Lecture 5 Our second generalization is to curves in higher-dimensional Euclidean space. A regular curve in Rn is a smooth map c : I → Rn , where I ⊂ R is an interval, such that or all t. The naive generalization of our twodimensional deﬁnition would be where det is the determinant of the matrix with given column vectors. This is reparametrization invariant. Physically it’s of type m−n(n−1)/2 , where m is the unit of distance in Rn . Frenet theory decomposes this as a product of curvatures, each carrying diﬀerent information. Lemma 5.1 (Gram-Schmidt orthogonalization). Let (v1 , . . . , vk ) be linearly independent vectors. There are unique orthonormal vectors (e1 , . . . , ek ) of the form where fii > 0. Note that in particular, each (e1 , . . . , ei ) spans the same subspace as (v1 , . . . , vi ). An explicit inductive formula is Lemma 5.2. Let E(t) be a family of orthogonal matrices, depending diﬀer entiably on t. Write d E(t) = E(t)A(t). dt Then the matrices A(t) are skewsymmetric, A(t)tr = −A(t). ei = Definition 5.3. c : I → Rn is a Frenet curve if for all t, the vectors are linearly independent. One then deﬁnes the Frenet frame (e1 (t), . . . , en (t)) as follows. First, ap ply Gram-Schmidt to , which yields (e1 (t), . . . , en−1 (t)). Then, take the unique vector en (t) which is orthogonal to (e1 (t), . . . , en−1 (t)) and satisﬁes det(e1 (t), . . . , en (t)) = 1. The components of the last vector are j-th unit vector en,j = det(e1 , . . . , en−1 , (0, . . . , 1, . . . , 0) ). Lemma 5.4. Frenet frames are reparametrization invariant. Explicitly, if c is a Frenet curve and d(t) = c(φ(t)) a reparametrization, then d is again Frenet, and its Frenet frame is related to that of c by fi (t) = ei (φ(t)).