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.