Notice that, for curves [gamma] isometrically immersed in a Riemannian manifold, H is nothing but the geodesic curvature
of [gamma], [kappa], and W(S) boils down to
In other words, its acceleration is normal to the manifold so that the geodesic curvature
is zero along the geodesic, and thus the two-point boundary value problem (TPBVP) arises from geodesic differential equations on Riemannian manifold.
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and 1/[T.sub.g], 1/[R.sub.n] and 1/[R.sub.g] are geodesic torsion, normal curvature and geodesic curvature
where [K.sub.g] is the geodesic curvature
of the timelike curve [alpha] on the [([s.sup.2.sub.1]).sub.H] and g (t, t) = -1, g ([alpha], [alpha]) = 1, g (s, s) = 1, g (t,[alpha])= g (t,s)= g ([alpha],s)= 0.
where [epsilon] = [+ or -]1, the curve [alpha] is timelike for [epsilon] = 1 and spacelike for [epsilon] = -1, and [[kappa].sub.g](s) is the geodesic curvature
of [alpha] on [S.sup.2.sub.1] which is given by [[kappa].sub.g](s) = det([alpha](s), t(s), t'(s)), where s is the arc length parameter of [alpha].
Normal and geodesic curvature
of variable lead helix
where [k.sub.g](s) = det([alpha](s), T(s), T'(s)) is the geodesic curvature
of [alpha] on [H.sup.2.sub.0] in [R.sup.3.sub.1] and s is arc length parameter of alpha.
In these formulae at (4) and (5) , the functions [[kappa].sub.n], [[kappa].sub.g] and [[tau].sub.g] are called the geodesic curvature
, the normal curvature and the geodesic torsion, respectively.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [[kappa].sub.g] (v) is the geodesic curvature
of the curve f on [S.sup.2]
We obtain the geodesic curvature
and the expressions for the Sabban frame's vectors of spacelike and timelike pseudospherical Smarandache curves.