A point p [member of] [M.sup.n] is called an umbilic point, if there exist real numbers [[delta].sup.1], [[delta].sup.2] such that
Then, p is an umbilic point if and only if rank[(df).sub.p] = 0 holds.
Since f has no umbilic point on U, the Gauss map v depends only on [u.sub.1] and[mathematical expression not reproducible] holds.
By Facts 3 and 5, q is not an umbilic point. By Proposition 8, we have that f is given by (3.10) on U := I x [R.sup.n-1].
A point p = X(u) is an umbilic point if all the principal curvatures coincide at p.
Let [[lambda].sub.0] [member of] [R.sup.n + 1.sub.2] and let M be a Lorentzian surface without any umbilic point satisfying Kp(u) # 0.
It can be shown that for a generic embedding of M, [K.sub.c](0) is a regular surface except by a finite number of points (u, v), which are singularities of type [[summation].sup.2, 0] of [L.sup.[+ or - ]] or equivalently umbilic points ([D.sup.[+ or -].sub.4]) of [h.sub.v] .