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