J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2019
This course is an introduction to the geometry of smooth if the velocity never vanishes). Then the speed is a (smooth) curves and surfaces in Euclidean space Rn (in particular for positive function of t. (The cusped curve β above is not regular n = 2, 3). The local shape of a curve or surface is described at t = 0; the other examples given are regular.) in terms of its curvatures. Many of the big theorems in the DE The lengthR [ : Länge] of a smooth curve α is defined as subject – such as the Gauss–Bonnet theorem, a highlight at the | | len(α) = I α˙(t) dt. (For a closed curve, of course, we should end of the semester – deal with integrals of curvature. Some integrate from 0 to T instead of over the whole real line.) For of these integrals are topological constants, unchanged under any subinterval [a, b] ⊂ I, we see that deformation of the original curve or surface. Z b Z b We will usually describe particular curves and surfaces |α˙(t)| dt ≥ α˙(t) dt = α(b) − α(a) . locally via parametrizations, rather than, say, as level sets. a a Whereas in algebraic geometry, the unit circle is typically be described as the level set x2 + y2 = 1, we might instead This simply means that the length of any curve is at least the parametrize it as (cos t, sin t). straight-line distance between its endpoints. Of course, by Euclidean space [DE: euklidischer Raum] The length of an arbitrary curve can be defined (following n we mean the vector space R 3 x = (x1,..., xn), equipped Jordan) as its total variation: with with the standard inner product or scalar product [DE: P Xn Skalarprodukt√ ] ha, bi = a · b := aibi and its associated norm len(α):= TV(α):= sup α(ti) − α(ti−1) . |a| := ha, ai. ··· ∈ t0<