bonnet formula -- 16.5.Metrics on manifolds of dimension 2 -- 16.6.Conformal changes -- 16.7.Sectional curvatures and universal covering spaces -- 16.8.The Jacobi field equation -- 16.9.Constant sectional curvature and the Jacobi field equation -- 16.10.Manifolds of dimension 3 -- 16.11.The Riemannian curvature of a compact matrix group -- Additional reading -- 17.1.Some basics concerning holomorphic functions on Cn -- 17.2.The definition of a complex manifold -- 17.3.First examples of complex manifolds -- 17.4.The Newlander[–]Nirenberg theorem -- 17.5.Metrics and almost complex structures on TM -- 17.6.The almost Kähler 2-form -- 17.7.Symplectic forms -- 17.8.Kähler manifolds -- 17.9.Complex manifolds with closed almost Kähler form -- 17.10.Examples of Kähler manifolds -- Appendix 17.1 Compatible almost complex structures -- Additional reading -- 18.1.Holomorphic submanifolds of a complex manifold -- 18.2.Holomorphic submanifolds of projective spaces -- 18.3.Proof of Proposition 18.2, about holomorphic submanifolds in CPn -- 18.4.The curvature of a Kahler metric -- 18.5.Curvature with no (0, 2) part -- 18.6.Holomorphic sections -- 18.7.Example on CPn -- Additional reading -- 19.1.Definition of the Hodge star -- 19.2.Representatives of De Rharn cohomology -- 19.3.A fairy tale -- 19.4.The Hodge theorem -- 19.5.Self-duality -- Additional reading.

Summary

Bundles, connections, metrics and curvature are the lingua franca of modern differential geometry and theoretical physics. Supplying graduate students in mathematics or theoretical physics with the fundamentals of these objects, this book would suit a one-semester course on the subject of bundles and the associated geometry.

Subject(s)

Geometry, Differential

ISBN

9780191774911 (ebook)

Bibliography Note

Includes bibliographical references and index.

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