In this chapter we will generalize the Laplacian on Euclidean space to an operator on differential forms on a Riemannian manifold. By a Riemannian manifold, we roughly mean a manifold equipped with a method for measuring lengths of tangent vectors, and hence of curves. Throughout this text, we will concentrate on studying the heat flow associated to these Laplacians. The main result of this chapter, the Hodge theorem, states that the long time behavior of the heat flow is controlled by the topology of the manifold.

In §1.1, the basic examples of heat flow on the one dimensional manifolds S1 and R are studied. The heat flow on the circle already contains the basic features of heat flow on a compact manifold, although the circle is too simple topologically and geometrically to really reveal the information contained in the heat flow. In contrast, heat flow on R is more difficult to study, which indicates why we will restrict attention to compact manifolds. In §1.2, we introduce the notion of a Riemannian metric on a manifold, define the spaces of L2 functions and forms on a manifold with a Riemannian metric, and introduce the Laplacian associated to the metric. The Hodge theorem is proved in §1.3 by heat equation methods. The kernel of the Laplacian on forms is isomorphic to the de Rham cohomology groups, and hence is a topological invariant. The de Rham cohomology groups are discussed in §1.4, and the isomorphism between the kernel of the Laplacian and de Rham cohomology is shown in §1.5.

In the first chapter we discussed heat flow on a compact manifold and the topological significance of the long time behavior of the heat flow. In contrast, the short time behavior of the heat flow might appear trivial, as we know the heat operator goes to the identity operator as t ↓ 0. However, we shall see in Chapter 3 that the way in which the heat kernel approaches the delta function (the kernel of the identity operator) is determined by the local Riemannian geometry of the manifold.

This chapter covers those parts of Riemannian geometry used to construct the heat kernel and its short time asymptotics in Chapter 3. We also prove Garding's inequality from Chapter 1, and develop some of the supersymmetric techniques used to prove the Chern-Gauss-Bonnet theorem in Chapter 4. The key concepts discussed are the various types of curvature in Riemannian geometry (§2.1), the Levi-Civita connection associated to a Riemannian metric (§2.2.1), the Weitzenböck formula and Gårding's inequality (§2.2.2), geodesies and Riemannian normal coordinates (§2.3). There is a technical section on the Laplacian in normal coordinates (§2.4). Other references for this material include [4], [27], [64, Vols. I, II].

Curvature

There is no better place to begin a discussion of curvature than with Gauss' solution to the question: when is a piece of a surface in R3 (such as the earth's surface) flat? By flat, we mean that there should exist a distortion free – i.e. isometric – map from the piece of the surface to a region in the standard plane.

From the basic definitions, differential topology studies the global properties of smooth manifolds, while differential geometry studies both local properties (curvature) and global properties (geodesies). This text studies how differential operators on a smooth manifold reveal deep relationships between the geometry and the topology of the manifold. This is a broad and active area of research, and has been treated in advanced research monographs such as [5], [30], [59]. This book in contrast is aimed at students knowing just the basics of smooth manifold theory, say through Stokes' theorem for differential forms. In particular, no knowledge of differential geometry is assumed.

The goal of the text is an introduction to central topics in analysis on manifolds through the study of Laplacian-type operators on manifolds. The main subjects covered are Hodge theory, heat operators for Laplacians on forms, and the Chern-Gauss-Bonnet theorem in detail. Atiyah-Singer index theory and zeta functions for Laplacians are also covered, although in less detail. The main technique used is the heat flow associated to a Laplacian. The text can be taught in a one year course, and by the conclusion the student should have an appreciation of current research interests in the field.

We now give a brief, quasi-historical overview of these topics, followed by an outline of the book's organization.

The only natural differential operator on a manifold is the exterior derivative d taking κ-forms to (κ + 1)-forms. This operator is defined purely in terms of the smooth structure.

In this chapter we will encode the spectral information of a Laplacian-type operator into a zeta function first introduced by Minakshisundaram and Pleijel [48] and Seeley [61]. While this is theoretically equivalent to the encoding of the spectrum given by the trace of the heat operator, the zeta function contains spectral information hard to obtain by heat equation methods. In particular, the important notion of the determinant of a Laplacian is given in terms of the zeta function.

In §5.1, we introduce the zeta function and use it to produce new conformal invariants in Riemannian geometry. In §5.2, we outline Sunada's elegant construction of isospectral, nonhomeomorphic four-manifolds. While the results in §5.1 are conceivably obtainable directly from the heat operator, the results in §5.2 depend on the zeta function for motivation. Finally, in §5.3 we discuss the determinants of Laplacians on forms and define analytic torsion, which we show is a smooth invariant subtler than the invariants produced in Chapter 4. We conclude with an overview of recent work of Bismut and Lott connecting analytic torsion with Atiyah-Singer index theory for families of elliptic operators. This last discussion is the most difficult part of the book and contains no proofs.

The Zeta Function of a Laplacian

By a Laplacian-type operator, we mean any symmetric second order elliptic differential operator Δ : Γ(E) → Γ(E) acting on sections ƒ of a Hermitian bundle E over a compact n-manifold M satisfying 〈Δƒ, ƒ〉 ≥ C〈ƒ, ƒ〉 for some C ∈ R. The basic examples are the Laplacians on forms, where C = 0.

The Atiyah-Singer index theorem is a deep generalization of the classical Gauss- Bonnet theorem, including as special cases the Chern-Gauss-Bonnet theorem, the Hirzebruch signature theorem, and the Hirzebruch-Riemann-Roch theorem. Although the index theorem is about 35 years old at this point, it continues to have new applications in areas as apparently diverse as number theory and mathematical physics. The index theorem and its various generalizations (families index theorem, K-theoretic versions, etc.) admit many interpretations. We will choose the point of view that the index theorem expresses topological quantities in terms of geometric ones, just as in the Gauss-Bonnet theorem. This viewpoint leads to a heat equation proof of the index theorem, suggested by McKean and Singer [43] in the late 1960s and established by Gilkey [29], Patodi [55, 56], and Atiyah, Bott and Patodi [1] in the early 1970s. The heat equation method has since been refined by Getzler [28] (cf. [5]).

In this chapter, we will give a complete heat equation proof for the Chern- Gauss-Bonnet theorem, and state without proof the Hirzebruch signature theorem, the Hirzebruch-Riemann-Roch theorem, and the Atiyah-Singer index theorem. Complete proofs can be found in [5] and [30]. We have also included a short introduction to characteristic classes.

The Chern-Gauss Bonnet Theorem

The key ideas in the heat equation method are (i) by Chapter 1, the long time behavior of the heat operator for the Laplacian on forms is controlled by the topology of the manifold in the form of the de Rham cohomology, (ii) the short time behavior is controlled by the geometry of the asymptotic expansion, as explained in Chapter 3, and (iii) certain combinations of heat operators will have time independent behavior.