## Geometry of Differential FormsSince the times of Gauss, Riemann, and Poincare, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. This book is a comprehensive introduction to differential forms. It begins with a quick presentation of the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results about them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated in the book is a detailed description of the Chern-Weil theory. With minimal prerequisites, the book can serve as a textbook for an advanced undergraduate or a graduate course in differential geometry. |

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### Contents

Manifolds | 1 |

11 What is a manifold? | 2 |

12 Definition and examples of manifolds | 11 |

13 Tangent vectors and tangent spaces | 23 |

14 Vector fields | 36 |

15 Fundamental facts concerning manifolds | 44 |

Differential Forms | 57 |

22 Various operations on differential forms | 69 |

42 Laplacian and harmonic forms | 153 |

43 The Hodge theorem | 158 |

44 Applications of the Hodge theorem | 162 |

Vector Bundles and Characteristic Classes | 169 |

52 Geodesics and parallel translation of vectors | 180 |

53 Connections in vector bundles and curvature | 185 |

54 Pontrjagin classes | 193 |

55 Chern classes | 204 |

23 Frobenius theorem | 80 |

24 A few facts | 89 |

The de Rham Theorem | 95 |

31 Homology of manifolds | 96 |

32 Integral of differential forms and the Stokes theorem | 104 |

33 The de Rham theorem | 111 |

34 Proof of the de Rham theorem | 119 |

35 Applications of the de Rham theorem | 133 |

Laplacian and Harmonic Forms | 145 |

56 Euler classes | 211 |

57 Applications of characteristic classes | 216 |

Fiber Bundles and Characteristic Classes | 231 |

62 𝓢¹ bundles and Euler class | 240 |

63 Connections | 257 |

64 Curvature | 265 |

65 Characteristic classes | 275 |

66 A couple of items | 285 |

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### Common terms and phrases

Ak(M Al(M arbitrary point bundle map called characteristic classes Chern classes class Cx closed form closed manifold cocycle cohomology class cohomology group commutative compact complex vector bundle condition connection form consider coordinate neighborhood coordinate system curvature form Cx function Cx map Cx(M defined definition denote diffeomorphism differential forms dimensional element equation Euler class example exterior differentiation exterior product fc-form fiber bundle FIGURE follows formula G bundle geometry given harmonic form Hence homology homomorphism Hopf induced inner product integral curve invariant polynomial isomorphism Lemma Lie group linear map matrix natural obtain open covering open neighborhood open set oriented Pontrjagin classes principal bundle principal G proof Proposition prove Rham cohomology Rham theorem Riemannian manifold Riemannian metric S1 bundle satisfies simplex simplicial complex singular Stokes theorem structure group submanifold subset subspace surface tangent bundle tangent space tangent vector theory trivial vector field vector space