Supplementary Notes for MM08 Geometry I 2.OntheFundamentalTheoremforCurvesinSpace Andrew Swann These notes are a supplement to Theorem2.3 in Pressley [1] which gives theFundamentalTheoremforCurvesinSpace. The theorem has two parts. The rst states that two curves in R 3 parameterised by arclength with the same curvature and torsion , with > 0, are equivalent under rigid motions of R 3 . The second concerns existence of curves in R 3 with prescribed curvature and torsion. The proof of the rst part is relatively direct and covered perfectly adequately by Pressley's text. The second part however relies on results from thetheory of linear dierential equations, in particular long-time existence of solutions, that are not covered by previous courses here in Odense. These notes will obtainthe desired result using only the local existence result proved in MM04. We begin by considering a smooth curve : (; ) ! R 3 , s 7! (s), which is parameterised by arclength and whose curvature is never zero. The latter assumption implies that the torsion is dened and we may write ft; n; bg forthe Frenet frame of . Then these vectors satisfy the Serret-Frenet equations 9>= t 0 = n n 0 >; = t +b (2.1) b 0 = n where 0 = d=ds. We wish to show that these equations essentially determine . The rst part of Pressley's Theorem2.3 shows that is unique up to rigid motions. We are thus left with: Theorem2.1 (Existence part of theFundamentalTheorem). Let k; t : (; ) ! R be smooth functions with k(s) > 0 for all s 2 (; ). Then there exists a smooth curve : (; ) ! R 3 which is parameterised by arclength, has curvature (s) = k(s) and has torsion (s) = t(s), for all s 2 (; ). 2.1