Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems. Special topics - Comparison geometry (theorems of Rauch, Toponogov, Bishop-Gromov), and Bochner techniques.
Suggested books and references:
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian geometry, Third edition., Universitext. Springer-Verlag, Berlin, 2004.
Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.
John Lee, Riemannian Geometry - An introduction to curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997.