Differential Geometry of Curves and Surfaces

Prerequisites: Basic topology of Rn and familiarity with the inverse function theorem in more variables.

Curves in the plane and in space. Length, curvature and torsion. Surfaces in R3. Definitions, charts and coordinate changes, tangent planes. First fundamental form, distances and areas. Curvature. The Gauss map, the second fundamental form, principal curvatures, Gaussian and mean curvatures, lines of curvature and asymptotic lines. The sign of the Gaussian curvature. Mean curvature and area. Sphere rigidity theorem and Alexandrov’s Theorem. Intrinsic geometry versus extrinsic geometry. Isometries and Gauss’ Egregium Theorem. Covariant derivatives and paralel transport. Geodesic curvature and geodesics. The Gauss-Bonnet Theorem and applications. Other global theorems and topics.

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

* Standard program. The teacher has the autonomy to make any changes.