Prerequisites: Familiarity with the statement of the inverse function theorem in Rn and with the basics of the topology of Rn space.

Curves in the plane and in space. Length, curvature and torsion. Surfaces in R3. Definition, parametrizations and coordinate changes, tangent planes. First fundamental form, distances and areas. Curvature. The normal application of Gauss, the second fundamental form, principal curvatures, Gaussian curvature and mean curvature, lines of curvature and asymptotics. The sign of Gaussian curvature. Mean curvature and area. The sphere rigidity theorem and Alexandrov’s theorem. Intrinsic geometry versus extrinsic geometry. Isometries and Gauss’s Egregious Theorem. Covariant derivative and parallel transport. Geodesic curvature and geodesics. The Gauss-Bonnet Theorem and applications. Other global theorems and other topics.

References:
CARMO, M. – Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.*
MONTIEL, S. and ROS, A. – Curves and Surfaces, Graduate Studies in Mathematics, vol. 69, AMS, 2005.

* Basic syllabus. The teacher has the autonomy to make any changes.