(B) When -[1 + (1/8[pi]G[l.sup.2][[rho].sub.m])[square root of (V/U)](V - U)] < [w.sub.m] < [1/3 + [LAMBDA]/6[pi]G[[rho].sub.m] + (1/8[pi]G[l.sup.2][[rho].sub.m])[square root of (V/U)]((1/3)V + U)], [N.sub.a][N.sup.a] > 0 that shows [N.sup.a] is a spacelike vector and [Y.sub.A] is the

timelike surface. [Y.sub.A] has the signature (-,+,+) that shares the signature of a quasi-local timelike membrane in black-hole physics [20,24],

When [epsilon] = -1, M is a timelike surface. In this case, we can assume that f'(u) = sinh t and g'(u) = cosh t, and using the same algebraic techniques as for [epsilon] = 1 easily prove that the 3rd kind of catenoid and the de Sitter pseudosphere satisfy condition (33).

The only spacelike or timelike surfaces of revolution in R3 whose Gauss map G: M [right arrow] [M.sup.2]([epsilon]) satisfies (2) are locally the following spaces:

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].

Ugurlu studied the geometry of timelike surfaces [12].

Then, if the surface M is an oriented timelike surface, the relations between these frames can be given as follows.

(i) If the surface M is a timelike surface, then the curve [alpha](t) lying on M can be a spacelike or a timelike curve.

Since ([U.sub.2], [U.sub.2]) = 1, the surface [M.sub.2] is a timelike surface. Hence the components of the second fundamental form of [M.sub.2] are given by

Since ([U.sub.3], [U.sub.3]) = 1, the surface [M.sub.3] is a timelike surface. Thus the components of the second fundamental form of [M.sub.3] are given by

We can extend the concept of constant angle surfaces for tangent developable timelike surfaces. Let M be a tangent surface generated by a curve [gamma] such that M is timelike.

This generalizes Theorem 4.1 for tangent timelike surfaces.

[2] Inoguchi J.-I.,

Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J.