## Modern Differential Geometry for PhysicistsThis edition of the invaluable text Modern Differential Geometry for Physicists contains an additional chapter that introduces some of the basic ideas of general topology needed in differential geometry. A number of small corrections and additions have also been made.These lecture notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course “Quantum Fields and Fundamental Forces” at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen bearing in mind the way in which differential geometry is applied these days to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma models and other types of nonlinear field systems that feature in modern quantum field theory.The volume is divided into four parts: (i) introduction to general topology; (ii) introductory coordinate-free differential geometry; (iii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds; (iv) introduction to the theory of fibre bundles. In the introduction to differential geometry the author lays considerable stress on the basic ideas of “tangent space structure”, which he develops from several different points of view — some geometrical, others more algebraic. This is done with awareness of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry. |

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### Contents

An Introduction to Topology | 1 |

Differentiable Manifolds | 59 |

Vector Fields and nForms | 97 |

Lie Groups | 149 |

Fibre Bundles | 199 |

Connections in a Bundle | 253 |

BIBLIOGRAPHY | 277 |

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associated bundle base space bijection bundle map closed sets cohomology groups commutator compact components convergence coordinate chart coordinate system cotangent cross-section defined denoted diffeomorphisms differentiable manifold differential forms differential geometry equation equivalence class Euclidean space example exists exterior derivative fibre bundle Figure filter base function G-action G-bundle gauge transformation generalised GL(m GL(n GL+(n global hence homomorphism horizontal lift idea implies important induced infinite-dimensional integral curve isomorphism lattice left-invariant vector field Lie algebra Lie group Lie group G linear map map h map TT matrix metric space n-form neighbourhood Note one-form open sets open subset pair particular precisely principal bundle product bundle Proof pull-back push-forward QED Comments real numbers real vector space relation respect satisfies sequence smooth spacetime subspace surjective tangent space tangent vector tensor theorem theoretical physics topological space topology trivialisation unique vector bundle Yang-Mills field Yang-Mills theory