The index theorem and the heat equation method (Q2743190)

This is a lovely book on the local version of the Atiyah-Singer index theorem and the heat equation method. By the standards of this branch of mathematics the book is elementary, as it makes no use of Sobolev spaces, pseudo-differential calculus, advanced theory of elliptic operators, Schwartz kernel theorem or advanced functional analysis, and does not attempt to touch upon such topics as eta-invariants, index theorem for manifolds with boundary etc. Moreover the book provides all the necessary differential-geometric background i.e. fundamentals of the theory of connection and Chern-Weil theory of characteristic classes. Thus the book is reasonably self-contained and could serve as the starting point for studies of the index theorem. However, in the case of the book, ``elementary'' does not mean simple, as the book comprises many lengthy and tricky calculations, among them the so-called Chern root algorithm (below described in more detail) which is of particular beauty. It is worth underlying here that everywhere in the book the author avoids using advanced general theories in favor of explicit computations. NEWLINENEWLINENEWLINENow a more detailed review of the book follows.NEWLINENEWLINENEWLINEIn the first chapter the author reviews fundamentals of Riemannian geometry: Levi-Civita connection, connection on a principal bundle and induced connections on associated bundles, orthonormal moving frame method, ``differential geometric'' elliptic operators, and some expressions using normal coordinates.NEWLINENEWLINENEWLINEThe second chapter provides basic knowledge of heat equation method. The author describes the Cauchy initial problem for the heat equation and defines the notion of fundamental solution, and next he turns to the problem of existence of fundamental solution. The author solves the problem using a Levi algorithm, which is a recurrent method of building the fundamental solution out of a suitable first approximation (called an initial solution, a proof of the existence of which is postponed to the next section). In particular he makes absolutely no use of general theory of 1-parameter semi-groups and pseudo-differential operators. The chapter is concluded by the proof of the Hodge decomposition theorem.NEWLINENEWLINENEWLINEThe third chapter starts with a proof of existence of fundamental solutions using the Minakshisundaram-Pleijel parametrix. Next the ``Asymptotic Expansion Theorem'', which expresses the heat kernel in terms of the Minakshisundaram-Pleijel parametrix, is proved. Finally the index of ``differential-geometric'' elliptic operators is expressed in terms of the trace of the fundamental solution, and the well known integral formula for the index is given (with the integrand called ``local index'').NEWLINENEWLINENEWLINEThe Chern-Weil theory of characteristic classes (of both vector and principal bundles) form the bulk of the fourth chapter. The presentation is self-contained and no preliminary familiarity with the topic is needed. In particular, in the first three paragraphs of the chapter, the author gives the standard construction of Chern, Euler and Pontryagin classes using invariant polynomials on appropriate Lie algebras, defines the \(L\)-genus and proves the naturality of characteristic classes of principal bundles.NEWLINENEWLINENEWLINEThe final two paragraphs of the fourth chapter are much less standard. In the fourth paragraph the author introduces a Chern root algorithm, which is later proved to be a powerful computational tool. Here is a brief explanation of the idea behind that algorithm.NEWLINENEWLINENEWLINEThe author starts with the well known formula for the Chern forms computed out of the curvature form \(\Omega\) of a connection \(D\): NEWLINE\[NEWLINE1+c_1 (\Omega)+c_2 (\Omega)+ \cdots +c_n(\Omega)= \left(\prod^n_{i=1} (1+u_i) \right) (\Omega).NEWLINE\]NEWLINE The right-hand side of this expression makes sense as the polynomial \(\prod^n_{j=1} (1+u_i)\) can be expressed in terms of fundamental symmetric polynomials, which are well defined on the whole Lie algebra of the group \(U(n) \). Now the author introduces new ``functions'' of the curvature form \(u_i (\Omega)\), and the formula above assumes the form NEWLINE\[NEWLINE1+p_1 (\Omega)+p_2 (\Omega) +\cdots +p_n(\Omega)= \prod^n_{i=1} \bigl(1+u_i (\Omega)\bigr).NEWLINE\]NEWLINE Of course \(u_i\) is defined only on the Lie algebra of the maximal torus \(U(1)^n\) and not on the whole Lie algebra of \(U(n)\), thus \(u_i(\Omega)\) is well defined only for connections reducible to the maximal torus \(U(1)^n\) (in particular the relevant bundle decomposes into the direct sum of linear bundles).NEWLINENEWLINENEWLINEPower of this simple idea is exhibited in the last paragraph of the fourth chapter, where the author gives an elegant formal proof of the local index theorem of the signature operator. The proof consists in identifying the relevant integrand (local index) in the integral formula for the index, and this is done by performing explicit calculations (using said Chern root algorithm) with the fundamental solution of a simplified version of the signature operator, which in a suitable local coordinate system can be expressed as the ordinary Laplacian plus some lower order explicitly described terms. Taking a simplified version of the signature operator is justified later in chapter 7, where a standard proof of the index theorem is given.NEWLINENEWLINENEWLINEThe fifth chapter provides basic knowledge of Clifford algebras and their links with exterior algebras and Clifford modules.NEWLINENEWLINENEWLINEThe sixth chapter deals with the Dirac operator. The author gives fundamentals on Spin-structures on manifolds and Spin-connections induced by the Levi-Civita connection on a Riemannian manifold, defines the Dirac operator and cheek its basic properties (self-adjointness). The chapter is concluded with a formal proof of the index theorem for the Dirac operator which uses the above-mentioned Chern root algorithm and a simplified version of the relevant operator.NEWLINENEWLINENEWLINEIn the seventh chapter the author gives a more traditional proof of the local index theorem for both the signature and Dirac operators and justifies simplifications assumed in the course of the formal proofs.NEWLINENEWLINENEWLINEThe last chapter is devoted to the index theorem for Riemann-Roch operator on Kaehler manifolds. Again the exposition is self-contained and the discussion of the index theorem is preceded by an exposition of fundamentals of Kaehler manifolds.NEWLINENEWLINENEWLINESome minor errors were observed (mainly misprints) but these would not mislead a careful reader.