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 <zksi6t9dvjdv@forum.swarthmore.edu>,
Harry Watson <Watson.Harry@corona.navy.mil> 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