The Lorentz group element (12) is a null rotation by the angle [??] with respect to the null axis n, in the plane spanned by the light-like vector (1,-1, 0,0) and

spacelike vector (0,0,1, 0).

The projective tensor [h.sub.ij] = [[bar.g].sub.ij] + [u.sub.i][u.sub.j] is used to project a tangent vector at a point p in [bar.M] into a

spacelike vector orthogonal to u at p.

Let X: U [right arrow] [H.sup.n.sub.1] be an embedding, where U [subset] [R.sup.n-1] is an open subset; if there exists i such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is timelike vector and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

spacelike vector, we call M = X(U) Lorentzian surface in anti-de Sitter space.

Let [alpha]: I [subset] R [right arrow] [S.sup.2.sub.1] be a regular unit speed curve lying fully on [S.sup.2.sub.1] for all s [member of] I and its position vector [alpha] a unit

spacelike vector; then [alpha]' = t is a unit timelike and so [eta] is a unit

spacelike vector.

Hence [c.sub.0] is a unit

spacelike vector. Without loss of generality, we may put [c.sub.0] = (0,0,0,0,1).

A surface in the 3-dimensional Minkowski space [R.sup.3.sub.1] is a timelike surface if and only if a normal vector field of surface is a

spacelike vector field [2].

Then its position vector [alpha] is timelike vector, which implies that tangent vector [??] = [alpha]' is the unit

spacelike vector for all s [member of] I.

(iii) Spacelike angle: Let [bar.x] and [bar.y] be

spacelike vectors in [E.sup.3.sub.1] that span a

spacelike vector subspace.

can be a

spacelike vector field or a null vector field.

where [U.sup.[mu]] is the unit timelike four-velocity vector and [[zeta].sub.[mu]] is the unit

spacelike vector along the radial direction r.

A regular spacelike curve [gamma] : I [right arrow] [H.sup.3] is called a new type slant helix provided the spacelike unit vector m2 of the curve [gamma] has constant angle Q with

spacelike vector u, that is

Hence, we have orthonormal Sabban frame {[alpha](s), T(s), [xi](s)} along the curve [alpha], where [xi](s) = [alpha](s)xT(s) is the unit

spacelike vector. The corresponding Frenet formulae of [alpha], according to the Sabban frame, read