Curves in the plane and in the space. Frenet formulas. The group of orthogonal transformations. Regular surfaces. Metrics on surfaces. The first and the second fundamental forms. Normal and mean curvature. Curvature lines. Gaussian curvature. Derivation formulas and Bonnet theorem. Gauss' Theorema Egregium. Covariant derivative and geodesic lines. Euler-Lagrange equations. Gauss-Bonnet formula, Euler characteristic. Minimal surfaces. Surfaces of constant curvature. Conformal parameterization. The Weierstrass representation. Smooth manifolds and smooth maps. Tensor calculus. Embedding of smooth manifolds into the Euclidean space. Tangent and cotangent bundle. Vector fields. Metric tensor. Affine connection and covariant derivative. Curvature and torsion. Riemannian connection (Levi-Civita). Geodesic lines. Examples: Lobachevsky plane, pseudo-Euclidean spaces with application to physics.