From: ikastan@alumni.caltech.edu (Ilias Kastanas) Newsgroups: sci.math Subject: Re: Frenet frame in n dimensions Date: 29 Jan 1998 07:14:40 GMT In article , Harry Watson wrote: >Hi jason! >There is an "extension" in *Tensor Calculus* by >J.L. Synge & A. Schild (NY: Dover Publications, 1949). >On page 72 one reads: > > 2.7. Frenet Formulae. With any point on a twisted >curve in Euclidean 3-space there is associated an orthogonal >triad consisting of the tangent, principal normal, and binormal, >and two numbers, the curvature and the torsion. We shall >now extend these ideas to the case of a curve in Riemannian >space of N dimensions. > [article continues] I don't have that book, but surely the idea is simple enough; I would describe it as follows. Assume everything (manifolds, fields, functions...) is C_infinity. Consider a Riemannian manifold whose connection (covariant differentiation) is D. To demonstrate notation, one property of D is D_Y (fX) = (Yf)X + fD_Y(X). If a curve has non-vanishing tangent vector field V define the unit tangent field T = V/|V|. The (geodesic) curvature vector field is D_T (T), and its length |D_T(T)| = k_1 is the (geodesic) curvature of our curve. Whenever k_1 is > 0 define the unit first-normal field N_1 by D_T (T) = k_1 N_1; of course it is orthogonal to T. For any interval where k_1 > 0 we have D_T (N_1) != 0; the vector field D_T (N_1) + k_1 T is orthogonal to both T and N_1; let its length be k_2, the second curvature (torsion) of our curve. We continue the same way: if k_2 > 0 define the unit second-normal field N_2 by D_T (N_1) + k_1 T = k_2 N_2, if this happens over an interval then we can define k_3, if k_3 > 0 we define N_3 ... and so on. The Frenet vectors are thus T, N_1, N_2, ... and the Frenet formulas are the expressions of D_T (N_i) in terms of Frenet vectors. Ilias