• Riemannian metrics
• Linear connection: Definition, geometrical interpretation, covariant derivation, parallel transport, torsion and curvature, the identity of Bianchi, Levi-Civita connection.
• Geodesics: Definition, local existence, exponential map, minimal geodesics, theorem of Hoph-Rinow, completeness and isometries.
• Metric properties of submanifolds of the Riemannian manifolds: normal field, Gauss equation, equations of Weingarton and of Gauss-Codazzi, Egreguim theorem, submanifolds totally geodesic.
• Calculus of variations: variations of C -curves, space of paths, critical paths, energy of a path, formula of the first and second variation, Jacobi field, index theorem.