4 - Basic Riemannian geometry

Summary

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: