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**Spin geometry.**
*(English)*
Zbl 0688.57001

Princeton Mathematical Series. 38. Princeton, NJ: Princeton University Press. xii, 427 p. $ 55.00 (1989).

The spin groups are among the most important groups in geometric and topological research of the last thirty years. The possibility of reducing the structure group of the tangent bundle of a compact smooth manifold to spin comprises the existence of certain elliptic differential operators on the manifold. These can be used to study the geometry and the topology of the manifold. In this book the authors present the subject as it was developed starting from the papers of Atiyah and Singer on index theory in 1963 and Atiyah, Bott, Shapiro on Clifford modules in 1964 up to recent results on the existence of Riemannian metrics with positive scalar curvature.

Chapter I deals with algebraic constructions. It contains the definition of Clifford algebras, spin groups and related groups, a description of their structure and their representations, and applications to vector fields on spheres. K-theory is introduced and some constructions in K- theory related to Clifford algebras are given.

Chapter II deals with differential geometric constructions. Here the authors introduce the concepts that are central in differential geometry, e.g. manifolds, vector bundles, connections, curvature. They examine the fundamental structures of spin geometry: spin structure of vector bundles, Clifford and Spinor bundles and eventually the Dirac operators.

Chapter III deals with the analysis that is used in spin geometry. Here the theory of pseudo differential operators is developed and various forms of the Atiyah-Singer index theorem are formulated and proved. These include the classical theorem and its cohomological formula for the index, the index theorem for G-operators, the index theorem for families of operators, and an index theorem for \(Cl_ k\)-linear operators, which have an index in KO-theory.

Chapter IV contains applications in geometry and topology. Here the applications in differential topology include some integrality and divisibility theorems for characteristic numbers, results on immersions of manifolds into the Euclidean space and on the maximal number of linearly independent vector fields on a manifold, and theorems on the existence of smooth group actions on a manifold. The applications in differential geometry include the study of Riemannian manifolds with positive scalar curvature, the topology of the space of positive scalar curvature metrics, Kähler manifolds, spinor cohomology and other topics.

There are four appendices on principal G-bundles, characteristic classes, Thom isomorphisms in K-theory, and \(Spin^ c\)-manifolds.

Chapter I deals with algebraic constructions. It contains the definition of Clifford algebras, spin groups and related groups, a description of their structure and their representations, and applications to vector fields on spheres. K-theory is introduced and some constructions in K- theory related to Clifford algebras are given.

Chapter II deals with differential geometric constructions. Here the authors introduce the concepts that are central in differential geometry, e.g. manifolds, vector bundles, connections, curvature. They examine the fundamental structures of spin geometry: spin structure of vector bundles, Clifford and Spinor bundles and eventually the Dirac operators.

Chapter III deals with the analysis that is used in spin geometry. Here the theory of pseudo differential operators is developed and various forms of the Atiyah-Singer index theorem are formulated and proved. These include the classical theorem and its cohomological formula for the index, the index theorem for G-operators, the index theorem for families of operators, and an index theorem for \(Cl_ k\)-linear operators, which have an index in KO-theory.

Chapter IV contains applications in geometry and topology. Here the applications in differential topology include some integrality and divisibility theorems for characteristic numbers, results on immersions of manifolds into the Euclidean space and on the maximal number of linearly independent vector fields on a manifold, and theorems on the existence of smooth group actions on a manifold. The applications in differential geometry include the study of Riemannian manifolds with positive scalar curvature, the topology of the space of positive scalar curvature metrics, Kähler manifolds, spinor cohomology and other topics.

There are four appendices on principal G-bundles, characteristic classes, Thom isomorphisms in K-theory, and \(Spin^ c\)-manifolds.

Reviewer: K.H.Mayer

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |

53C20 | Global Riemannian geometry, including pinching |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

57R20 | Characteristic classes and numbers in differential topology |

57R42 | Immersions in differential topology |

57R22 | Topology of vector bundles and fiber bundles |

58J32 | Boundary value problems on manifolds |

58J20 | Index theory and related fixed-point theorems on manifolds |

15A66 | Clifford algebras, spinors |

58D17 | Manifolds of metrics (especially Riemannian) |