Introduction

These are informal notes of talks I gave at the Instructional Conference on Spectral Theory and Geometry, International Centre for Mathematical Sciences, Edinburgh, March 29–April 9, 1998. The first three days featured three introductory mini-courses consisting of three lectures each: (1) E.B. Davies on Friedrichs extensions of densely defined symmetric operators, and max-min methods and their computational aspects, (2) F. Burstall on introductory Riemannian geometry, and (3) myself on the Laplacian on Riemannian manifolds.

Burstall's course started from the definition of a manifold, and surveyed the basic definitions and theorems concerning (with slightly different order) connections, parallel translation, geodesies, exponential map, torsion and curvature, Jacobi fields, Riemannian metrics, Levi-Civita connections, geodesic spherical and Riemann normal coordinates, the conjugate and cut loci of a point, Riemann measure, divergence theorems, and the Laplacian, culminating in an elegant proof of Bishop's volume comparison theorem for Ricci curvature bounded from below (by way of the Lichnerowicz formula).

I will pick up the story from this point, and consider some elementary examples and theorems which are pleasing in their own right, and which are suitable and appropriate for presentation in such a course. We pick and choose in the presentation of detail, if any at all, in the proofs. Also, I have tried, in some strictly Riemannian topics, to complement Burstall's elegant treatment with a more classical approach to some of the same material. In particular, I devote more time than might be warranted to the calculation of the Laplacian in geodesic spherical coordinates.

Introduction

My mission was to describe the basics of Riemannian geometry in just three hours of lectures, starting from scratch. The lectures were to provide background for the analytic matters covered elsewhere during the conference and, in particular, to underpin the more detailed (and much more professional) lectures of Isaac Chavel. My strategy was to get to the point where I could state and prove a Real Live Theorem: the Bishop Volume Comparison Theorem and Gromov's improvement thereof and, by appalling abuse of OHP technology, I managed this task in the time alloted. In writing up my notes for this volume, I have tried to retain the breathless quality of the original lectures while correcting the mistakes and excising the out-right lies.

I have given very few references to the literature in these notes so a few remarks on sources is appropriate here. The first part of the notes deals with analysis on differentiate manifolds. The two canonical texts here are Spivak [5] and Warner [6] and I have leaned on Warner's book in particular. For Riemannian geometry, I have stolen shamelessly from the excellent books of Chavel [1] and Gallot-Hulin-Lafontaine [3]. In particular, the proof given here of Bishop's theorem is one of those provided in [3].

What is a manifold?

What ingredients do we need to do Differential Calculus? Consider first the notion of a continuous function:

yIntroduction

These are notes of some lectures on wave invariants and quantum normal form invariants of the Laplacian Δ at a closed geodesic γ of a compact boundaryless Riemannian manifold (M, g). Our purpose in the lectures was to give a survey of some recent developments involving these invariants and their applications to inverse spectral theory, mainly following the references [G.1] [G.2] [Z.1] [Z.2] [Z.3]. Originally, the notes were intended to mirror the lectures but in the intervening time we wrote another expository account on this topic [Z.4] and also extended the methods and applications to certain plane domains which were outside the scope of the original lectures [Z.5]. These events seemed to render the original notes obsolete. In their place, we have included some related but more elementary material on wave invariants and normal forms which do not seem to have been published before and which seem to us to have some pedagogical value. This material consists first of the calculation of wave invariants on manifolds without conjugate points using a global Hadamard-Riesz parametrix. Readers who are more familiar with heat kernels than wave kernels may find this calculation an easy-toread entree into wave invariants. A short section on normal forms leads the reader into this more sophisticated – and more useful – approach to wave invariants. We illustrate this approach by putting a Sturm-Liouville operator on a finite interval (with Dirichlet boundary conditions) into normal form.

Abstract Spectral Theory

The material in this section is standard theory which may be found in many textbooks, for example [4] which I follow closely. Much more comprehensive accounts are given in [11, 8], which are rightly regarded as classic accounts of the subject. My goal in these lectures is not to describe new research, but to provide students with the basic knowledge needed to follow the later and more advanced courses. However, in the third lecture I indulge myself somewhat by describing spectral theory from a computational point of view which will be familiar to numerical analysts, but not to most mathematicians and mathematical physicists. This lecture contains recent research material.

Let H be a separable Hilbert space, such as L2(U) where U is a region in RN, and let A be a differential operator acting in H it is common to abuse language and say that A acts in U. Since not all functions in H are differentiate the domain of A cannot be the whole of H, and we assume that it is a dense linear subspace Dom(H) of H. The precise choice of this subspace is both important and difficult in many cases. One often starts with a domain smaller than the final domain, which consists of suitably regular functions obeying the boundary conditions relevant to the operator in question, and then passes to a slightly larger domain by the following closure procedure.