## Applicable Differential GeometryThis is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students. |

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### Contents

The background vector calculus | 1 |

Affline spaces | 8 |

Curves functions and derivatives | 29 |

Vector fields and flows | 53 |

Volumes and subspaces exterior algebra | 85 |

Calculus of forms | 117 |

Frobeniuss theorem | 140 |

Metrics on affline spaces | 164 |

9 Geometry of surfaces | 216 |

10 Manifolds | 236 |

11 Connections | 268 |

12 Lie groups | 298 |

13 The tangent and cotangent bundles | 327 |

14 Fibre bundles | 353 |

15 Connections revisited | 371 |

8 Isometries | 188 |

### Common terms and phrases

affine coordinate system affine coordinates affine map affine space affine transformations argument bilinear form bracket calculus called Chapter chart commute components connection constraint 1-forms construction corresponding cotangent covariant derivative covector curvature decomposable defined definition denote determined diffeomorphism differential equations dimension directional derivative dual element example Exercise expression exterior derivative exterior product fibre flow follows formula generalisation geodesic given grad homomorphism horizontal hyperplane identity infinitesimal integral curve integral submanifold isometry isomorphism left-invariant Lie algebra Lie derivative Lie group Lie transport linear map manifold matrix multilinear n-dimensional neighbourhood non-singular non-zero null obtained one-parameter group open set open subset operator orbit orthogonal orthonormal basis p-form parallel parametrisation properties respect rotation satisfies scalar product Section Show smooth functions smooth map structure subgroup subspace symmetric tangent space tangent vector tensor field timelike topology translation vanishes vector bundle vector field vector space vertical volume form zero