Differential Geometry

An introductory course in differential geometry, focusing on the concepts of subvarieties of^Rn, vector fields, and flows.

Master the basic tools of differential geometry.

A bachelor's degree program in mathematics.

Recommended prerequisites: the material covered in the third-year (L3) course “Differential Calculus and Differential Equations” in the Bachelor’s program in Mathematics at the University of Montpellier.

1. Curves in the Plane and in Space: Curvature of a Curve in the Plane; Curvature and Torsion of a Curve in Space.
2. Review of differential calculus in^Rn: finite increments, local inversion, implicit functions, normal forms of immersions and submersions. Applications: subvarieties of^Rn, standard examples, tangent space, orientation.
3. Surfaces in^R3, second fundamental form, curvature.
4. Differentiable functions, regular values, Brown's theorem, and applications.
5. Vector fields and flows.

The course will be illustrated with applications chosen by the instructor. Examples (non-exhaustive list):
- lower bound on the total curvature of knotted curves;
- proof of Jordan’s theorem in the plane;
- Gauss-Bonnet theorem on surfaces;
- concept of an abstract manifold with standard examples: projective spaces, Grassmannian spaces.