Textbook: Manfredo do Carmo, Differential Geometry of Curves and Surfaces.
Prentice-Hall, 1976.
Intrinsic geometry of surfaces (continued):
- review of Math 345;
- the Gauss-Bonnet theorem;
- the exponential map;
- shortest geodesics.
Global differential geometry of surfaces:
- rigidity of the sphere;
- completeness {\it vs.} geodesic completeness;
- the Bonnet theorem (compactness of surfaces of positive
curvature);
- Jacobi fields and conjugate points; Jacobi's theorem;
- Hadamard theorem (on surfaces of negative curvature);
- surfaces of zero Gaussian curvature;
- abstract surfaces;
- Hilbert's theorem on the hyperbolic plane;
- ... (whatever time permits).