Lecture Notes Differential Geometry I Urs Lang ETH Zürich Autumn Semester 2019 Preliminary and incomplete version 17 August 2020 Contents Differential Geometry in Rn 1 1 Curves . 1 2 Surfaces . 9 3 Intrinsic geometry of surfaces . 13 4 Curvature of hypersurfaces . 19 5 Special classes of surfaces . 25 6 Global surface theory . 30 7 Hyperbolic space . 33 Differential Topology 37 8 Differentiable manifolds . 37 9 Transversality . 44 10 Vector bundles, vector fields and flows . 52 11 Differential forms . 59 12 Lie groups . 67 Appendix 75 A Analysis . 75 B General topology . 77 C Multilinear algebra . 79 Bibliography 83 i ii Differential Geometry in Rn 1 Curves Parametrized curves In the following, the symbol I will always denote an interval, that is, a connected subset of R. A continuous map c : I ! X into a topological space X is called a (parametrized) curve in X. A curve defined on »0;1¼ is also called a path. Now let X = ¹X; dº be a metric space. The length L¹cº 2 »0;1¼ of the curve c : I ! X is defined as k Õ L¹cº := sup d¹c¹ti−1º; c¹tiºº; i=1 where the supremum is taken over all finite, non-decreasing sequences t0 ≤ t1 ≤ ::: ≤ tk in I. The curve c is rectifiable if L¹cº < 1, and c has constant speed or is parametrized proportionally to arc length if there exists a constant λ ≥ 0, the speed of c, such that for every subinterval »a; b¼ ⊂ I, L¹cj»a;b¼º = λ¹b − aº; if λ = 1, then c has unit speed or is parametrized by arc length. The curve c : I ! X is a reparametrization of another curve c˜: I˜ ! X if there exists a continuous, surjective, non-decreasing or non-increasing map ': I ! I˜ (thus a < b implies '¹aº ≤ '¹bº or '¹aº ≥ '¹bº, respectively) such that c = c˜ ◦ '. Then clearly L¹cº = L¹c˜º. The following lemma shows that every curve of locally finite length is a reparametrization of a unit speed curve. 1.1 Lemma (reparametrization) Suppose that c : I ! ¹X; dº is a curve with L¹cj»a;b¼º < 1 for every subinterval »a; b¼ ⊂ I. Pick s 2 I, and define ': I ! R such that '¹tº = L¹cj»s;t¼º for t ≥ s and '¹tº = −L¹cj»t;s¼º for t < s. Then ' is continuous and non-decreasing, and there is a well-defined unit speed curve c˜: '¹Iº ! X such that c˜¹'¹tºº = c¹tº for all t 2 I. Proof : Whenever a; b 2 I and a < b, then d¹c¹aº; c¹bºº ≤ L¹cj»a;b¼º = '¹bº − '¹aº: (∗) 1 Thus ' is non-decreasing. Moreover, given such a; b and > 0, there exists a sequence a = t0 < t1 < : : : < tk = b such that k Õ L¹cj»a;b¼º − ≤ d¹c¹ti−1º; c¹tiºº ≤ d¹c¹aº; c¹rºº + L¹cj»r;b¼º i=1 for all r 2 ¹a;t1¼, and there is a δ > 0 such that d¹c¹aº; c¹rºº < for all r 2 ¹a; a+δº; thus L¹cj»a;r¼º = L¹cj»a;b¼º − L¹cj»r;b¼º < 2 for r > a close enough to a. It follows that ' is right-continuous, and left-continuity is shown analogously. By (∗) there is a well-defined 1-Lipschitz curve c˜: '¹Iº ! X such that c˜¹'¹tºº = c¹tº for all t 2 I. Then L¹c˜j»'¹aº;'¹bº¼º = L¹cj»a;b¼º = '¹bº − '¹aº for all »a; b¼ ⊂ I, hence c˜ is parametrized by arc length. We now turn to the target space X = Rn, endowed with the canonical inner product n Õ hx; yi = ¹x1;:::; xnº;¹y1;:::; ynº := xi yi i=1 and the Euclidean metric p d¹x; yº := jx − yj := hx − y; x − yi: In the following we will tacitly assume that the interior of the interval I is non-empty. For q 2 f0g [ f1;2;::: g [ f1g we write as usual c 2 Cq¹I;Rnº if c is continuous or q times continuously differentiable or infinitely differentiable, respectively. In the case that q ≥ 1 and I is not open, this means that c admits an extension c¯ 2 Cq¹J;Rnº to an open interval J ⊃ I. Suppose now that c 2 Cq¹I;Rnº for some q ≥ 1. Then ¹ b 0 L¹cj»a;b¼º = jc ¹tºj dt < 1 a for every subinterval »a; b¼ ⊂ I (a not easy exercise), and thus the function ' from ¹ º ¯ t j 0¹ ºj 2 Lemma 1.1 satisfies ' t = s c r dr for all t I. The curve c is called regular if c0¹tº , 0 for all t 2 I; then '0 = jc0j > 0 on I, and both ': I ! '¹Iº and the inverse '−1 : '¹Iº ! I are also of class Cq, that is, ' is a Cq diffeomorphism. Note also that c 2 C1¹I;Rnº has constant speed λ ≥ 0 if and only if jc0¹tºj = λ for all t 2 I. Local theory of curves The following notions go back to Jean Frédéric Frenet (1816–1900). 1.2 Definition (Frenet curve) The curve c 2 Cn¹I;Rnº is called a Frenet curve if for all t 2 I the vectors c0¹tº; c00¹tº;:::; c¹n−1º¹tº are linearly independent. The n corresponding Frenet frame ¹e1;:::; enº, ei : I ! R , is then characterized by the following conditions: 2 n (1) ¹e1¹tº;:::; en¹tºº is a positively oriented orthonormal basis of R for t 2 I; 0 ¹iº ¹iº (2) span¹e1¹tº;:::; ei¹tºº = span¹c ¹tº;:::; c ¹tºº and hei¹tº; c ¹tºi > 0 for i = 1;:::; n − 1 and t 2 I. Condition (2) refers to the linear span. The vectors e1¹tº;:::; en−1¹tº are ob- 0 ¹n−1º tained from c ¹tº;:::; c ¹tº by means of the Gram–Schmidt process, and en¹tº is n−i n then determined by condition (1). Note that ei 2 C ¹I;R º for i = 1;:::; n − 1, in 1 n particular e1;:::; en 2 C ¹I;R º. 1.3 Definition (Frenet curvatures) Let c 2 Cn¹I;Rnº be a Frenet curve with Frenet frame ¹e1;:::; enº. For i = 1;:::; n − 1, the function κi : I ! R, 1 κ ¹tº := he 0¹tº; e ¹tºi; i jc0¹tºj i i+1 is called the i-th Frenet curvature of c. n−i−1 Note that κi 2 C ¹Iº; in particular κ1; : : : ; κn−1 are continuous. Suppose now that c = c˜ ◦ ' for some curve c˜ 2 Cn¹I˜;Rnº and a Cn diffeomor- phism ': I ! I˜ with '0 > 0. For i = 1;:::; n − 1, the i-th derivative c¹iº¹tº is a Íi ¹kº 0 i linear combination k=1 ak¹tº c˜ ¹'¹tºº with ai¹tº = ¹' ¹tºº > 0, thus spanc0¹tº;:::; c¹iº¹tº = span¹c˜ 0 ◦ 'º¹tº;:::;¹c˜ ¹iº ◦ 'º¹tº; c is Frenet if and only if c˜ is Frenet, and the corresponding Frenet vector fields then satisfy the relation ei = e˜i ◦ '. Likewise, for the Frenet curvatures, 1 1 κ = he 0; e i = ¹e˜ 0 ◦ 'º'0; e˜ ◦ ' = κ˜ ◦ ': i jc0j i i+1 jc˜ 0 ◦ 'jj'0j i i+1 i Thus the curvatures are invariant under sense preserving reparametrization. 1.4 Proposition (Frenet equations) Let c 2 Cn¹I;Rnº be a Frenet curve with Frenet frame ¹e1;:::; enº and Frenet curvatures κ1; : : : ; κn−1. Then κ1; : : : ; κn−2 > 0, and 8 >κ1e2 if i = 1, 1 0 ><> e = −κ e + κ e if 2 ≤ i ≤ n − 1, jc0j i i−1 i−1 i i+1 > >−κn−1en−1 if i = n. : Proof : Since ¹e1¹tº;:::; en¹tºº is orthonormal, n 0 Õ 0 ei ¹tº = hei ¹tº; ej¹tºiej¹tº j=1 3 0 0 0 for i = 1;:::; n, and since hei ; eji + hei; ej i = hei; eji = 0, the coefficient matrix 0 K¹tº = ¹hei ¹tº; ej¹tºiº is skew-symmetric. For i = 1;:::; n − 1, 0 0 hei ; ei+1i = jc jκi: Now let i ≤ n − 2, and recall condition (2) of Definition 1.2. The vector ei¹tº Íi ¹kº 0 is a linear combination k=1 aik¹tºc ¹tº with aii¹tº > 0, so ei ¹tº is of the form Íi ¹kº ¹i+1º k=1 bik¹tºc ¹tº + aii¹tºc ¹tº, and it follows that 0 0 hei ; ei+2i = ::: = hei ; eni = 0 0 ¹i+1º and hei ; ei+1i = aii hc ; ei+1i > 0. This gives the result. In the case n = 2, a curve c 2 C2¹I;R2º is Frenet if and only if c is regular. Then the sole Frenet curvature 1 κ := κ = he 0; e i or 1 jc0j 1 2 0 0 is called the oriented curvature (or signed curvature) of c. Note that e1 = c /jc j 0 and hc ; e2i = 0, thus hc00; e i det¹e ; c00º det¹c0; c00º κ = 2 = 1 = : or jc0j2 jc0j2 jc0j3 The Frenet equations may be written in matrix form as 0 1 e1 0 κor e1 0 0 = : jc j e2 −κor 0 e2 The osculating circle (Schmiegkreis) of c at a point t with κor¹tº , 0 is the circle with center c¹tº+¹1/κor¹tººe2¹tº and radius 1/jκor¹tºj, which approximates the curve at t up to second order (exercise). In the case n = 3, c 2 C3¹I;R3º is a Frenet curve if and only if c0 and c00 are everywhere linearly independent.