A student who successfully completes this course will be able to:
1. State the definition of a metric and an isometry.
2. Use metrics to define arc-length, distance and volume.
3. Give proofs of and use the Divergence Theorem and Green's Theorem on Riemannian manifolds.
4. State definitions and properties of geometric invariants of Riemannian manifolds: Riemannian curvature, Ricci curvature, scalar curvature.
5. Define and compute Riemannian connection, covariant derivatives, and parallel translation.
6. Give a definition of geodesics (as curves with zero acceleration) and write their defining equations. Prove that geodesics are length-minimizing curves.
7. State and prove Hopf-Rinow completeness theorem.
8. State definitions and properties of geometric invariants of Riemannian submanifolds and hyperserfaces.
9. Distinguish between intrinsic and extrinsic invariants of Riemannian submanifolds.
10. State the Gauss-Bonnet Theorem. Explain the main ideas behind its proof and its importance as a global-local result