## Modern Differential Geometry for Physicists |

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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 |

277 | |

### Common terms and phrases

Af(x associated bundle base space bijection bundle map closed sets commutator compact components converges coordinate chart coordinate system cotangent cross-section defined definition denoted diffeomorphisms differentiable manifold differential forms differential geometry equation equivalence class example exists exterior derivative fibre bundle filter base follows function G-action G-bundle gauge transformation generalised GL(m GL(n hence homomorphism horizontal lift idea implies important induced infinite-dimensional integral curve IRn+1 isomorphism lattice left-invariant vector field Lie algebra Lie group Lie group G linear map map h matrix metric space n-form neighbourhood Note one-form open sets open subset pair particular precisely principal bundle product bundle pull-back push-forward QED Comments real numbers real vector space relation respect satisfies sequence smooth space IRm spacetime subspace tangent space tangent vector tensor theorem theoretical physics topological space topology trivial trivialisation unique vector bundle vector field Yang-Mills field Yang-Mills theory