This is approximate. I will attempt to keep track of all the references used in class.
Date  Material  Problems/Handouts 
Sept. 5 10  What is a manifold? References (doC, Chapter 0), (Tau, Chapter 1) 
Homework 1 
Sept. 1217  Vector fields, the Tangent bundle, Lie derivative, metrics (doC. Chapter 0, 1)  
Sept. 19 Sept. 26  Metrics, connections, geodesics: (doC, Chapter 13)  Homework 2 
Sept. 26 Oct. 10  Tensors, Curvature, Jacobi Fields: (doC, Chapter 35 )  Homework 3 
Oct. 15 22  Covering spaces, theorems of HopfRinow and Hadamard (doC, Chapter 7),(Hat, Chapter 1)  Homework 4 
Oct. 24Oct. 31  The cut locus, embedded submanifolds, mean curvature (doC, Chapter 6), (Pet, Sec. 5.7.3) 

Nov. 5  Nov. 19  Space with constant curvature, comparison geometry (do C, Chapter 8), (Pet, Chapters 1, 3)  Homework 5 
Nov. 21Dec. 3  BishopGromov, Cheng's sphere theorem, CheegerGromoll splitting(Pet) (ScYa)  Homework 6 
Dec. 3  Dec. 9  The segment inequality, ChengYau gradient estimate (Pet, Chapter 7) (ScYa)  Homework 7 