##
**Invariance theory, the heat equation and the Atiyah-Singer index theorem.
2nd ed.**
*(English)*
Zbl 0856.58001

Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. ix, 516 p. (electronic reprint) (1995).

The book under review is the second, largely revised, edition of the monograph ‘Invariance theory, the heat equation, and the Atiyah-Singer index theorem’, Publish or Perish (1984; Zbl 0565.58035) which itself was an expanded version of a series of lecture notes published as ‘The index theorem and the heat equation’, Publish or Perish (1974; Zbl 0287.58006). This new edition summarizes a large part of the work that the author has done over the past 25 years, and documents the strength of the heat equation method in the study of the global (and local) structure of manifolds. The main theme is still the Atiyah-Singer index theorem. Besides the \(K\)-theoretical approach that is now part of the larger area of operator \(K\)- and \(KK\)-theory, there are essentially two different ways to use the heat equation in the proof of the index theorem. The more recent one by E. Getzler using a scaling method that interpolates between Clifford and exterior multiplication is presented in N. Berline, E. Getzler and M. Vergne, ‘Heat kernels and Dirac operators’, Springer, Berlin (1992; Zbl 0744.58001); the other one developed by V. Patodi and the author is used in the present monograph. The author prefers this second approach not only by personal bias but also since here the interaction of analysis and topology already appears on an elementary level reminiscent of H. Weyl’s pioneering work (on the asymptotic behavior of the spectrum) dating from the beginning of the century.

The first two chapters provide the background that is necessary to understand the ingredients of the cohomological version of the index formula. Chapter 1 presents the functional analytic tools. It gives an introduction to the theory of pseudodifferential operators on \(\mathbb{R}^n\) and on compact manifolds, the construction of the parametrix for elliptic operators in the general context of Fredholm theory, its application to the Hodge decomposition, and the special theory and functional calculus of self-adjoint operators necessary for the solution of the heat equation. The main result is the McKean-Singer trace formula for a formally self-adjoint elliptic partial differential operator \(P\) of positive order with spectrum \(\{\lambda_n\}_{n\geq 0}\) bounded from below: \[ \text{tr}_{L^2}(e^{- tP})= \int_M \text{tr}_{V_x}k(t, x, x) dx= \sum_{n\geq 0} e^{- t\lambda_n}. \] More precisely, \(P\) acts on the smooth sections of a vector bundle \(V\) over the closed manifold \(M\) of dimension \(m\), and \(k\) denotes the kernel function which evaluated on the diagonal allows an asymptotic expansion. This is also extended to non self-adjoint operators and to boundary value problems. Chapter 2 introduces the differential geometric/topologic notions. Characteristic classes of vector bundles are defined and computed for canonical bundles over complex projective spaces. As a special case of the index theorem, the Gauss-Bonnet theorem is deduced from the trace formula: Taking \(P= \Delta_{\text{ev/odd}}\), the Laplacian restricted to even and odd forms, resp., the difference of traces, \(\text{tr}_{L^2}(e^{- t\Delta_{\text{ev}}}- e^{-t\Delta_{\text{odd}}})\) is independent of \(t\), hence the corresponding right-hand side after asymptotic expansion takes the form \(\sum_{n\geq 0} t^{(n- m)/2} \int_M a_n(x) dx= \int_M a_m(x) dx\), and can be identified as the Euler-Poincaré characteristic by invariance theory. Again, this is also extended to the more complicated case of manifolds with boundary. The Atiyah-Singer index theorem is stated and proved in Chapter 3 along with the various special index theorems for the signature, Dolbeault, and spin Dirac operators. Moreover, the author proves Lefschetz fixed point formulas and the general index theorem for manifolds with boundary. These index theorems show the influence that the topological structure of a manifold has on the solvability of partial differential equations. In Chapter 4 the opposite direction is taken. Here, the author discusses geometrical and topological invariants that are encoded in the spectrum of natural differential operators on a compact manifold (possibly with boundary). This is done by explicitly calculating the coefficients (for small \(n\)) in the asymptotic expansion of \(k(t, x, x)\). The variety of results and applications is too complex to be stated here. In the final (new) chapter there is given (by the reviewer) a short historical survey on index theory and its ramifications, supplemented by a comprehensive bibliography which contains most of the existing literature (up to 1993).

The first two chapters provide the background that is necessary to understand the ingredients of the cohomological version of the index formula. Chapter 1 presents the functional analytic tools. It gives an introduction to the theory of pseudodifferential operators on \(\mathbb{R}^n\) and on compact manifolds, the construction of the parametrix for elliptic operators in the general context of Fredholm theory, its application to the Hodge decomposition, and the special theory and functional calculus of self-adjoint operators necessary for the solution of the heat equation. The main result is the McKean-Singer trace formula for a formally self-adjoint elliptic partial differential operator \(P\) of positive order with spectrum \(\{\lambda_n\}_{n\geq 0}\) bounded from below: \[ \text{tr}_{L^2}(e^{- tP})= \int_M \text{tr}_{V_x}k(t, x, x) dx= \sum_{n\geq 0} e^{- t\lambda_n}. \] More precisely, \(P\) acts on the smooth sections of a vector bundle \(V\) over the closed manifold \(M\) of dimension \(m\), and \(k\) denotes the kernel function which evaluated on the diagonal allows an asymptotic expansion. This is also extended to non self-adjoint operators and to boundary value problems. Chapter 2 introduces the differential geometric/topologic notions. Characteristic classes of vector bundles are defined and computed for canonical bundles over complex projective spaces. As a special case of the index theorem, the Gauss-Bonnet theorem is deduced from the trace formula: Taking \(P= \Delta_{\text{ev/odd}}\), the Laplacian restricted to even and odd forms, resp., the difference of traces, \(\text{tr}_{L^2}(e^{- t\Delta_{\text{ev}}}- e^{-t\Delta_{\text{odd}}})\) is independent of \(t\), hence the corresponding right-hand side after asymptotic expansion takes the form \(\sum_{n\geq 0} t^{(n- m)/2} \int_M a_n(x) dx= \int_M a_m(x) dx\), and can be identified as the Euler-Poincaré characteristic by invariance theory. Again, this is also extended to the more complicated case of manifolds with boundary. The Atiyah-Singer index theorem is stated and proved in Chapter 3 along with the various special index theorems for the signature, Dolbeault, and spin Dirac operators. Moreover, the author proves Lefschetz fixed point formulas and the general index theorem for manifolds with boundary. These index theorems show the influence that the topological structure of a manifold has on the solvability of partial differential equations. In Chapter 4 the opposite direction is taken. Here, the author discusses geometrical and topological invariants that are encoded in the spectrum of natural differential operators on a compact manifold (possibly with boundary). This is done by explicitly calculating the coefficients (for small \(n\)) in the asymptotic expansion of \(k(t, x, x)\). The variety of results and applications is too complex to be stated here. In the final (new) chapter there is given (by the reviewer) a short historical survey on index theory and its ramifications, supplemented by a comprehensive bibliography which contains most of the existing literature (up to 1993).

Reviewer: H.Schröder (Dortmund)

### MSC:

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

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

58J70 | Invariance and symmetry properties for PDEs on manifolds |

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

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |