Chapter 1:
Regular curve, unit speed paramatrisation.
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 (with proof). The isoperimetric inequality (with proof, Wirtinger's lemma may be assumed).
Chapter 4 and 5:
Regular surface. Thm 5.1.1 (with proof): 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 (with proof): 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-Bonnets Theorem for curvilinear polygons and for compact surfaces.