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