Now we calculate the partial derivatives of (22) with respect to s and k; using

Frenet formulas, we get

Then, we have the following

Frenet formulas, (TURHAN; KORPINAR, 2011b):

If [alpha] is a pseudo null curve, the

Frenet formulas have the form [2, 3]

If [k.sub.1] and [k*.sub.1] are the natural curvatures of base curves of M and M* and [k.sub.2] and [k*.sub.2] are the torsions of base curves of M and M*, then from the

Frenet formulas we have [k*.sub.i] = [k.sub.i] [dt/dt*], 1 [less than or equal to] i [less than or equal to] 2.

If [alpha] is a null space curve with a spacelike principal normal [??], then the following Frenet formulas hold

If [alpha] is a pseudonull curve, that is, [alpha] is a spacelike curve with a null principal normal N, then the following Frenet formulas hold:

Finally, Chapter 7 is dedicated to various aspects of extrinsic geometry, such as

Frenet formulas for curves, ruled and developable surfaces, the shape operator, principal curvatures and curvature lines, etc.

We confine ourselves to the case n = 4, but it is obvious that similar Frenet formulas hold for every dimension n.

We study the Frenet frames and Frenet formulas in the Minkovski spacetime [E.sup.4.sub.1], that is [R.sup.4] with the metric g = -d[x.sup.2.sub.1] + d[x.sup.2.sub.2] + d[x.sup.2.sub.3] + d[x.sup.2.sub.4].

(The

Frenet formulas for non-unit speed curves in [E.sup.3.sub.1]) For a regular curve [alpha] with speed

It is clear to show that there exists a function [tau](s)on D such that the

Frenet formulas hold.

For their derivatives the following

Frenet formulas hold [9]