Course Material
Course Structure (2013)
(31 lectures : 7+9+10+5)
Part I : Curves
Part II : Surfaces
Part III : Curvature
Part IV : Geodesics
Lectures
(Curves)
Lecture 1 : What is a curve ? Arc-length
Lecture 2 : Reparametrisation
Lecture 3 : Level curves vs. parametrised curves
Lecture 4 : Curvature
Lecture 5 : Plane curves
Lecture 6 : Space curves I
Lecture 7 : Space curves II
(Surfaces)
Lecture 8 : What is a surface ?
Lecture 9 : Smooth surfaces. Smooth maps
Lecture 10 : Tangents and derivatives. Normals (and orientability)
Lecture 11 : Level surfaces. Quadric surfaces
Lecture 12 : Ruled surfaces and surfaces of revolution
Lecture 13 : Compact surfaces
Lecture 14 : Length of curves on surfaces
Lecture 15 : Isometries of surfaces
Lecture 16 : Conformal mappings on surfaces
(Curvature)
Lecture 17 : The second fundamental form
Lecture 18 : The Gauss and Weingarten maps
Lecture 19 : Normal and geodesic curvatures
Lecture 20 : Parallel transport (and covariant derivative)
Lecture 21 : Gaussian and mean curvatures
Lecture 22 : Principal curvatures of a surface
Lecture 23 : Surfaces of constant Gaussian curvature
Lecture 24 : Flat surfaces
Lecture 25 : Surfaces of constant mean curvature
Lecture 26 : Gaussian curvature of compact surfaces
(Geodesics)
Lecture 27 : Definition and basic properties
Lecture 28 : Geodesic equations I
Lecture 29 : Geodesics equations II
Lecture 30 : Geodesics on surfaces of revolution
Lecture 31 : Geodesic as shortest paths
OPTIONAL
- The Gauss and Codazzi-Mainardi equations
- Gauss' remarkable theorem
- Surfaces of constant Gaussian curvature I
- Surfaces of constant Gaussian curvature II
