Differential Geometry

Introductory course in differential geometry, focusing on the concepts of submanifolds of^Rn, vector fields, and flow.

Master basic differential geometry tools.

A Bachelor's degree in Mathematics.

Recommended prerequisites: the content of the L3 course "Differential Calculus and Differential Equations" from the Bachelor's degree 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. Revisions of differential calculus in^Rn: finite increments, local inversion, implicit functions, normal forms of immersions and submersions. Applications: submanifolds of^Rn, standard examples, tangent space, orientation.
3. Surfaces in^R3, second fundamental form, curvature.
4. Differentiable applications, 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):
- minimization of the total curvature of knotted curves;
- proof of Jordan's theorem in the plane;
- Gauss-Bonnet theorem on surfaces;
- concept of abstract manifolds with standard examples: projective spaces, Grassmannian manifolds.