*(English)*Zbl 1090.53001

From the preface: The first edition of the present book was published in 1990 (Hilger, Bristol) and reviewed in Zbl 0764.53001. There has been incredible progress in geometry and topology applied to theoretical physics and vice versa since then. The boundaries among these disciplines are quite obscure these days.

I found it impossible to take all the progress into these fields into account in this second edition and decided to make the revision minimal. Besides correcting typos, errors and miscellaneous small additions, I added the proof of the index theorem in terms of supersymmetric quantum mechanics. There are also some rearrangements of material in many places. I have learned from publications and internet homepages that the first edition of the book has been read by students and researchers from a wide variety of fields, not only in physics and mathematics but also in philosophy, chemistry, geodesy and oceanology among others. This is one of the reasons why I did not specialize this book to the forefront of recent developments. I hope to publish in the near future, possibly in collaboration with a mathematician or two, a separate book on the recent fascinating application of quantum field theory to low-dimensional topology and number theory.

##### MSC:

53-01 | Textbooks (differential geometry) |

57-01 | Textbooks (manifolds) |

53C80 | Applications of global differential geometry to physics |

81T13 | Yang-Mills and other gauge theories |

83Cxx | General relativity |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) |

58J20 | Index theory and related fixed point theorems (PDE on manifolds) |