Differential Geometry

In this course we present the basic concepts of differential geometry (metric, curvature, connection, etc.). The main goal of our study is a deeper understanding of the geometrical meaning of all notions and theorems.

Prerequisites: Multivariable Calculus; Linear Algebra; Ordinary Differential Equations.


Curriculum:

  1. Plane and space curves. Curvature, torsion, Frenet frame. Curves in pseudo-Euclidean spaces.
  2. Surfaces in 3-space. Metrics and the second quadratic form. Curvature. `Theorema egregium’ of Gauss.
  3. Parallel translations. Gauss-Bonnet formulas.
  4. Fibrations. Topological connections as parallel translations. Curvature of a topological connection. Frobenius criterion.
  5. Vector bundle. Tangent, cotangent, and tensor bundles. Sections.
  6. Differential forms on manifolds. Integrals of differential forms. Stokes’ theorem.
  7. Connections as covariant derivatives. Curvature and torsion tensors.
  8. Riemannian manifolds. Symmetries of the curvature tensor. Geodesics. Extremal properties of geodesics.
  9. Sard’s lemma. Transversality theorem. Applications.

Textbooks

  • Monfredo Do Carmo, Differential Geometry of Curves and Surfaces.