In these lectures, we introduce some fundamental notions of differential geometry such as submanifolds, tangent spaces, riemannian metrics, curvature, covariant derivatives, geodesics, etc.

At the end of the series of lectures, we will prove the Gauss-Bonnet theorem. This classical result, which relates local and global aspects of the geometry of surfaces, is the model for many important developments in geometry in the 20th and 21st centuries.


1- Global results on curves
2- Differential calculus on submanifolds
3- Vector fields. Orientation
4- Transversality. Jordan-Brouwer separation theorem.
5. Curvatures on surfaces. Second fundamental form.
6. Intrinsic geometry of surfaces. Gauss Theorema Egregium
7. Goedesics.
8. Integration on surfaces. Grenn-Ostrogradski formula.
9. Gauss-Bonnet* theorem

Last Modification : Tuesday 30 April 2013