Course description: Riemannian geometry is a higher dimensional generalization of the intrinsic geometry of surfaces. The course introduces basic notions and theorems of Riemannian geometry. By the end of the course, students will learn how to handle the most important techniques of Riemannian geometry that will enable them to follow contemporary research and start their own research in the area of Riemannian geometry.

Topics:

• Connections on manifolds, torsion of a connection, parallel transport, Levi-Civita connection of a Riemannian manifold.
• Curvature tensor, Riemannian curvature tensor and its symmetries.
• Sectional curvature, Ricci curvaturre, scalar curvature, Weyl tensor.
• Length of a curve, first variation of the length, geodesics, exponential map, Gauss lemma.
• Second variation of the length. Jacobi fields, conjugated points, Morse index theorem.
• Geodesic completeness, Hopf-Rinow theorem.
• Curvature and topology: Myers' theorem, Hadamard manifolds.
• Rauch comparison theorem.
• Submanifolds of a Riemannian manifold.
• Integration on Riemannian manifolds.