Differential Geometry

Introductory course in differential geometry, focusing on the notions of subvarieties of ^Rn, vector fields and flows.

Master the basic tools of differential geometry.

A Bachelor's degree in Mathematics.

Recommended prerequisites: the content of the L3 course "Calcul différentiel et équations différentielles" (Differential calculus and differential equations) in the Licence de Mathématiques (Mathematics degree) at the Université de Montpellier.

1. Plane and space curves: curvature of a plane curve, curvature and torsion of a space curve.
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 applications, regular values, Brown's theorem and applications.
5. Vector fields and streams.

The course will be illustrated by applications chosen by the teacher. Examples (not exhaustive):
- minimization of the total curvature of knotted curves;
- proof of Jordan's theorem in the plane;
- Gauss-Bonnet theorem on surfaces;
- notion of abstract variety with standard examples: projective spaces, grassmanians.