This is a continuation of the course 532: Introduction to Differential Geometry. Depending on time and progress, topics may include:
1. Jacobi Fields and conjugate points
2. Isometric Immersions
3. Hopf-Rinow, Hadamard Theorems
4. Variations of energy, Bonnet-Myers theorem
5. Rauch Comparison Theorem
6. Harmonic forms, Hodge Theorem
7. Holonomy groups
References:
do Carmo: Riemannian Geometry
Jost: Riemannian Geometry and Geometric Analysis
Besse: Einstein manifolds
do Carmo, Riemannian Geometry
point set topology, multivariable calculus, 532: introduction to differential geometry