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.