Differential Geometry

Prerequisites: Variety analysis (varieties and tensors), Basic theorem of ordinary differential equations (can be run in parallel).

Plane curves; isoperimetric inequality. Curves in space, curvature and torsion, Frenet frames, theorem of existence and uniqueness of curves.
Surfaces; in R3. First fundamental form, area. Gauss map; principal directions, Gaussian curvature and mean curvature, lines of curvature.
Intrinsic geometry, classical examples of surfaces. Covariant derivative, the theorem egregium; geodesic curvature, the geodesic equations, calculation of geodesics on surfaces, the exponential map, the Gauss-Bonnet theorem. Introduction to manifolds. Other topics.

References:
CARMO, M. – Differential Geometry of Curves and Surfaces. Englewood Cliffs, Prentice-Hall, 1976.
DUPONT, J. – Fiber Bundles in Gauge Theory, Arhus Universitet, 2003.
DUPONT, J. – Curvature and Characteristic Classes, Springer, 1978.
KOBAYASHI, S. e NOMIZU, K. – Foundations of Differential Geometry, Wiley  – Interscience, 1996.
MADSEN, H. – From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Cambridge University Press, 1997.
MONTIEL, S. e ROS, A. – Curves and Surfaces, Graduate Studies in Mathematics, vol. 69, AMS, 2005.
POOR, W. A. – Differential Geometric Structures, Dover Publications; Dover Ed edition, 2007.
SPIVAK, M. – A Comprehensive Introduction to Differential Geometry, vol.3, Berkeley, Publish or Perish, 1979.

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