2. On the Fundamental Theorem for Curves in Space

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To start the proof, let us look at the Serret-Frenet system (2.1) in matrix form. Regarding t, n and b 0 as row 1 vectors 0 we have 1 1 B@ t n b CA0 We therefore consider the equations for V : (; ) ! M 3 (R) with A(s) = = B@ 0 CA 0 B@ 0 t n 0 0 0 b CA : (2.2) V 0 = AV (2.3) 0 B@ 0 (s) 0 (s) 0 (s) 0 (s) 0 1 CA : (2.4) One may get local existence of solutions to (2.3) with prescribed initial data, by applying the following theorem from MM04 (Supplerende noter til punktmængdetopologi, Sætning 2.2, side 6): Theorem 2.2. Let B R n+1 be an open set. Suppose f : B ! R n , (s; x) 7! f (s; x) is a continuous function whose partial derivatives @f=@x j , j = 1; : : : ; n, exist and are continuous in B. Fix a point (s 0 ; x 0 ) in B. Then the dierential equation x 0 (s) = f (s; x(s)) with initial condition x(s 0 ) = x 0 ; has a unique solution x(s) with s in an interval (s 0 "; s 0 + ") for some " > 0. To apply this to (2.3), note that M 3 (R) = R 9 , so take n = 9 and put f (s; V ) = A(s)V; B = (; ) ¢ M 3 (R): As the entries of A are smooth in s, the required partial derivatives of f exist and are smooth. Thus, we have local solutions to (2.3) around any given point s 0 2 (; ) with any given initial data V 0 2 M 3 (R). 2.2
F or two solutions V 1 and V 2 to (2.3) on some subinterval of ( 0 ; 0 ) (; ), consider the matrix V T 2 V 1 obtained by multiplying the transpose of V 2 onto V 1 . Dierentiating we have (V T 2 V 1 ) 0 = (V 0 2 )T V 1 + V T 2 V 0 1 = (AV 2) T V 1 + V T 2 (AV 1 ) = V T 2 (AT + A)V 1 = 0; since A in (2.4) skew-symmetric. We have thus proved Lemma 2.3. Suppose V 1 and V 2 satisfy V 0 i = AV i , i = 1; 2 on ( 0 ; 0 ) with A as in (2.4). Then V T 2 V 1 is constant on ( 0 ; 0 ). Fix a point s 0 2 (; ) and let V be the maximal solution to (2.3) with initial condition V (s 0 ) = Id, where Id is the (3 ¢ 3) identity matrix. Here maximal means that V is dened on some subinterval ( 0 ; 0 ) of (; ) and that there is no such solution dened on a larger subinterval. Our aim is to prove that ( 0 ; 0 ) = (; ). Lemma 2.4. V (s) is an orthogonal matrix for each s 2 ( 0 ; 0 ), i.e., V T (s)V (s) = Id : Proof. By Lemma 2.3, V T V is constant on ( 0 ; 0 ). At s 0 it has value Id T Id = Id. Lemma 2.5. The set O(3) = f X 2 M 3 (R) : X T X = Id g of orthogonal matrices is a compact subset of M 3 (R). Proof. If X is orthogonal, then each column x i of X is a unit vector in R 3 . Thus as vector in R 9 , X has length kx 1 k 2 + kx 2 k 2 + kx 3 k 2 = 3. So O(3) is bounded in M 3 (R). The function F : M 3 (R) ! M 3 (R) given by F (X) = X T X is continuous, so O(3) = F 1 (fIdg) is closed. Therefore, O(3) is closed and bounded and hence compact. Suppose 0 < . Let (~s i ) 1 i=1 be a sequence of points in ( 0; 0 ) converging to 0 . Then as V (~s i ) lie in the compact set O(3), there is a subsequence (s n ) 1 n=0 and a point ~V 2 O(3) such that V (s n ) ! ~ V as n ! 1: Let W be a solution of the system (2.3) with initial condition W ( 0 ) = V ~ . Then W (s) is dened on some interval ( 1 ; 1 ) around 0 . We may assume 1 > 0 . Then on ( 1 ; 0 ) we have by Lemma 2.3 that W T V is constant. However, (W T V )(s n ) ! V ~ T ~V = Id as n ! 1, so W T V = Id on ( 1 ; 0 ). Now V is orthogonal, so W = (V T ) 1 = V on this interval. Putting ^V (s) =( V (s) 0 < s < 0 ; W (s) 0 6 s < 1 ; 2.3
Page 1: Supplementary Notes for MM08 Geomet

2. On the Fundamental Theorem for Curves in Space

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