##
**Elliptic operators, topology and asymptotic methods.**
*(English)*
Zbl 0654.58031

Pitman Research Notes in Mathematics Series, 179. Harlow (UK): Longman Scientific & Technical; New York: John Wiley & Sons. 184 p. $ 15.50 (1988).

This book grew out of lecture notes, and presents a proof of the Atiyah- Singer index theorem for Dirac operators in the spirit of E. Getzler [Topology 25, 111-117 (1986; Zbl 0607.58040)]. The second approach alluded to in Getzler’s paper which combines his scaling argument and Patodi’s generalization of the well-known Minakshisundaram- Pleijel expansion of the heat kernel is completely worked out. Since there is more in this book let us briefly survey its content.

Chapter 1 recalls the basic tools from differential geometry: the definition of connexions (connections) on vector bundles and principle bundles, the existence of “good” geodesic coordinates (which give simple representations for the metric tensor and the Christoffel symbols locally around a point), and the definitions of the Hodge-*-operator, the differential, and the codifferential. Chapter 2 introduces Clifford bundles and Dirac operators as developed by M. Gromov and H. B. Lawson jun. [Publ. Math., Inst. Hautes Etud. Sci. 58, 295-408 (1983; Zbl 0538.53047)]. The two main results are the formal self-adjointness of Dirac operators and the Bochner-Weitzenböck formula.

Chapter 3 gives the elliptic theory of Dirac operators using Gårding’s inequality and classical a priori estimates in Sobolev spaces. Here the author follows the common approach via analysis on the tori \(T^ n\) [cf. R. S. Palais et al., Seminar on the Atiyah- Singer index theorem (Ann. Math. Studies 57) (1965; Zbl 0137.170)]. The chapter ends with the functional calculus for bounded functions on the spectrum. This is applied in the next chapter to prove the Hodge decomposition theorem.

Chapter 5 is central with its treatment of the heat and the wave equation. The main results are the existence and uniqueness of the heat kernel and the construction of its asymptotic expansion. The integral formula for the trace of the associated trace class operator \(e^{- t\Delta}\) is given in chapter 6, and the famous Weyl formula for the asymptotic distribution of eigenvalues of the Laplacian on a compact manifold is deduced in chapter 7.

Chapter 8 presents a proof of the Atiyah-Bott-Lefschetz-formula, and chapter 9 gives the supertrace formula for the index of a graded Dirac operator (which goes back to McKean and Singer).

Chapter 10 collects the rudiments of characteristic classes which are necessary to state the index theorem properly, introduces the notion of spin manifolds together with Lichnerowicz’ specialization of the Bochner- Weitzenböck formula in this case, and provides the last ingredient for the proof of the index theorem, i.e. Mehler’s formula.

The proof of the index theorem for Dirac operators makes up chapter 11. Also included is a proof of the Hirzebruch signature theorem and a sketch of a proof of the Chern-Gauß-Bonnet theorem as an exercise. It should be mentioned that each of these chapters is supplemented with a set of exercises which are all quite manageable for the graduate student.

The last two chapters contain Witten’s approach to Morse theory and Atiyah’s \(L^ 2\)-index theorem for Galois coverings.

This book can very well serve as a guide to a one-semester course introducing and proving the Atiyah-Singer index theorem for Dirac operators. It requires some background in differential geometry and functional analysis since the corresponding parts in the book are too selective on purpose.

Chapter 1 recalls the basic tools from differential geometry: the definition of connexions (connections) on vector bundles and principle bundles, the existence of “good” geodesic coordinates (which give simple representations for the metric tensor and the Christoffel symbols locally around a point), and the definitions of the Hodge-*-operator, the differential, and the codifferential. Chapter 2 introduces Clifford bundles and Dirac operators as developed by M. Gromov and H. B. Lawson jun. [Publ. Math., Inst. Hautes Etud. Sci. 58, 295-408 (1983; Zbl 0538.53047)]. The two main results are the formal self-adjointness of Dirac operators and the Bochner-Weitzenböck formula.

Chapter 3 gives the elliptic theory of Dirac operators using Gårding’s inequality and classical a priori estimates in Sobolev spaces. Here the author follows the common approach via analysis on the tori \(T^ n\) [cf. R. S. Palais et al., Seminar on the Atiyah- Singer index theorem (Ann. Math. Studies 57) (1965; Zbl 0137.170)]. The chapter ends with the functional calculus for bounded functions on the spectrum. This is applied in the next chapter to prove the Hodge decomposition theorem.

Chapter 5 is central with its treatment of the heat and the wave equation. The main results are the existence and uniqueness of the heat kernel and the construction of its asymptotic expansion. The integral formula for the trace of the associated trace class operator \(e^{- t\Delta}\) is given in chapter 6, and the famous Weyl formula for the asymptotic distribution of eigenvalues of the Laplacian on a compact manifold is deduced in chapter 7.

Chapter 8 presents a proof of the Atiyah-Bott-Lefschetz-formula, and chapter 9 gives the supertrace formula for the index of a graded Dirac operator (which goes back to McKean and Singer).

Chapter 10 collects the rudiments of characteristic classes which are necessary to state the index theorem properly, introduces the notion of spin manifolds together with Lichnerowicz’ specialization of the Bochner- Weitzenböck formula in this case, and provides the last ingredient for the proof of the index theorem, i.e. Mehler’s formula.

The proof of the index theorem for Dirac operators makes up chapter 11. Also included is a proof of the Hirzebruch signature theorem and a sketch of a proof of the Chern-Gauß-Bonnet theorem as an exercise. It should be mentioned that each of these chapters is supplemented with a set of exercises which are all quite manageable for the graduate student.

The last two chapters contain Witten’s approach to Morse theory and Atiyah’s \(L^ 2\)-index theorem for Galois coverings.

This book can very well serve as a guide to a one-semester course introducing and proving the Atiyah-Singer index theorem for Dirac operators. It requires some background in differential geometry and functional analysis since the corresponding parts in the book are too selective on purpose.

Reviewer: H.Schröder

### MSC:

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

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

58A30 | Vector distributions (subbundles of the tangent bundles) |

58A10 | Differential forms in global analysis |

58A14 | Hodge theory in global analysis |

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

58C50 | Analysis on supermanifolds or graded manifolds |