In general one should of course be able to define and use the concepts from the course. Particularly important concepts and results are listed below. In an exam you may be asked to prove one (or more) of these results.

Chapter 1:

Regular curve, unit speed parametrisation.

Chapter 2:

Curvature and torsion for unit speed and arbitrary parametrisation. Frenet-Serret equations. Thm 2.2.6 and 2.3.6: curvature and torsion determine essentially the curve.

Chapter 3:

Four vertex theorem. The isoperimetric inequality.

Chapter 4 and 5:

Regular surface. Thm 5.1.1: a level surface f = 0 is regular if the gradient of f does not vanish in every point of the surface. A surface is locally representable as graph. Differentiable map between regular surfaces. Tangent plane. Prop 4.4.2: the tangent plane is a vector space of dimension 2. Normal vector. Orientability. Surfaces of revolution. Ruled surfaces.

Chapter 6:

First fundamental form. (Local) isometries. Conformal maps.

Chapter 7 and 8:

Gauss and Weingarten maps. Normal and geodesic curvatures. Normal section. Gaussian and mean curvature. Principal curvatures and principal vectors. Umbilic points. Asymptotic lines, lines of curvature (defined in exercises 7.3.6 and 8.2.2).

Chapter 9:

Geodesics. A normal section is a geodesic. Differential equation for geodesics. Local isometries preserve geodesics. Geodesics on a surface of revolution.

Chapter 10:

Theorema egregium: Gaussian curvature is intrinsic.

Chapter 13:

The Gauss-Bonnet Theorem for curvilinear polygons and for compact surfaces.