UC Berkeley

We first review a number of well known theorems in Riemannian geometry, and we discuss in detail some of their proofs. We then present, in chapters 2, 3 and 4, proofs of three results: a local $L_p$ bound on $||\text{Ric}||$ for $p<\frac{1}{2}$ under a lower Ricci curvature bound, the lower semicontinuity of volume on surfaces of bounded Euler characteristic, and a construction for metrics of nonpositive sectional curvature that develop a positive sectional curvature somewhere with respect to the Ricci flow.