## Modern Differential Geometry for Physicists |

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### Contents

An Introduction to Topology | 1 |

Differentiable Manifolds | 59 |

Vector Fields and nForms | 97 |

ix | 110 |

Lie Groups | 149 |

Fibre Bundles | 199 |

### Common terms and phrases

associated bundle base space bijection bundle map class of curves commutator compact components converges coordinate chart coordinate system cotangent cross-section defined denoted diffeomorphisms differentiable manifold differential forms differential geometry differential structure element equation equivalence class Euclidean space example exists fibre bundle Figure filter base finite G-action G-bundle generalised given GL(n GL+(n global hence homomorphism horizontal lift idea implies important induced infinite-dimensional integral curve intersection isomorphism lattice left-invariant vector field Lie algebra Lie group Lie group G linear map m-dimensional map h matrix metric space n-form neighbourhood Note one-form one-parameter one-to-one open sets open subset operation pair particular precisely principal bundle product bundle Proof pull-back push-forward QED Comments real numbers real vector space relation respect satisfies sense sequence smooth space IRm spacetime subspace surjective tangent space tangent vector tensor theorem theoretical physics topological space topology trivial unique vector bundle