285G, Lecture 0: Riemannian manifolds and curvature

Next week (starting on Wednesday, to be more precise), I will begin my class on Perelman’s proof of the Poincaré conjecture. As I only have ten weeks in which to give this proof, I will have to move rapidly through some of the more basic aspects of Riemannian geometry which will be needed throughout the course. In particular, in this preliminary lecture, I will quickly review the basic notions of infinitesimal (or microlocal) Riemannian geometry, and in particular defining the Riemann, Ricci, and scalar curvatures of a Riemannian manifold. (The more “global” aspects of Riemannian geometry, for instance concerning the relationship between distance, curvature, injectivity radius, and volume, will be discussed later in this course.) This is a review only, in particular omitting any leisurely discussion of examples or motivation for Riemannian geometry; it is impossible to compress this subject into a single lecture, and I will have to refer you to a textbook on the subject for a more complete treatment (I myself am using the text “Riemannian geometry” by my colleague here at UCLA, Peter Petersen).