A selection of the following topics:
● Exponential map and the Hopf-Rinow theorem
● Connection between curvature and topology (e.g. Myer's theorem, Hadamard-Cartan, Klingenberg, rigidity theorems)
● Closed geodesics
● Stokes' theorem and Cohomology
● Spaces of constant curvature, Lie groups, homogeneous spaces
● Conformal geometry, geometric differential equations
● Basic notions from differential topology

• Lee, John M., Introduction to Smooth Manifolds, Springer, 2012
• Lee, John M., Riemannian Manifolds: An Introduction to Curvature, Springer, 1997
• Kühnel, Wolfgang, Differentialgeometrie:Kurven - Flächen - Mannigfaltigkeiten, Springer, 2012
• O'Neill, Barret: Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983

Further Reading:

• Minimum number of charts for RP^n: http://link.springer.com/chapter/10.1007%2FBFb0085228
• C1-embedding of the flat torus in R^3: http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html