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 j j 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 jα˙(t)j 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 Skalarproduktp ] ha; bi = a · b := aibi and its associated norm len(α):= TV(α):= sup α(ti) − α(ti−1) : jaj := ha; ai. ··· 2 t0< <tn I i=1 This is the supremal length of inscribed polygons. (One can A. CURVES show this Jordan length is finite over finite intervals if and only if α has a Lipschitz reparametrization, e.g., by arclength. For Given any interval I ⊂ R, a continuous map α: I ! Rn a Lipschitz curve, the velocity is defined almost everywhere, is called a (parametrized) curve [DE: parametrisierte Kurve] so the integrals we used above – giving displacement as the n in R . We write α(t) =: α1(t); : : : ; αn(t) : integral of velocity and length as the integral of speed – exist We say α is Ck if it has continuous derivatives of order up in the sense of Lebesgue.) to k. Here of course C0 means nothing more than continuous, If J is another interval and ': J ! I is an orientation- while C1 is a minimal degree of smoothness, which is insuffi- preserving homeomorphism, i.e., a strictly increasing surjec- cient for many of our purposes. Indeed, for this course, rather tion, then α◦': J ! Rn is a parametrized curve with the same than tracking which results require, say, C2 or C3 smoothness, image (or trace) as α, called a reparametrization of α. (Note we will use smooth [DE: glatt] to mean C1 and will typically that the reverse curveα ¯ : − I ! Rn, defined byα ¯(t):= α(−t), assume that all of our curves are smooth. traces the same image in reverse order; this could be called an Examples (parametrized on I = R): orientation-reversing reparametrization.) 3 When studying arbitrary continuous curves, it’s sometimes • α(t):= (a cos t; a sin t; bt) is a helix in R (for a; b , 0); helpful to allow more general reparametrizations via ': J ! I • β(t):= (t2; t3) is a smooth parametrization of a plane which is monotonic but not strictly monotonic. That is, we curve with a cusp; allow a reparametrization that stops at one point for a while – or that removes such a constant interval. • γ(t):= (sin t; sin 2t) is a figure-8 curve in R2; We instead focus on regular smooth curves α. Then if ': J ! I is a diffeomorphism [DE: Diffeomorphismus] 2 n n • µ(t):= (t; t ;:::; t ) is called the moment curve in R . (a smooth map with nonvanishing derivative, so that '−1 is also smooth) then α ◦ ' is again smooth and regular. A simple curve [DE: einfache Kurve] is one where the map We are interested in properties invariant under such smooth α: I ! Rn is injective. A closed curve [DE: geschlossene reparametrizations. Declaring a (regular smooth) curve to Kurve] is one where α: R ! Rn is T-periodic for some T > 0, be equivalent to any smooth reparametrization, this gives an meaning α(t + T) = α(T) for all t 2 R. Of course no closed equivalence relation on the space of all parametrized curves. curve is simple in the above sense; instead we define a sim- Formally, we could define an unparametrized (smooth) curve ple closed curve [DE: einfach geschlossene Kurve] as a closed as an equivalence class. These are really the objects we curve where α is injective on the half-open interval [0; T). want to study, but we do so implicitly, using parametrized A smooth (or even just C1) curve α has a velocity vector curves and focusing on properties that are independent of [DE: Geschwindigkeitsvektor]α ˙(t) 2 Rn at each point. The parametrization, switching to a different parametrization fundamental theorem of calculus says this velocity can be in- when convenient. tegrated to give the displacement vector For a fixed t0 2 I we define the arclength function s(t):= R t Z b jα˙(t)j dt. Here s maps I to an interval J of length len(α). If t0 α˙(t) dt = α(b) − α(a): α is a regular smooth curve, then s(t) is smooth, with positive a derivatives ˙ = jα˙j > 0 equal to the speed. Thus it has a smooth The speed [DE: Bahngeschwindigkeit] of α is jα˙(t)j ≥ 0. We inverse function ': J ! I. We say β = α ◦ ' is the arclength say α is regular [DE: regulär] if the speed is positive (that is, parametrization [DE: Parametrisierung nach Bogenlänge] (or 1 J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2019 unit-speed parametrization) of α. We have β(s) = α('(s)), In particular, if v ? w (i.e., v · w ≡ 0) then v0 · w = −w0 · v. so β(s(t)) = α('(s(t))) = α(t). It follows that β has constant And if jvj is constant then v0 ? v. (Geometrically, this is just speed 1, and thus that the arclength of βj[a;b] is b − a. saying that the tangent plane to a sphere is perpendicular to The arclength parametetrization is hard to write down ex- the radius vector.) In particular, we have ~κ ? T. plicitly for most examples – we have to integrate a square root, Example: the circle α(t) = (r cos t; r sin t) of radius r then invert the resulting function. (There has been some work (parametrized here with constant speed r) has in computer-aided design on so-called “pythagorean hodo- graph curves”, curves with rational parametrizations whose −1 T = (− sin t; cos t); ~κ = (cos t; sin t); κ ≡ 1=r: speed is also a rational function, with no square root. But this r still doesn’t get us all the way to a unit-speed parametrization.) The fact that the arclength parametrization always exists, Given regular smooth parametrization α with speed σ := however, means that we can use it when proving theorems, s˙ = jα˙j, the velocity is σT, so the acceleration vector is and this is usually easiest. (Even when considering curves with less smoothness, e.g., Ck, there is a general principle α¨ = σT· = σ˙ T + σT˙ = σ˙ T + σ2T 0 = σ˙ T + σ2~κ: that no regular parametrization is smoother than the arclength parametrization.) Solving for ~κ we get the formula Although for an arbitrary parameter we have used the name t (thinking of time) and written d=dt with a dot, when we use hα,¨ α˙i α˙ jα˙j2 ~κ = α¨ − hα,¨ Ti T = α¨ − the arclength parametrization, we’ll call the parameter s and jα˙j2 write d=ds with a prime. Of course, for any function f along the curve, the chain rule says for the curvature of a curve not necessarily parametrized at unit speed. d f ds d f n = ; i.e., f 0 = f˙=s˙ = f˙=jα˙j: Any three distinct points in R lie on a unique circle (or ds dt dt line). The osculating circle to α at p is the limit of such circles through three points along α approaching p. It is also the limit Suppose now that α is a regular smooth unit-speed curve. of circles tangent to α at p and passing through another point Then its velocity α0 is everywhere a unit vector, the (unit) tan- of the curve approaching p. Again, one can investigate the gent vector [DE: Tangenten(einheits)vektor] T(s):= α0(s) to exact degree of smoothness required to have such limits exist. the curve. (In terms of an arbitrary regular parametrization, we have of course T = α/˙ jα˙j.) Note that the second-order Taylor series for a unit-speed curve around the point p = α(0) (we assume without further End of Lecture 8 Apr 2019 comment that 0 2 I) is: We should best think of T(s) as a vector based at p = α(s), perhaps as an arrow from p to p + T(s), rather than as a point s2 α(s) = p + sT(0) + ~κ(0) + O(s3): in Rn.