## Manifolds, Groups, Bundles, and SpacetimeManifolds, Groups, Bundles, and Spacetime was written for those who are interested in modern differential geometry and its applications in physics. The primary material is suitable for a graduate level course in the theory of differentiable manifolds, Lie groups, and fiber bundles. The first two chapters are an introduction to concepts from linear algebra and tensors and can be read to establish familiarity with the notation and conventions of the text by those who are already familiar with these topics. The third and fourth chapters are a review of topics from advanced calculus and topology and are included primarily as a convenient reference. |

### Contents

TOPICS FROM LINEAR ALGEBRA | 1 |

MULTILINEAR ALGEBRA | 25 |

DIFFERENTIAL CALCULUS ON VECTOR SPACES | 57 |

DIFFERENTIABLE MANIFOLDS | 127 |

TOPOLOGY | 149 |

DIFFERENTIAL CAL CULUS ON MANIFOLDS | 165 |

LIE GROUPS AND LIE ALGEBRAS | 207 |

FIBER BUNDLES | 235 |

THE DIFFERENTIAL GEOMETRY OF SPACETIME | 291 |

INTEGRATION ON MANIFOLDS | 317 |

INTRODUCTION TO GENERAL RELATIVITY | 353 |

PICARDS THEOREM | 379 |

399 | |

### Common terms and phrases

arbitrary atlas bundle isomorphism called change of basis compact components converges coordinate functions coordinate system Corollary countable defined Definition denoted Df(x diffeomorphism differentiable manifold domain dual basis Ea'ample equation Example Exercise f is continuous finite dimensional function f geodesic gives Gl(n Hausdorff Hence homomorphism inner product space integral curve Lemma Let f Let G let m e Let X1 Lie algebra Lie group linear mapping linearly independent manifold and let matrix maximal integral curve metric tensor n-dimensional manifold non-empty normed vector space notation open covering open set open subset orientation p-form principal fiber bundle proof of Theorem Prove Remark respectively right-hand side scalars sequence Show space and let spacetime structure subspace summation surjective tangent vector tensor field tensor of type topological space topology transformation U1 O U2 unique vector bundle