# Topics - Riemannian manifolds: basic examples of Riemannian metrics, Levi-Civita connection. - Geodesics: definition, first variation formula, exponential map, minimizing properties of geodesics. - Curvature: Riemann curvature tensor, sectional curvature, Ricci curvature, scalar curvature. - Riemannian submanifolds: examples, second fundamental form, Gauss–Codazzi equations. - Jacobi fields: Jacobi equation, conjugate points. - Completeness: Hopf–Rinow and Cartan–Hadamard theorems - Constant curvature: classification of complete manifolds with constant curvature. - Second variation and applications: second variation formula, Bonnet–Myers and Synge’s theorems.